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]]>A digital system must store binary numbers in addition to performing logic. To take care of this requirement, a memory cell, called a FLIP-FLOP, is introduced. Theoretically any digital system can be constructed entirely from NAND gates and FLIP-FLOPs. These integrated circuits form the practical (commercially available) basic building blocks for a digital system.

**These chips perform the following functions: **

1. Binary addition;

2. Decoding (demultiplexing);

3. Data selection (multiplexing);

4. Counting;

5. Storage of binary information (memories and registers);

6. D/A and A/D conversion; and

7. A few other related operations.

Those building blocks depend upon combinational logic. The standard combinations are small- scale integration (SSI). Less than 100 individual circuit components (about 12 gates) on a chip are considered SSI. The flip-flops are also SSI packages. Most other functions (using BJTs) are examples of medium scale integration (MSI), defined to have more than 100 but less than 1000 components (about 100 gates per chip). The BJT memories and MOSFET arrays may contain in excess of 1000 components and are defined as large scale integration (LSI).

SSI < 100 individual circuits (about 12 gates) – (e.g., flip-flop).

MSI > 100 but < 1000 (about 100 gates) – (e.g., most circuit using BJT).

LSI > 1000 – (e.g., BJT memories and MOSFET arrays).

It is the basic operation. Multiplication can be obtained by programming.

A two-input adder is called a half adder, because to complete an addition requires two such half adders.

**Half Adder****: **

A half adder has two inputs A and B representing the bits to be added and two outputs D and C, where D is for the digit of the same significance as A and B represent and C for the carry bit. Parallel operation of binary adder is illustrated in Fig. 6.2.

The symbol for the n-th hill adder (FA) is shown in Fig. 6.3(a). The circuit has three inputs: the addend A_{n}, the augend B_{n} and the input carry C_{n-1} (from the next lower bit). The output is the sum S_{n} and the output carry C_{n}.

A parallel 4-bit adder is shown in Fig. 6.3(b). As FAO represents the least significant bit (LSB), it has no input carry; hence C_{n} = 0.

The circuitry within the block FA is determined from Fig. 6.4, which is the truth table for adding 3 binary bits.

From this table one can verify that the Boolean expressions for S_{n} and C_{n} are given by-

It may be noted that the first term of S_{n} corresponds to line 1 of the table, the second term to line 2, the third term to line 4 and the last term to line 7 (these are the only rows where S_{n} = 1). In a similar way the first term of C_{n} corresponds to line 3 (where C_{n} = 1), the second term to line 5, etc.

The AND operation ABC is also called the product of A, B and C. Further, the OR operation + is referred to as summation. So expressions shown in equations (6.1) and (6.2) represent a Boolean sum of products. Such an equation is said to be in a standard, or canonical form and each term in the equation is called a minterm which contains the product of all Boolean variables or their components. Only one full adder is required for serial arithmetic, while in parallel addition we must use a full adder for each bit. Thus, parallel addition becomes more expensive than serial operation.

A serial binary full adder is indicated in Fig. 6.5. The time delay unit TD is a type D FLIP- FLOP and the serial numbers A_{n}, B_{n} and S_{n} are stored in shift registers.

In a digital system, instructions as well as numbers are conveyed by using binary levels or pulse trains. If 4 bits of a character are set aside to convey instructions, then 16 different instructions are possible. This information is coded in binary form.

Frequently a need arises for a multiposition switch which is operated in accordance with this code. It means that for each of the 16 codes, one and only one line is to be excited. This process of identifying a particular code is known as decoding.

**Binary-Coded-Decimal (BCD) System****: **

The code translates decimal numbers by replacing each decimal digit with a combination of 4 binary digits. As there are 16 distinct ways in which the 4 binary digits can be arranged in a row, any 10 combinations can be used to represent the decimal digits from 0 to 9. In fact, we have a wide choice of BCD codes. One of these, called the “natural binary-coded decimal” is the 8421 code. This is illustrated in Table 6.1.

This is a weighted code as the decimal digit in 8421 code is equal to the sum of the products of the bits in the coded words times the successive powers of two starting from the right (LSB). We require N 4-bit sets to represent in BCD notation as N-digit decimal number. The first 4-bit set on the right represents units, the second represents tens, the third hundreds and so on.

For example, the decimal number 264 requires three 4-bit sets, as shown in Table 6.2. This three-decade BCD code can represent any number between 0 and 999. Thus, it has a resolution of 1 part in 1000 or 0.1%. It needs 12 bits which in a straight binary code can resolve one part in 2^{12 }= 4096 or 0.025%.

Let we like to decode a BCD instruction representing one decimal digit, say 5. The operation is carried out with a four-input AND gate excited by the 4 BCD bits. The output of the AND gate in Fig. 6.6 is I, if and only if the BCD inputs are A = 1 (LSB), B = 0, C = 1 and D = 0. As this code represents the decimal number 5, the output is labelled “line 5”.

A BCD-to-decimal decoder is shown in Fig. 6.7. The MSI unit has four inputs, A, B, C and D and ten output lines. Moreover, there must be a ground and a power supply connection and so a 16-pin package 5 is needed. The complementary inputs A, B, C and D are found from inverters on the chip.

As NAND gates are used, output is 0 (low) for the correct BCD code and is 1 (high) for any other invalid code. The system shown in Fig. 6.7 is also referred to as a “4-to-10-line decoder” designating that a 4-bit input code selects 1 of 10 output lines. Thus, the decoder works as a 10-position switch which responds to a BCD input instruction.

It is required to decode only during certain intervals of time. In such applications an additional input, known as a strobe, is added to each NAND gate. All strobe inputs are tied together and are excited by a binary signal S. This is shown by the dashed lines in Fig. 6.7. If S = 1, a gate is enabled and decoding takes place, while if S = 0, no coincidence is possible and decoding is inhibited. The strobe input is used with a decoder having any number of inputs or outputs.

**[Multiplexer (Data Selector): **

The function performed by a multiplexer is to select 1 out of N input data sources and thereby to transmit the selected data to a single information channel.

A decoder is a system which accepts an M- bit word and establishes the state 1 on one and only one of 2^{M} output lines. A decoder thus identifies a particular code. The inverse process is known as encoding. An encoder has a number of inputs, only one of which is in the 1 state. An N-bit code is generated in an encoder depending upon which of the inputs is excited.

We consider that a binary code is transmitted with every stroke of an alphanumeric keyboard. There are 26 lower case and 26 capital letters, 10 numerals and about 22 special characters on a keyboard. Thus, the total number of codes necessary is approximately 84. This condition is satisfied with a minimum of 7 bits (2^{7} = 128, but 2^{6} = 64).

Let us assume that the keyboard is modified so that if a key is depressed, a switch is closed, hence connecting a 5-V supply corresponding to the 1 state to an input line. A block diagram of an encoder is shown in Fig. 6.8. Inside the shaded block there is a rectangular array of wires and we determine how to interconnect these wires so as to generate the desired codes.

In order to explain the design procedure for constructing an encoder, we simplify the above example by limiting the keyboard to only 10 keys, the numerals 0, 1, …, 9. A 4-bit output code is sufficient in this case and we choose BCD words for the output codes. The truth table defining this encoding is given in Table 6.3.

Input W_{n} (n = 0, 1,…, 9) indicates the n-th key. When W_{n} = 1, key n is depressed. As it is considered that not more than one key is activated simultaneously, then in any row every input except 1 is a 0. From this truth table we find that Y_{0} = 1, if W_{1} = 1, W_{3} = 1, W_{5} = 1, W_{7} = 1, W_{9} = 1.

Thus, in Boolean notation,

The OR gates in equations 6.3 and 6.4 are implemented in the form of array and is known as a rectangular diode matrix.

Let us consider the problem of converting one binary code into another. Such a code-conversion system is known as ROM [Fig. 6.9(a)]. It has M inputs (X_{0}, X_{1},…, X_{M-1},) and N outputs (Y_{0}, Y_{1},…, Y_{N-1}), where N may be greater than, equal to or less than M. A definite M-bit code is to result in a specific output code of N bits.

The code translation is obtained by first decoding the M inputs onto 2^{M} = µ word lines (W_{0}, W_{1} …, W_{1-µ}) and then encoding each line into the desired output word. If the inputs assume all possible combinations of 1s and 0s, then µ N-bit words are “read” at the output.

The functional relationship between output and input words is built into hardware in the encoder block of Fig. 6.9. As this information is stored permanently, we may say that the system has “memory”. The memory elements are the diodes or the emitters of transistors.

The output word for any input code may be read as often as desired. As the stored relationship between output and input codes cannot be modified without adding or subtracting memory elements (hardware), this system is termed a read only memory and abbreviated as ROM.

A typical bipolar ROM (MM 6280) is available from monolithic M = 2^{10 }= 1024 words of 8 bits each. This size is referred to as an 8 x 1024 = 8192-bit memory. This is a good example of a large scale integration (LSI).

The random access memory, abbreviated as RAM, is an array of storage cells that memorize information in binary form. In this memory, information can be randomly written into or read out of each storage element as needed and hence it is called as the random access or read/write memory.

**Linear Selection:**

In order to explain how the RAM operates we consider 1-bit S-R FLIP- FLOP circuit shown in Fig. 6.10, with data input and output lines. From the figure we find that to read data out of or to write data into the cell, it is required to excite the address line (X = 1). To perform the write operation, the write enable line must be excited. If the write input is a logic 1 (0), then S = 1 (0) and R = 0 (1). Hence Q = 1 (0) and the data read out is 1 (0), corresponding to that written in.

Let us assume that we like to read/write 16 words of 8 bits each. This system needs eight data inputs and eight data outputs lines. A total of 16 x 8 = 128 storage cells must be used. Of this number, 8 cells are arranged in a horizontal line, all excited by the same address line. There are 16 such lines, each excited by a different address. This means that addressing is provided by exciting 1 of 16 lines. This type of addressing is known as linear selection.

**Coincident Selection: **

An RAM memory of sixteen 8-bit words has 16 lines with 8-storage cells per line, when linear addressing is used. A commonly used topology is to arrange 16 memory elements in a rectangular 4 x 4 array, each cell now storing one bit of one word. Bight such matrix planes are necessary, one for each of the 8 bits in each word.

One plane of the arrangement of cells is indicated in Fig. 6.11. Each bit as indicated as a shaded rectangle, is located by addressing an X-address line and a Y- address line; the intersection of the two lines locates a point in the two-dimensional matrix and thus identifies the storage cell under consideration. This two-dimensional addressing is known as X-Y or coincident selection.

**Basic RAM Elements: **

In the 1-bit memory of Fig. 6.10 separate read and write leads are essential. For either the bipolar or the MOS RAM it is possible to construct a FLIP-FLOP which has a common terminal for both writing and reading, such as terminals 1 and 2 in Fig. 6.12. This configuration needs the use not only on the write data W (write 1) but also of its complement W (write 0). At the cell terminal to which W (W) is applied, there is achieved the read or sense data output S (S). Such a memory unit is indicated in Fig. 6.12.

Here a total of four input/output leads to the storage cell is required, two for X-Y addressing and two for read/write data necessary. The base elements on which an RAM is constructed are shown in Fig. 6.13. These include the rectangular array of storage cells, the X and Y decoders, the write amplifiers to drive the memory and also the sense amplifiers to detect the stored digital information.

Some RAMs include a write enable input, wherein the write amplifiers of Fig. 6.13 are two input AND gates as in Fig. 6.12. Each word is identified by the matrix number X-Y in the (shaded) memory cell. For M-bit words there will be M planes, as in Fig. 6.13. Since in Fig. 6.12 the output of the write amplifier is connected to the input of the read amplifier, indicating that the sense amplifiers must not be used to supply information on the state of a memory cell when a write amplifier is excited.

An example of a 16-bit bipolar RAM of the pattern of Fig. 6.13 is the TI 7481. Average power dissipation is 275 mW and reading propagation delay is typically 20 ns. A larger RAM is the IM 5503 (Intersil Memory Corporation) having 16 x 16 = 256 words by 1-bit organization and has an excess time of 75 ns.

Fig. 6.14 shows a three-input, direct-coupled transistor logic (DCTL) NOR gate. The name direct-coupled comes from the fact that the inputs are coupled directly to the transistor bases.

A 1 input on A or B or C will turn on the transistor whose base has the 1 on it (assuming positive logic). This will cause the output to be low or a 0. When all inputs are zero, all three transistors will be OFF and the output will be a 1. In fact, as we know that this is the condition for a NOR gate.

DCTL gates are susceptible to what is called “current hogging”. Let the circuit in Fig. 6.14 is loaded with several other DCTL NOR gates. This means that several base-emitter junctions are being driven from the same point and due to inevitable differences in their base-to-emitter characteristics, one junction will turn on first. It is quite possible for the first junction to “hog” sufficient current to prevent some of the other junctions from turning ON.

The problem of current hogging can be easily solved by placing a resistor in series with each input base. The result is the resistor-transistor logic (RTL) NOR circuit. The current hogging is avoided as the resistors isolate the bases from the common driving point, thereby permitting the base-emitter voltages to individually adjust to the levels necessary for turn-ON. The addition of the input series resistors also increases the circuit input impedance, so that the fan-out of the driving circuit is increased.

RTL circuits were the first type of logic to be integrated in the early 1960s. The basic circuit had been proved in discrete-circuit form and was readily adaptable to integrated circuit fabrication. Thus, designers were familiar with RTL and lots of reliability and evaluation data were available.

**RTL circuits have three main disadvantages:**

(i) Relatively low speed,

(ii) Low fan-out and

(iii) Temperature sensitivity.

The low speed arises because the external series base resistor combines with the input capacity of the transistor to form a low-pass filter. This degrades the rise and fall times of any input pulse.

The low fan-out occurs as the input current to a given transistor is limited by having to flow through R_{L} and the series base resistor of the next gate.

Speed in RTL circuits is a function of the size of the resistors used, since inherent transistor and stray capacities must be charged and discharged through collector and base resistors. In order to increase speed, resistors can be reduced in value but this increases the circuit power requirements.

To achieve high speed with less power, RTL circuits can be modified to the RCTL (resistor- capacitor-transistor logic). Here, the base resistors have been paralleled by speed-up capacitors, so that fast rise and fall can be achieved, even with relatively large base resistors. Apart from a high ratio of speed to power, RCTL is not significantly different from RTL and has not become widely accepted.

Another type of logic which was very popular as a discrete circuit and which was quickly translated into integrated form is DTL (diode-transistor logic).

As shown in Fig. 6.17, this usually consists of an input-diode AND gate (D_{1}, D_{2}, D_{3} and D_{4}) followed by a transistor inverter which results in a NOT-AND or NAND gate. If any of the inputs A, B, C and D are low (a logic 0), point X will be approximately +0.7 V and the transistor will be turned OFF because its V_{BE} will be less than 0.7 owing to D_{5} and D_{6}. The output is, therefore, high (a logic 1).

However, if inputs A, B, C and D are all high (logic 1), all of the four input diodes turn OFF and the values of the two base bias resistors are such that the transistor turns ON. Its output is, therefore, low (a logic 0) only when all inputs are high (1). This is, in fact, the condition for a NAND gate.

The two extra diodes, D_{5} and D_{6}, serve two purposes. With any input low they cause the base of the transistor to be well below +0.7 V, which ensures that the transistor is solidly off. Also, all inputs must rise above + 1.4 V before the transistor can turn ON, because point X in the figure is three diode drops (D_{5}, D_{6} and the transistor base emitter) or 3 x 0.7 V = 2.1 V above ground as the transistor turns ON.

DTL is well-suited to monolithic integrated circuit fabrication. No capacitors are required and component values are not critical. Also, some monolithic versions of the DTL NAND gate in Fig. 6.4 replace D_{5} with a transistor. This improves performance. Replacing a diode with a transistor in a monolithic circuit would improve performance.

A form of logic that is related to DTL is shown in Fig. 6.18(a). It is called transistor-transistor logic, and abbreviated as TTL or T^{2}L.

In the DTL circuit of Fig. 6.17, D_{4} and D_{5} form an np-pn combination. In TTL this combination is replaced with an npn transistor. To get multiple-input AND gate similar to the four-diode AND gate in the DTL circuit, TTL circuit uses multiple emitters on the npn transistor.

The logic action of the TTL ckt can be understood comparing the explanation for the DTL circuit. TTL is an example of the application of monolithic integrated circuit technology. When a DTL circuit is integrated, it cannot take advantage of the technique in the same way as TTL can.

Since the multiemitter transistor is smaller in area than the number of diodes it replaces, the yield from a wafer is improved. Moreover, the smaller area results in a lower capacitance to the substrate, thus, reducing circuit rise and fall times and increasing speed. Fig. 6.18(b) shows how a TTL NAND gate might be integrated.

The TTL circuit of Fig. 6.18(a) is practically not used in its basic form due to its limited noise immunity. The higher speeds and fan-out are possible with a modified version as shown in Fig. 6.18(c).

TTL input circuits need higher drive currents than DTL. This is why TTL circuits usually have high-power output stages. The output circuit in Fig. 6.18(c) is called a totem-pole output, as the three output components Q_{3}, D_{1} and Q_{4} are stacked one on top of another in the manner of a totem-pole.

The more common (non-memory) digital subsystems which are made up of combinations.

Medium scale integrated circuits (MSI) are introduced here as many of these subsystems are available on a single chip of silicon.

One of the functions frequently required in digital system is the ability to count. Such counting is not done in the decimal system but in binary system of 1s and 0s. The simplest type of counter is the binary ripple counter which contains three cascaded flip-flops, with the output of each flip- flop triggering the next, with a total count capacity of 0 to 7.

There are many variations of this basic binary counter. The maximum count can be extended to 15, by adding one more flip-flop.

The total number of allowed states is called as the modulus of the counter. For example, one that is capable of counts 0, 1, 2, 3, 4, 5, 6 and 7 is a mod-8 counter as in Fig. 6.19. A similar counter with one disallowed state (say, it skips 2) is a mod-7 counter.

The decade counter has a base or modulus of 10. The binary counter of Fig. 6.19 can be converted to a decade counter by adding one more flip-flop and eliminating 6 of the 16 possible states. One method of eliminating states is to leave off the last 6 counts so that the remaining 10 counts have the same weighting as the binary counter of Fig. 6.19.

Considering the waveforms and truth table of Fig. 6.20, we find that to limit the binary counter to a total of 10 counts we have to stop at the count of 9 instead of advancing to 10.

The status of the flip-flops at the counts of 9 and 10 would normally be-

**If we return the counter to 0 after the count of 9, we have to do the following: **

1. Prevent flip-flop B from setting to 1.

2. Reset D to 0.

A decade counter is similar to the binary counter of Fig. 6.19, but it has the extra stage, like the master-slave RST flip-flop, and the interconnection changes are made to stop the count at 9.

Considering Fig. 6.20(a) the operation of the decade counter can be explained as follows. When the counter attains 9, that is-

Q_{D} = 0 and this is fed back to the set input of flip-flop B. As flip-flop B is 0, the first requirement is satisfied. At the count of 9, Q_{B} = 1 and Q_{C} = 1, and the NOR gate output becomes 0. The NOR gate is connected to the set input of flip-flop D. As R_{D} is normally high, flip-flop D is reset to 0 after the count of 9.

The waveforms of Fig. 6.20(b) illustrate the inherent frequency division capability of digital counters. Thus, the input or clock frequency is divided in half at each successive flip-flop.

Fig. 6.21 is an example of a digital circuit. The decoded output of a single Decade Counter Unit (DCU) represents only one decimal digit (varying from 0 through 9); therefore, additional DCU stages must be cascaded to increase the capacity to numbers greater than 9, i.e., tens and hundreds.

A typical three-digit DCU capable of counting to 999 is shown in Fig. 6.21. Note that a carry line connects the highest order BCD output of each DCU to each succeeding higher power DCU. The carry is necessary to advance the count from one power to the next; i.e., after the units DCU has progressed through the count of 9, the tens DCU must start at 0001.

Its circuit is similar to that of the multiple flip-flop counters of Figs. 6.19 and 6.20, but its function is somewhat different. Registers are a very important part of most digital systems. They are used to temporarily store binary information, especially before or after conversion or encoding/decoding operations. They allow a simple means of converting from serial to parallel (or vice versa) format.

Registers are also fundamental to the basic arithmetic operations such as multiplication, division, complementation and analogue-to-digital (and vice versa) conversion. Since, a flip-flop can store only one binary number (1 or 0), a shift register must contain one flip- flop for each bit of a binary number. There must also be some provision for entering (shifting) the binary number into, out of, and from one stage to another of the shift register. This brings up the two general methods of shifting data into shift registers, serial and parallel.

The first involves serially shifting one bit at a time into the register, whereas all the data are entered at the same time in a parallel shift register. The more universal MSI shift registers have designed so that data can be entered serially or in parallel so that the register can be used for a variety of sequential operations.

Fig. 6.22 demonstrates the basic serial shift-register technique. This is a four-stage (four-bit) serial shift register. Note that each Q output is directly coupled to the/input of the adjacent flip- flop, and the triggering inputs are all in parallel to provide simultaneous shifting of data from A to B, etc.

After all flip-flops are initially reset to 0, the/and K inputs of the first flip-flop (A) determine whether a 1 or a 0 is inserted into the register. At each clock pulse the state (Q = 0 or 1) of flip-flop A is shifted to the next higher order stage (B) and so on until the four-bit word is entered in the register. Four more clock pulses will then shift the word completely out of the register.

We shall now discuss the important operations of digital-to-analogue (D/A) and analogue-to- digital (A/D) conversion. The best examples of D/A include conversion from binary machine language to analogue signals for plotters, recorders or meters. A/D conversion is often needed to convert analogue transducer outputs to digital form for input into a digital system.

D/A conversion is much simpler than A/D conversion and in fact a D/A converter is usually included as part of an A/D converter. For this reason, we shall discuss D/A conversion first. The basic problem in D/A conversion is to change a string of digital 0s and 1s to an equivalent analogue voltage. For example, consider the binary numbers 0 through 9 and the equivalent weighted analogue voltages shown in Fig. 6.23(a).

In this case we have made an arbitrary decision that the maximum analogue voltage is 9 V. Since there are 10 different decimal equivalents to express, the lowest (0000) is made equal to 0V and the highest (1001) to 9V, with 1V increments. Therefore, each binary number is expressed as a discrete analogue voltage. Note that, as we progress from one bit to the next, the equivalent analogue voltage doubles; i.e., 2^{1} = 2 V, 2^{2} = 4 V, 2^{3} = 8 V.

A resistive divider and operational amplifier [Fig. 6.23(b)] is all that is needed to perform the D/A conversion indicated in Fig. 6.23(a). Note the common output summing point and the fact that the weighted resistor values decrease by a factor of 2 for each bit increase. This means that with a high 2^{3} input the current through R/8 is twice that through R/4 with a high 2^{2} input. Hence, the network fulfills the requirement of doubling the analogue current or voltage for every succeeding bit.

There is more hardware involved in an actual D/A converter than just the precision resistive divider. Fig. 6.24 shows a block diagram of a typical D/A converter.

Proceeding from the parallel digital input, we see that some form of storage is required for the digital information, plus some means of inputs controlling the data read-in and read-out. Next, level amplifiers and a precision voltage reference are needed to ensure that the normally imprecise digital signals are equal and constant, regardless of environmental or ladder loading effects.

These amplifiers operate in the analogue comparator mode, i.e., they have two inputs- the reference voltage and one of the shift-register outputs. Hence the output of a particular level amplifier is either low or high (≈0 or equal to the reference voltage), depending on the state of its input from the shift register. Thus, the resistive divider is provided with the proper digital inputs and it performs the actual digital-to-analogue conversion. ~~ ~~

In this system a continuous sequence of equally spaced pulses is passed through a gate. The gate is normally closed and is opened at the instant of the beginning of a linear ramp.

The A/D converter using a counter is shown in Fig. 6.25(a) while the counter ramp waveform is shown in Fig. 6.25(b).

In Fig. 6.25(a), the clear pulse resets the counter to the zero count. In binary form the number of pulses are then recorded by the counter from the clock line. The clock is nothing but a source of pulses equally spaced in time. The binary word representing this count is used as the input of a D/A converter whose output is represented in Fig. 6.25(b).

So long input V_{s} is greater than V_{d}, the comparator output is high and the AND gate is open for the transmission of the clock pulses to the counter. When V_{d} becomes greater than V_{s}, the output of the comparator changes to the low value and then the AND gate is disabled. This stops the counting at the moment when V_{s }≈ V_{d}. The count can be read out as the digital word represents the analogue input voltage.

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]]>Resistors are those circuit elements which introduce electrical resistance into the circuit. There are many types of resistors, each one suitable for its particular application. They vary with regard to their resistance expressed in ohms, their power rating in watts, the material of the resistance element and whether they are of the fixed or variable resistance type.

**Ratings of Resistors****: **

The standard power ratings for resistors are 0.125, 0.25 0.5, 1, 2, 3, 4, 5, 10, 20, 25, 30, 40, 50, 75, 100, 150, 200, watts and higher.

**The standard values of resistances available are: **

Ohms and Kilo Ohms: 1, 1.2, 1.5, 1.8, 2.2, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2, 10, 12, 15, 18, 22, 33, 39, 47, 56, 68, 82, 100, 120, 150, 180, 220, 330, 390, 470, 560, 680, 820.

Meghaohms: 1, 1.2, 1.5, 1.8, 2.2, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2, 10, 12, 15, 18, 22.

**Colour Coding of Resistors****: **

Colour codes are used to designate the value of a resistor placed in any position in an equipment board. Rings or band of different colours are usually placed around the resistor. The colour of first three bands determine the total resistance value, while the fourth band is used to indicate tolerance as illustrated in figure 8.1 Table 8.1 gives the complete colour code of electronic resistors.

A. Determines the first digit

B. Determines the second digit

C. Determines the number of zeroes following the first two digits.

D. Determines the tolerance.

A capacitors is a device capable of storing an electric charge (static electricity). It consists of two metal plates separated by a dielectric material. A capacitor is designed for deliberately providing a known amount of capacitance in circuit. Capacitors are available in values ranging from less than one Pico farad to thousands of microfarads. While using a capacitor its ratings must be carefully observed to be certain that the potential to be applied across the capacitor is not greater than the rated value.

It is an electronic component designed especially for introducing a required amount of inductance in the circuit. An inductor consists of a two-terminal, single winding wound on air or iron core. An air core inductor employs a non-magnetic core whereas an iron core may employ a magnetic material other than iron.

The unit of measurement of inductance is the henry. Inductors commonly employed in electronic circuits range in values from less than one mH to about 20 H. Small inductors are used in radio frequency tuned circuits and as radio frequency chokes. Larger inductors are employed at audio-frequency; the largest inductors are used as filter chokes in power supplies.

**Construction of Inductors****: **

The constructional features an inductor are determined largely by the frequency range in which it is to operate. In general, low frequency inductors have many turns and employ an iron core. In contrast high frequency inductors have fewer turns and often employ an air core. The stray capacitance between the turns and between the layers of turns is an important factor in higher frequency coils, and therefore special winding configurations may be employed to minimise this capacitance.

Current and voltage consideration also determine the constructional features of an inductor. The gauge of the wire for example, is selected in accordance with the amount of current the coil must carry. Receiver coils, which normally carry only a few mA of current, are wound from fine wires.

Transmitter coils, which normally carry much greater current, are wound using heavier wires. An inductor designed for use in high voltage circuits will have heavier insulation than the one designed for low voltage applications. Inductors may be made of iron core, powder core, ferrite core on air core depending upon the frequency range of their operation.

**Iron-Core Inductors: **

Low-frequency inductors are normally iron core inductors. These are generally large, both in inductance and in physical dimensions. Such inductors include filter reactors and audio frequency chokes. Low frequency iron-cored chokes are largely used to smooth out the ripple voltage in rectified ac power supplies. The core materials most commonly used are silicon-iron laminations and grain oriented silicon steel. Windings are usually of enamelled solid copper wire with interlayer insulation and impregnated with suitable material.

**Power Core Inductors: **

Powder core inductors are used at much higher frequencies than iron-core inductors. The techniques of reducing eddy current loss by dividing the core into smaller segments, accounts for use of the ‘powdered-iron’ core. The iron dust in these cores is often ferrite (a mixture of ferric oxide and other substances such as nickel and cobalt).

The magnetic powder is mixed with an insulating binder material so that each magnetic particle is electrically insulated from one another. The magnetic powder and insulating binder substance (usually rod shaped) are moulded to fit into the coil. This rod of core material (known as slug) is normally mounted so that it can be moved into or out of the coil, usually by means of a screw.

**Ferrite Core Toroids: **

A toroid is a coil wound on a round shaped solid of circular cross-section ferromagnetic core. The main advantage of the toroidal coil is that all of the magnetic flux is contained within the core material; therefore remains unaffected by surrounding components. Another advantage of toroids is that it has smaller physical size, for a given amount of inductance, than solenoid coil.

The core is usually made of ferrites. As they are electrically insulators, they do not suffer from the effect of eddy current. The toroidal winding is placed in the annular space. An air gap is introduced in the centre core and by choosing a suitable length for this gap, the properties of the core may be changed to suit a wide range of design requirements.

**Air Core Inductor: **

High frequency inductors, such as RF (Radio Frequency) chokes and the tank coils are generally more critical than low frequency types with respect to core-losses and stray capacitance. Iron cores are seldom used in RF inductors because the core losses would be excessive at high frequencies. At higher carrier and radio frequencies air core and powdered iron slugs are used.

Transformers are used in electronic equipments not only to step-up or step down the voltage but also to electrically isolate the electronic circuit from the mains. Transformers are made in great variety to meet application requirements.

**Following types of transformers are generally used in electronic equipments: **

(i) Isolation transformation;

(ii) Current transformers;

(iii) Auto transformers;

(iv) Pulse transformers;

(v) Audio transformers;

(vi) Radio-frequency transformers;

(vii) Intermediate frequency transformers;

(viii) Rectifier transformers;

(ix) Constant voltage transformers.

A relay is a device that functions as an electrically operated switch. Relays are used to open or close the contacts in the circuits. The contacts may be either in the same circuit as the operating signal or in another circuit or combination of circuit. Relays are widely used in industrial applications. When the coil of a relay is energised or de-energised, its contacts are On or OFF. Various types of relays are available in the market. The selection of a particular relay depends on the type of application.

Connectors are usually described as a piece of equipment used in joining two pieces of electronic equipment. There are many types of connectors available in the market.

**Various types of connectors used in electronic equipments are: **

(i) Audio connectors

(ii) RF connectors

(iii) PC board connectors

(iv) Standard rack and panel connectors

(v) Standard threaded circular connectors.

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]]>The post Power Factor: Causes and Disadvantages | AC Circuit | Electricity appeared first on Engineering Notes India.

]]>The cosine of the angle between voltage and current in an ac circuit is known as power factor.

In an ac circuit, there is generally a phase difference between voltage and current. In an inductive circuit, the current lags behind the applied voltage and the power factor of the circuit is referred to as lagging. In a capacitive circuit the current leads the applied voltage and therefore, the power factor of the circuit is said to be leading.

Consider an inductive circuit, which draws a current I from the supply mains lagging behind the supply voltage V by an angle ɸ, known as phase angle, the phasor diagram is shown in Fig. 15.6.

The current I can be resolved into two components, one along the voltage phasor and the other perpendicular to it. The component along the voltage phasor, I cos ɸ is called the in- phase or active component of current, and the one perpendicular to the voltage phasor, I sin ɸ is called the out of phase or wattless or reactive component of current.

If all these components are multiplied by voltage V, the product of voltage V and in-phase component of current I cos ɸ i.e., VI cos ɸ will represent the true power of the circuit in watts or kW, whereas the product of voltage V and the quadrature component of current I sin ɸ i.e., VI sin ɸ will represent the reactive power in VARs or kVARs and the product of voltage V and current I i.e., VI will represent the apparent power in volt-amperes or kVA. Thus we get a power triangle, as shown in Fig. 15.7.

For leading currents the triangle becomes reversed. This fact provides a key to the power factor improvement. If a device drawing leading reactive power is connected in parallel with the inductive load, then the lagging reactive power of the load will be partly neutralised, resulting in improvement of the power factor of the system.

(i) All ac motors (except overexcited synchronous motors and certain type of commutator motors) and transformers operate at lagging power factor. The power factor falls with the decrease in load. For example an induction motor has a reasonable higher power of 0.85 at full load, 0.8 at 75% of full load, 0.7 at half-full load, 0.5 at 25% of full load and as low as 0.1 on no load.

(ii) Arc lamps and electric discharge lamps operate at low lagging power factor.

(iii) Due to increased supply mains voltage, which usually occurs during low-load periods such as lunch hours, night hours etc., the magnetizing current of inductive reactances increase and power factor of the electrical plant as a whole comes down.

(iv) The power factor at which motors operate falls due to improper maintenance and repairs of motors. In repaired motors, less wire is sometimes used than originally wound motors, therefore, in such motors leakage of magnetic flux increases and power factor of the motor decreases.

In case of heavily worn-out bearings, the rotor may catch at the stator.

Some metal is sometimes removed from the rotor by turning instead of replacing the defective bearings. In doing so, the length of air gap between stator and rotor increase, due to which greater magnetising current is required and, therefore, power factor drops.

(v) Industrial heating furnaces such as arc and induction furnaces operate on very lagging power factor.

The current for a given load supplied at constant voltage will be higher at a lower power factor and lower at higher power factor.

**For example if load P is to be supplied at terminal voltage V and at power factor cos ɸ by a 3-phase balanced system then load current is given by: **

If P and V are constant, the load current, I_{L} is inversely proportional to power factor, cos ɸ i.e., lower the power factor, higher the current and vice versa.

**The higher current due to poor power factor affects the system and results in following disadvantages:**

(i) Rating of generators and transformers are proportional to their output current hence inversely proportional to power factor, therefore, large generators and transformers are required to deliver same load but at low power factor.

(ii) The cross-sectional area of the bus-bar, and the contact surface of the switchgear is required to be enlarged for the same power to be delivered but at low power factor.

(iii) For the same power to be transmitted but at low power factor, the transmission line or distributor or cable have to carry more current.

The size of the conductor will have to be increased if current density in the line is to be kept constant. Thus more conductor material is required for transmission lines, distributors and cables to deliver the same load but at low power factor.

(iv) Energy losses are proportional to the square of the current hence inversely proportional to the square of the power factor i.e., more energy losses incur at low power factor, which results in poor efficiency.

(v) Low lagging power factor results in large voltage drop in generators, transformers, transmission lines and distributors which results in poor regulation. Hence extra regulating equipment is required to keep the voltage drop within permissible limits.

(vi) Low lagging power factor reduces the handling capacity of all the elements of the system.

Thus we see that the low power factor leads to a high capital cost for the alternators, switchgears, transformers, transmission lines, distributors and cables etc.

Keeping in view the various drawbacks associated with the low power factor, the power suppliers insist on a power factor of 0.8 or above for industrial establishments. The power tariffs are devised to penalize the consumers with low lagging power factor and to encourage them to install power factor correction devices or equipment.

The post Power Factor: Causes and Disadvantages | AC Circuit | Electricity appeared first on Engineering Notes India.

]]>The post Improvement of Power Factor in AC Circuit | Electrical Engineering appeared first on Engineering Notes India.

]]>The low power factor is almost invariably due to inductive nature of load and, therefore, the logical corrective is to connect such devices across the load, which takes leading reactive power such as static capacitors, synchronous machines or synchronous condensers.

The leading reactive component of current drawn by power factor correcting device neutralises the lagging reactive component of current drawn by the load partly or completely. Power factor of the system will become unity when lagging reactive component of load current is completely neutralised by the leading reactive component of current drawn by power factor correcting device.

Let the current drawn by an inductive circuit be I lagging behind the applied voltage by an angle ɸ. The leading current required to neutralize the lagging reactive component of current drawn by the inductive circuit (equipment) to give unity power factor.

**By Use of Static Capacitors****: **

Power factor can be improved by connecting the capacitors in parallel with the equipment operating at lagging power factor such as induction motors, fluorescent tubes. Static capacitors have the advantages of small losses (less than ½ per cent) or higher efficiency (say 99.6%), low initial cost, little maintenance owing to absence of rotating parts, easy installation being lighter in weight and capability to operate under ordinary atmospheric conditions.

However, they have drawbacks of short service life (8 to 10 years), getting damaged on over-voltages and uneconomical repair. The current drawn by induction motors or fluorescent tubes can be resolved into two components; the active component, which is in phase with the supply voltage and the quadrature or wattless component of constant magnitude.

The capacitors draw current leading the supply voltage by 90° approximately and neutralise the quadrature or wattless component of current drawn by the equipment across which these are connected. These capacitors remain connected permanently across the equipment and are across the supply mains, whenever the equipment is switched on.

**The value of the static capacitors for the improvement of the power factor can be determined as follows: **

**The leading current required to neutralise the lagging reactive component of the current drawn by the equipment to give unity power factor is expressed as: **

**The value of capacitance in star bank is given by: **

Where V is the phase voltage, I is the phase current and / is supply frequency.

For given kVAR and line voltage the delta value will be one-third of star value.

Power factor can also be improved by connecting static capacitors in series with the line, as shown in Fig. 15.10. Capacitors connected in series with the line neutralize the line reactance. The capacitors, when connected in series with the line, are called the series capacitors, and when connected in parallel with the equipment, are called the shunt capacitors.

Shunt capacitors are used in factories, plants and also on transmission lines.

Series capacitors are used on long transmission lines as they provide automatic compensation with the variations in load.

**The capacity of the capacitors to neutralize the line reactance is given by: **

Where f is the supply frequency and L is the inductance of the line per phase.

The value of reactance required is usually very large but reduced to reasonable value by use of a transformer, as shown in Fig. 15.11.

Shunt capacitors are used in ratings from 15 kVAR to 10,000 kVAR. Small capacitors, up to a few hundred rating are used on individual distribution circuits of customers. Capacitors banks of 500 – 3,000 kVAR ratings are employed in small distribution substations and those with larger rating at big substations.

Three phase capacitor banks can be connected in star earthed, star unearthed or in delta arrangements. Ungrounded star connection is preferred because of easier protection. In this method, the fault current in case of a fault in any unit in one of the phases is restricted by the capacitors in the sound phases. This results in the use of smaller fuses and less protection materials.

The capacitor must be provided with a suitable discharge device to dissipate the stored energy, and to reduce the residual voltage to a safer value within a short period (50 V or less within one minute in case of medium voltage capacitors and within five minutes in case of high voltage capacitors as per ISS 2834-1963).

The discharge resistance is usually incorporated within the ‘unit’ itself in the case of medium voltage capacitors and in case of high voltage capacitors, potential transformers of the circuit breakers are generally utilised as a discharge device.

**The reactive output of the capacitors in kVAR is given by: **

Where V_{L} is the line voltage, f is the supply frequency in Hz and C is the capacitance in µF between the line terminals.

Thus we see that the corrective capacity of the capacitors is a function of the line voltage and supply frequency, varying in accordance with the square of the voltage and directly with the supply frequency. The units as manufactured are designed for a variation of voltage of ±10% of normal voltage. It is, therefore, impossible to overload these units so long as normal voltage and frequency are maintained.

The characteristics of capacitors, in general, are similar to those of synchronous condensers except that the corrective kVAR of the synchronous condenser is adjustable and may be controlled automatically whereas that of the capacitor is fixed unless there is a possibility of change in the number of units connected.

**By Use of Synchronous or High Power Factor Machines****: **

Synchronous machines are excited by dc, and the power factor may be controlled by controlling the field excitation. The various synchronous machines available for power factor correction comprise synchronous motors, synchronous condensers, synchronous converters, synchronous phase modifiers, phase advancers, and synchronous-induction motors.

**1. By Use of Synchronous Motors:**

These motors have characteristics that make them adaptable for a wide range of applications. The speed is constant, the efficiency is high and uniform from light loads up to considerable overloads, and the starting characteristics compare favourably with those of induction motors.

Another desirable characteristic of the synchronous motor is its tendency to maintain a constant load voltage even if there are variations in the supply voltage. When the line voltage increases, the leading reactive kVA falls and when the line voltage falls, the leading reactive kVA increases.

The usual practice is to keep the field excitation constant at a value corresponding to normal full-load rating as regards output and power factor. Synchronous motors are designed for 1.0 – 0.8 leading power factors at full load. The unity power factor motor costs less and has a higher efficiency, but if fully loaded, it cannot furnish leading reactive kVA to compensate for lagging reactive kVA in the system.

**2. By Use of Synchronous Condensers:**

An overexcited synchronous motor running on no load is called the synchronous condenser or synchronous phase advancer and behaves like a capacitor, the capacitive reactance of which depends upon the motor excitation. Power factor can be improved by using synchronous condensers like shunt capacitors connected across the supply.

In phasor diagram (Fig. 15.12), phasor I_{L} represents the current drawn by the industrial load, lagging behind the applied voltage V by a large angle ɸ_{L} and phasor I_{M} represents the current drawn by the synchronous condenser leading the applied voltage V by the angle ɸ_{M}. The resultant current I is the phasor sum of I_{L} and I_{M} and now angle of lag ɸ is much smaller than ɸ_{L}. Thus overall power factor is improved from cos ɸ_{L} to cos ɸ by the use of the synchronous condenser. In this way the power factor can be made unity even.

Synchronous condensers are usually built in large units and are employed where a large quantity of corrective kVAR (say 5,000 kVAR or more) is required.

**The advantages of synchronous condensers over static capacitors as a power factor correction devices are:**

(i) A finer control can be obtained by variation of field excitation;

(ii) Inherent characteristic of synchronous condensers of stabilizing variations in the line voltage and thereby automatically aid in regulation,

(iii) Possibility of overloading a synchronous condenser for short periods, and

(iv) Improvement in the system stability and reduction of the effect of sudden changes in load owing to inertia of synchronous condenser.

By use of synchronous condensers at intermediate stations, the voltage of the line can be kept constant at various points along its length, thereby, increasing the current carrying capacity of the line and improvement of power factor.

**The disadvantages of synchronous condensers over static capacitors as power factor correcting devices are: **

(i) Except in size above about 5,000 kVAR, the cost is higher than that of static capacitors of the same rating;

(ii) Comparatively higher maintenance and operating costs;

(iii) Comparatively lower efficiency (say 97%) due to losses in rotating parts and heat losses,

(vi) Noise is produced in operation,

(v) An auxiliary equipment is required for starting synchronous condensers;

(vi) Possibility of synchronous condensers falling out of synchronism causing in interruption of supply; and

(vii) Increase of short-circuit currents when the fault occurs near the synchronous condenser.

Synchronous condensers are largely employed by utilities at large substations for improving the power factor and voltage regulation. Machines up to 100 MVAR rating or even higher have been used. The excitation current is regulated automatically to give a desired voltage level.

**3. By Use of Phase Advancers:**

The power factor of an induction motor falls mainly due to its exciting current drawn from the ac supply mains, because exciting current lags behind the voltage by π/2. It may be improved by equipping the set with an ac exciter or phase advancer which supplies this exciting current to the rotor circuit at slip frequency. Such an exciter may be mounted on the same shaft as the main motor or may be suitably driven from it.

Use of phase advancer is not generally economical in connection with motors below 150 kW output but above this size, phase advancers are frequently employed. Shunt and series type of phase advancers are available according to whether the exciting winding of the advancer is connected in parallel or series with the rotor winding of the induction motor.

**There are two main advantages of phase advancers:**

(i) Lagging kVAR drawn by the motor are considerably reduced due to supply of exciting ampere-turns at slip frequency, and

(ii) The phase advancers can be conveniently employed where the use of synchronous motor is inadmissible.

**4. By Use of Synchronous-Induction Motors:**

These are special types of motors which operate at certain loads as synchronous motors and at other loads as induction motors.

**5. By Use of High Power Factor Motors:**

Besides synchronous motors or synchronous-induction motors there are other several types of motors which operate at a power factor of approximately unity such as compensated induction motors, and Schrage motors. These motors are more expensive and have higher maintenance cost than ordinary induction motors.

**Location of Power Factor Correction Equipment****: **

The best location for the power factor correction equipment to be installed is where the apparatus or equipment responsible for low power factor is operating. Synchronous condensers are used at load centres where considerable corrective kVAR is required whereas static capacitors are justifiably used in smaller units and may be placed closer to the point where the load of inductive nature is installed and thereby relieving the distributors and feeders from carrying excessive currents owing to low power factor.

In cases of transmission system, if synchronous condensers are to be employed for power factor improvement then these should be installed at the receiving end so that not only the generators but also the transmission lines are relieved of carrying excessive current due to poor power factor. However, if synchronous condensers are installed near the generators then only generators will be relieved from the excessive current component and the transmission lines will have to carry it.

Modern alternators are designed to have high reactance in order that alternators may not get damaged at the time of short circuit and normally reactance is 20 times of resistance. Any change in current and power factor causes the change in terminal voltage, so voltage can be kept fairly constant by power factor control.

If the power factor of the supply or power station is raised to unity, the current for the same amount of power to be supplied is reduced to minimum. This results in reduction of transmission line copper losses, and reduction of voltage drop in transmission line and in alternator windings, as copper losses are directly proportional to the square of supply current and voltage drop is directly proportional to the current.

The terminal voltage of an alternator is given by the phasor difference of induced emf and voltage drop in synchronous impedance. Neglecting the resistance as compared to synchronous reactance, the synchronous impedance can be considered purely inductive, so synchronous impedance voltage drop leads the load current by π/2.

From the phasor diagrams shown in Figs. 15.13 (a), 15.13 (b) and 15.13 (c) for lagging, unity and leading power factor respectively it is obvious that as the power factor is raised from lagging to unity, the difference of terminal voltage and induced emf is reduced and this difference can be reduced to zero by making the power factor, leading and less than unity. Hence to have zero regulation the power factor should be made leading one, so that no other regulating equipment is required.

**It is not economical to raise the power to unity or leading one due to the following reasons: **

1. In case the power factor is improved to unity for full- load conditions, it would become leading for loads less than full load (unless some capacitors are switched off which is usually difficult). At power factors lower than unity (lagging or leading) the current for the same amount of power to be supplied is increased and thereby energy losses in generators, transformers, transmission lines and distribution lines will be increased.

2. As the power factor approaches unity, the capacity of power factor correction device increases more rapidly i.e., the power factor of an installation can be improved from 0.7 or 0.8 to 0.8 or 0.9 by a much smaller capacitive kVAR than that required for raising the power factor from 0.9 to unity.

**The advantages of good (or improved) power factor are:**

(i) Reduction in load current;

(ii) Increase in voltage level across the load;

(iii) Reduction in energy losses in the system (generators. transformers. transmission lines and distributors) due to reduction in load current;

(iv) Reduction in kVA loading of the generators and transformers which may relieve an overloaded system or release capacity for additional growth of load, and

(v) Reduction in kVA demand charge for large consumers.

When the power factor is improved it involves an expenditure on account of the power factor correcting equipment. Improvement of power factor will result in reduction of maximum demand and thus affect an annual saving over the maximum demand charge but on the other hand an expenditure is to be incurred every year in the shape of interest and depreciation on account of the investment made over the power factor correcting equipment.

The limit of the power factor at which the net saving (saving in annual maximum demand charges less annual expenditure incurred on power factor correcting equipment) is maximum is known as economical limit of power factor correcting. It will be seen that the economical limit of power factor correction is governed by the relative costs of the supply and power factor correcting equipment.

Consider a consumer taking a peak load of P kW at a power factor of cos ɸ_{1}, and charged at the rate of Rs x per kVA of maximum demand per annum. Let the expenditure per kVAR per annum of the power factor correction equipment be Rs y.

From, the above expression, value of most economical power factor cos ɸ_{2} can be determined which is independent of original factor cos ɸ_{1} and is governed by the relative costs of supply and power factor correction equipment.

**Example 1:**

**Calculate the value of the new power factor when the tariff is Rs 1,350 per kVA of maximum demand plus a flat rate paise 80 per kWh. Assume additional cost of condensers etc. at Rs 1,050 per kVA of such plant. Rate of interest and depreciation together is taken as 10%. **

**Solution: **

Maximum demand charges,

x = Rs 1,350 per kVA/annum

Cost of phase advancing plant = Rs 1,050 per kVA

Expenditure on phase advancing plant,

y = Annual interest and depreciation

= Rs 1,050 × (10/100) = Rs 105/kVAR/annum

Most economical power factor.

**Example 2:**

**A factory operates at 0.8 pf lagging and has a monthly demand of 750 kVA. The monthly power rate is Rs 8.50 per kVA. To improve the power factor, 250 kVA capacitors are installed in which there is negligible loss. The installed cost of equipment is Rs 20,000 and fixed charges are estimated as 10% per year. Calculate the annual savings affected by the use of capacitors. **

**Solution: **

Maximum demand = 750 kVA

Load power factor, cos ɸ = 0.8

Power factor angle ɸ = cos^{-1} 0.8 = 36.87°

kW component of load, P = kVA cos ɸ = 750 x 0.8 = 600 kW

kVAR component of load, Q_{1} = kVA sin ɸ = 750 sin 36.87° = 450 (lagging)

Leading kVAR supplied after pf improvement,

Q = Q_{1} – kVAR supplied by capacitors = 450 – 250 = 200

kVA demand after pf improvement,

Reduction in kVA demand = 750 – 632.45 = 117.55

Annual savings on kVA charge = Rs 8.50 × 12 × 117.55 = Rs 11,990

Fixed charges per annum on the capacitors = Rs 20,000 × (10/100) = Rs × 2,000

Annual savings = Rs 11,990 – Rs 2,000 = Rs 9,900 Ans.

The increase in power demand on the generating station can be met either by increasing the capacity of the generating plant working at the same pf or by raising the power factor of the system by installation of phase advancers.

Owing to improvement of power factor in the beginning the saving in the generating and distributing plant outweighs the extra cost of the pf correction equipment in most of the cases but as the power factor is raised further its cost begins to approximate to the saving and finally any saving over the plant is obtained by incurring a greater expenditure on the pf correcting equipment.

Thus there is a limit beyond which it is not economical still further to improve the power factor. The maximum value to which the power factor can be economically raised entirely depends upon the relative costs of the generating plant and phase advancing plant.

Let the rating of generating station be S in kVA, supplying load at power factor cos ɸ_{2}.

Assume new demand can be met by improving the power factor from cos ɸ_{1} to cos ɸ_{2} with the same capacity of the plant.

Existing load in kW, P_{1} = S cos ɸ_{1}

New load in kW, P_{2} = S cos ɸ_{2}

**(i) Cost by increasing the capacity of the generating station: **

Increase in load = P_{2} – P_{1} = S (cos ɸ_{2} – cos ɸ_{1})

Increase in the capacity of the generating plant operating at pf cos ɸ_{1}

Increase in annual cost due to increase in capacity of the plant-

where x is the annual cost per kVA of generating plant

**(ii) Cost on power factor correction equipment: **

Reactive power drawn by load operating at the old power factor, cos ɸ_{1} = P_{2} tan ɸ_{1} = S cos ɸ_{2} tan ɸ_{1}

Reactive power supplied by the plant = Capacity of the plant in kVA sin ɸ_{2} = S sin ɸ_{2 }since kVA rating of the plant remains the same

kVAR rating of the pf correction equipment = Reactive power drawn by load – reactive power supplied by the plant = S cos ɸ_{2} tan ɸ_{1} – S sin ɸ_{2} = S (cos ɸ_{2} tan ɸ_{1} – sin ɸ_{2})

Annual cost on pf correction equipment-

= Rs y S (cos ɸ_{2} tan ɸ_{1} – sin ɸ_{2}) …(15.9)

where y is the annual cost per kVAR rating of power factor correction equipment.

Power factor correction equipment will be cheaper if annual cost on it is less than annual cost on account of increasing the generating capacity.

In the limiting case, the maximum cost of pf correction equipment, which would justify its installation, should be equal to cost on account of increasing the generating plant capacity.

**Example 3:**

**A power plant is working at its maximum kVA capacity with a lagging pf of 0.7. It is now required to increase its kW capacity to meet the demand of additional load. **

**This can be done: **

**(i) By increasing the pf to 0.85 (lagging) by pf correction equipment. **

**Or **

**(ii) By installing additional generation plant costing Rs 800 per kVA.**

**What is the maximum cost per kVA of pf correction equipment to make its use more economical than the additional plant? **

**Solution: **

Let the rating of the power plant be S kVA Increase in load

= P_{2} – P_{1} = S(cos ɸ_{2} – cos ɸ_{1}) = S(0.85 – 0.7) = 0.15 S

Increase in capacity of the generating plant-

Cost of additional plant = Rs 800 × 0.2143 S = Rs 171.43 S

Let the rate of interest and depreciation be m per cent per annum Increase in annual cost due to increase in capacity of power plant-

**Alternative Method:**

Annual cost per kVA of generating plant-

where m is the rate of interest and depreciation per annum.

Original power factor,

cos ɸ_{1} = 0.707 and therefore ɸ_{1} = Cos^{-1} 0.7 = 45.573°

Improved power factor,

cos ɸ_{2} = 0.87 and therefore ɸ_{2} = Cos^{-1} 085 = 31.788˚

Annual cost per kVAR of pf correction equipment-

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]]>The post Graphical Representation of an Electric Circuit | Engineering appeared first on Engineering Notes India.

]]>An energy source (or source), such as a primary or secondary cell, a generator, and the like, is a device that converts chemical, mechanical, thermal or some other form of energy into electrical energy.

An energy convertor, also called the load, (such as lamp, heating appliance, or an electric motor) converts electrical energy into light, heat, mechanical work and so on.

Events in an electrical circuit may be defined in terms of emf (or voltage) and current.

When electrical energy is generated, transmitted and converted under conditions such that the currents and voltages involved remain constant with time, the electric circuit is identified as direct current (dc) circuit. If the currents and voltages do change with time, the circuit is defined as alternating current (ac) circuit.

A graphic representation of an electric circuit is called a circuit diagram (Fig. 2.1). Such a diagram consists of an interconnected symbol called circuit elements or circuit parameters. Two elements are necessary to represent processes in a dc circuit. These are source of emf E_{S} and of internal (or source) resistance R_{S} and the load resistance (which includes the resistance of the conductors) R.

In any electric circuit the energy convertor (or load) and the conductors connecting it to the source make up the external circuit in which current flows from the + ve side to the – ve side of the source whereas inside the source, current flows in the opposite direction, i.e., from the – ve side to the + ve side. The source emf is directed from the terminal at a lower potential to that at a higher one. In diagrams this is shown by arrows.

The source emf (or open-circuit voltage) is the voltage that appears across the source when no load is connected across it.

When a load is connected to the source terminals and the circuit is closed, an electric current starts flowing through the circuit. Now voltage across source terminals (called the terminal voltage) is not equal to source emf. It is due to voltage drop inside the source, i.e., across the source resistance.

Voltage drop inside the source = I R_{S}.

The relationship between the current through a resistance and the voltage across the same resistance is called its volt-ampere (or voltage-current) characteristic. When represented graphically, voltages are laid off as abscissae and currents as ordinates.

There are two types of volt-ampere characteristics-straight line and non-linear (curve), as shown in Figs. 2.2 (a) and 2.2 (b) respectively.

Resistive elements for which the volt-ampere characteristic is a straight line [Fig. 2.2 (a)] are called linear, and the electric circuits containing only linear resistances are called linear circuits.

Resistive elements for which the volt-ampere characteristic is other than a straight line are termed nonlinear, and so the electric circuits containing them are called non-linear circuits. Examples of non-linear elements are tungsten lamps, vacuum tubes and transistors, etc.

An electric circuit, whose characteristics or properties are same in either direction (e.g., a distribution or transmission line), is called the bilateral circuit. The distribution or transmission line can be made to perform its function equally well in either direction.

An electric circuit, whose characteristics or properties change with the direction of its operation (e.g., a diode rectifier), is called the unilateral circuit. A diode rectifier cannot perform rectification in both directions.

A network is said to be passive if it contains no source of emf in it. The equivalent resistance between any two terminals of a passive network is the ratio of potential difference across the two terminals to the current flowing into (or out of) the network. When a network contains one or more sources of emf and/ or current, it is said to be active.

In case, a branch is removed from an electric network, the remainder of the network is left with a pair of terminals. The part of the network, which is considered with respect to the removed branch or terminal pair or port, is termed as one-port network.

When two branches are removed so that the network is left with four terminals or two pairs of terminals, the remainder network is called the two-port network. Usually one port accepts a source and the other port is coupled to a load, so that there is an input port and an output port in any two-port system.

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]]>When the resistors are connected end to end, so that they form only one path for the flow of current, then resistors are said to be connected in series and such circuits are known as series circuits.

Let resistors R_{1}, R_{2}, and R_{3} be connected in series as shown in Fig. 2.5, and the potential difference of V volts be applied between extreme ends A and D to cause flow of current of I amperes through all the resistors R_{1}, R_{2}, and R_{3}.

**Now according to Ohm’s law:**

Voltage drop across resistor R_{1}, V_{1} = I R_{1}

Voltage drop across resistor R_{2}, V_{2} = I R_{2}

Voltage drop across resistor R_{3}, V_{3} = I R_{3}

Voltage drop across whole circuit,

V = Voltage drop across resistor R_{1} + voltage drop across resistor R_{2 }+ voltage drop across resistor R_{3}

i.e. V = I R_{1} + I R_{2} + I R_{3} = I (R_{1} + R_{2} + R_{3})

Or V/I = R_{1} + R_{2 }+ R_{3} …(2.4)

And according to Ohm’s law V/I gives the whole circuit resistance, say R

.** ^{.}**. Effective resistance of the series circuit,

R = R_{1} + R_{2 }+ R_{3} …(2.5)

Thus when a number of resistors are connected in series, the equivalent resistance is given by the arithmetic sum of their individual resistances.

i.e., R = R_{1} + R_{2} + R_{3} + … R_{n} …(2.6)

**From the above discussions for a series circuit we conclude that: **

1. Same current flows through all parts of the circuit,

2. Applied voltage is equal to the sum of voltage drops across the different parts of the circuit,

3. Different resistors have their individual voltage drops,

4. Voltage drop across individual resistor is directly proportional to its resistance, current being the same in each resistor,

5. Voltage drops are additive,

6. Resistances are additive, and

7. Powers are additive.

Series circuits are common in electrical equipment. The tube filaments in small radios are usually in series. Current controlling devices are wired in series with the controlled equipment. Fuses are in series with the equipment they protect. A thermostat switch is in series with the heating element in an electric iron. Automatic house-heating equipment has a thermostat, electro-magnet coils, and safety cut-outs in series with a voltage source. Rheostats are placed in series with the coils in large motors for motor current control.

When a number of resistors are connected in such a way that one end of each of them is joined to a common point and the other ends being joined to another common point, as shown in Fig. 2.9, then resistors are said to be connected in parallel and such circuits are known as parallel circuits. In these circuits current is divided into as many paths as the number of resistances.

Let the resistors R_{1}, R_{2} and R_{3} be connected in parallel, as shown in Fig. 2.9, and the potential difference of V volts be applied across the circuit.

Since potential difference across each resistor is same and equal to potential difference applied to the circuit i.e. V

Thus when a number of resistors are connected in parallel, the reciprocal of the equivalent resistance is given by the arithmetic sum of the reciprocals of their individual resistances.

**In general if n resistors of resistances R _{1}, R_{2}, R_{3} …Rn are connected in parallel, then equivalent resistance R of the circuit is given by the expression:**

**From the above discussions for a parallel circuit we conclude that:**

1. Same voltage acts across all branches of the circuit,

2. Different resistors (or branches) have their individual currents,

3. Total circuit current is equal to the sum of individual currents through the various resistors (or branches),

4. Branch currents are additive,

5. Conductance are additive,

6. Powers are additive, and

7. The reciprocal of the equivalent or combined resistance is equal to the sum of the reciprocals of the resistances of the individual branches.

Parallel circuits are very common in use. Various lamps and appliances in a house are connected in parallel, so that each one can be operated independently. A series circuit is an “all or none” circuit, in which either everything operates or nothing operates. For individual control, devices are wired in parallel.

**Current Distribution in Parallel Circuits: **

**Let two resistors of resistances R _{1} and R_{2} be connected in parallel across a pd of V volts, According to Ohm’s law:**

Hence current flowing through each resistor, when connected in parallel, is inversely proportional to their respective resistances.

**Since conductance is reciprocal of resistance and if G _{1} and G_{2} are the respective conductance’s of resistors R_{1} and R_{2} then:**

**Adding 1 on both sides of the above expression we have:**

Hence current in any branch of a parallel circuit is directly proportional to its respective conductance and is equal to the total current flowing through the circuit multiplied by the ratio of the conductance of the branch to that of the circuit.

The same relation holds good for parallel circuit consisting of more than two resistors and is very useful for its solution.

Let us consider a circuit consisting of resistances R_{1}, R_{2}, R_{3}, and R_{4} ohms respectively connected in parallel across a potential difference of V volts, as shown in Fig. 2.12.

So far, only simple series and simple parallel circuits have been considered. Practical electric circuits very often consist of combinations of series and parallel resistances. Such circuits may be solved by the proper application of Ohm’s law and the rules for series and parallel circuits to the various parts of the complex circuit.

There is no definite procedure to be followed in solving complex circuits; the solution depends on the known facts concerning the circuit and the quantities which one desires to find. One simple rule may usually be followed, however — reduce the parallel branches to an equivalent series branch and then solve the circuit as a simple series circuit.

For example consider a series-parallel circuit shown in Fig. 2.14 for solution.

First of all equivalent resistances of all parallel branches are determined separately e.g. of branches AB and CD by the law of parallel circuits,~~ ~~

~~ ~~

**Now the circuit shown in Fig. 2.14 gets reduced to a simple series circuit shown in Fig. 2.15 consisting of three resistances:**

**After knowing I, potential differences across branches AB, BC and CD are determined from the relations:**

**After determination of potential difference across each parallel branch, the currents in the various resistances are determined from the relations:**

Thus equivalent resistance of the whole circuit, voltage drop across each branch and currents in the various resistors may be determined.

**Network Simplification (Or Reduction): **

Sometimes we come across so complicated circuits that they cannot be solved simply by applying Ohm’s law. Hence for solution of such a circuit first of all the circuit is reduced to simple series or simple parallel or series-parallel circuit and then solved by applying Ohm’s law.

Network reduction or simplification is a process by which currents and voltages acting in a circuit composed of resistors having a series, parallel or series-parallel combinations can be determined.

It may be seen from Fig. 2.18 (a) that resistances R_{3}, R_{4} and R_{5} are connected in series and the same current I_{3} flows through them because the circuit does not branch off at points C and D.

**By virtue of this fact, the three resistances R _{3}, R_{4} and R_{5} can be combined into a single equivalent resistance given by the expression:**

R’ = R_{3} + R_{4} + R_{5}

Thus a simpler equivalent circuit shown in Fig. 2.18 (b) is obtained. From circuit diagram shown in Fig. 2.18 (b) it can be seen that resistance R_{2} is connected in parallel with the series combination of resistances R_{3}, R_{4} and R_{5} i.e., equivalent resistance R’.

**Resistances R _{2}, and R’ can be combined into a single equivalent resistance given by the expression:**

Thus the circuit is further simplified [Fig. 2.18 (c)]. Resistances R_{1}, R” and R_{0} in the circuit of Fig. 2.18 (c) are seen to be connected in series.

**Replacing all the three resistances by a single equivalent resistance connected across the battery (supply source), a simple series circuit [Fig. 2.18 (d)] is obtained where in:**

R” = R_{1} + R_{6} + R”

In a similar way most of the series-parallel circuits containing a single supply source can be reduced to a simple series circuit.

Now the current in the simple series circuit, shown in Fig. 2.18 (d), can be readily determined by applying Ohm’s law. Now the branch currents can be determined by restoring the circuit step by step to its original form in the reverse order.

It is evident from circuit diagram shown in Fig. 2.18 (c)

I_{1} = I_{2} + I_{3}

**Furthermore, the voltage across points B and E will be I _{1}R”. Knowing the voltage across points B and E, currents I_{2}, and I_{3} can be easily determined from the expressions:**

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]]>The following points highlight the two theorems used for determining the current flowing in a circuit. The theorems are: 1. Maxwell Circulating Current Theorem 2. Node Voltage Theorem.

If a network with several sources has more than two nodes the current in it may be determined by Maxwell circulating current theorem. This is one of the most universal methods for solving networks.

In a number of cases, a network may be considered as consisting of a set of adjoining loops, each of which forms a polygon made up of several branches of the network (without any diagonals). Some branches of the network are common to two adjacent loops, while others form an external circuit where each branch occurs in one loop only.

This theorem involves representing a current that is assumed to circulate around a closed loop by a curved arrow and labelling the arrow with its identifying current symbol I with a subscript.

By this theorem the current flowing through the branch common to two meshes will be equal to the algebraic sum of the two loop currents flowing through it. The direction of any loop current may be taken either as clockwise or counter-clockwise but for systematic solution the directions of all loop currents are assumed to be the same (say clockwise).

Then Kirchhoff’s second law is applied to each mesh and algebraic equations are obtained. The total number of independent equations is equal to the number of meshes (i.e. there are fewer equations than in a purely Kirchhoffian solution). Therefore, they can be solved as simultaneous equations to give the circulating currents and then the branch currents. Thus, this method eliminates a great deal of tedious calculation work involved in the branch current method.

**Application of Maxwell circulating current theorem will be more clear from the following illustrations:**

**Example 1:**

**Solve the network shown in Fig. 2.65 by mesh current method.**** **

**Solution: **

The network is redrawn, as illustrated in Fig. 2.66. There are two independent loops. The loop currents have been taken clockwise, as marked in the circuit diagram. The individual branch currents along with their directions of flow are also shown in the circuit diagram.

**Applying Kirchhoff’s voltage law to meshes I and II we have:**

For application of node voltage theorem one of the node is taken as reference or zero potential or datum node and the potential difference between each of the other nodes and the reference node is expressed in terms of an unknown voltage (symbolized as V_{1}, V_{2} or V_{A}, V_{B} or V_{x}, V_{y} etc.) and at every node Kirchhoff’s first (or current) law is applied assuming the possible directions of branch currents.

This assumption does not change the statement of problem, since the branch currents are determined by the potential difference between respective nodes and not by absolute values of node potentials.

Like Maxwell’s circulating current theorem, node-voltage theorem reduces the number of equations to be solved to determine the unknown quantities. If there are n number of nodes, there shall be (n – 1) number of nodal equations in terms of (n – 1) number of unknown variables of nodal voltages. By solving these equations, nodal voltages are known to compute the branch currents.

When the number of nodes minus one is less than the number of independent meshes in the network, it is, in fact more advantageous. Moreover, it is particularly suited for networks having many parallel circuits with common ground connected node such as in electronic circuits.

**Example 2: **

Consider, for example, a two node network, as illustrated in Fig. 2.72.

Node C has been taken as reference node. Let respectively with respect to node C. Let the current distribution be as shown on the circuit diagram (Fig. 2.72) arbitrarily. Now let us get independent equations for these two nodes.

**Node A is the junction of resistors R _{1}, R_{2}, and R_{4}. So current equation for node A is:**

**Node B is the junction of resistors R _{2}, R_{3}, and R_{5}. So current equation for node B is:**

The Eqs. (2.34) and (2.35) can now be solved to get the values of V_{A} and V_{B} and then the values of currents I_{1}, I_{2}, I_{3}, I_{4} and I_{5} can be computed easily.

**Example 3:**

**Two batteries A and B are connected in parallel to a load of 10 ohm. Battery A has an emf of 12 V and an internal resistance of 2 ohm and battery B has an emf of 10 V and internal resistance of 1 ohm. Using nodal analysis, determine the currents supplied by each battery and load current. **

**Solution: **

Taking node C as reference node and the potentials of nodes A and B be V_{A} and V_{B} respectively and current distribution as shown in Fig. 2.73 arbitrarily.

From circuit diagram shown in Fig. 2.73.

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]]>**The current-voltage relationship involving the capacitance component given as:**

Above equation describes a situation in which the current through the capacitance component is proportional to the derivative of voltage across it.

**Solving Eq. (1.11) for the voltage yields:**

The proportionality constant C expresses the charge-storing property of the element and is called the capacitance of the component. Any circuit component showing the property of yielding a current which is directly proportional to the rate of change of the voltage across it terminals is called a capacitor.

A capacitor is basically meant to store electrons (or electrical energy), and release them whenever required. Capacitance is a measure of a capacitor’s ability to store charge and is measured in farads (F). Farad, the unit of capacitance is very large, so micro-farad (µF) or micro-micro-farad (µµF) is usually used 1 µ F = 1^{-6} F and 1 µ µ F, also called the Pico-farad (p F), = 10^{-12} F.

A capacitor offers low impedance to ac but very high impedance to dc. So capacitors are used to couple alternating voltage from one circuit to another circuit and at the same time to block dc voltage from reaching the next circuit. It is also employed as a bypass capacitor where it passes the ac through it without letting the dc to go through the circuit across which it is connected. A capacitor forms a tuned circuit in series or in parallel with an inductor.

A capacitor consists of two conducting plates, separated by an insulating material, called the dielectric. Capacitors, like resistors and inductors, can either be fixed or variable. Some of the commonly used fixed capacitors are mica, ceramic, paper, plastic-film and electrolytic capacitors.

Variable capacitors are mostly air-gang capacitors. An elementary variable capacitor consists of two sets of copper or aluminum plates (which may have the shape of half discs).

Each set is mounted on a common shaft, one set being fixed and the other, which interleaves with the former, being movable. The capacitance of the capacitor can be easily varied by varying the degree of interleaving, which is possible by rotation of movable plate shaft. Such variable-capacitance, air-capacitors, shown in Fig. 1.13 is widely used in radio work.

Symbols for fixed type and variable type capacitors are given in Figs. 1.14 (a) and 1.14 (b) respectively.

Capacitors employed in power supplies of electronic and radio equipment may have values ranging from a few micro-farads to several thousand micro-farads. The capacitors used in the tuning circuits of the TV receivers may be as small as 10-20 pF.

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]]>Consider an ac circuit consisting of resistance of R ohms and inductance of L henrys connected in series, as shown in Fig. 4.5 (a).

Let the supply frequency be f and current flowing through the circuit be of I amperes (rms value).

Now voltage drop across resistance, V_{R} = I R in phase with the current.

Voltage drop across inductance, V_{L} = I X_{L} = I ω L leading I by π/2 radians, as shown in Fig 4.5 (c).

The applied voltage, being equal to the phasor sum of V_{R} and V_{L}, will be given by the diagonal of the parallelogram.

Quantity √R^{2} + XL^{2} is known as impedance, denoted by Z and is expressed in ohms.

**From phasor diagram it is also evident that the current lags behind the applied voltage V by angle which is given by:**

Since X_{L} and R are known, the value of phase angle ɸ can be computed.

**If the applied voltage v = V _{max} sin **

If a triangle ABC is drawn so that AB = V_{R}/I = R, BC = V_{L}/I = X_{L} and AC = V/I = Z, it is a triangle similar to that produced by the voltage triangle. Such a triangle is called an impedance triangle, which is most useful in letting one see at a glance how R, X, and Z are related to each other. The angle between Z and R sides of the impedance triangle is known as phase angle of the circuit and cos of this angle is known as power factor of the circuit.

Power factor = Cos ɸ = R/Z

Where V and I are the rms values of voltage and current and ɸ is the phase angle between applied voltage V and circuit current I.

**Alternatively power consumed by the circuit: **

p = Power consumed by resistance

** ^{.}**.

= I^{2} R = I (I R) = V/Z. I R = V I R/Z = V I cos ɸ

Since from impedance triangle cos ɸ = R/Z

So the power in an ac circuit is given by the product of rms values of current and voltage and cosine of the phase angle between voltage and current. Cosine of the phase angle between the voltage and current, cos ɸ is known as the power factor of the circuit, and is equal to R/Z which is obvious from impedance triangle [Fig. 4.6 (b)]

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]]>Let the current flowing through the circuit be of I amperes and supply frequency be f Hz.

Voltage drop across resistance, V_{R} = IR in phase with I

Voltage drop across inductance, V_{L}= IωL leading I by π/2 radians or 90°

Voltage drop across capacitance, V_{C} = I/ω C or IX_{C} lagging behind I by π/2 radians or 90°

V_{L} and V_{C} are 180″ out of phase with each other (or reverse in phase), therefore, when combined by parallelogram they cancel each other. The circuit can either be effectively inductive or capacitive depending upon which voltage drop (V_{L} or V_{C}) is predominant. Let us consider the case when V_{L} is greater than V_{C}.

**The applied voltage V, being equal to the phasor sum of V _{R}, V_{L} and V_{C} is given in magnitude by:**

**Phase angle ****ɸ**** between voltage and current is given by:**

ɸ will be + ve i.e. applied voltage will lead the current if X_{L} > X_{c} and 3> will be – ve i.e., applied voltage will be behind the current if X_{L} < X_{c}.

**Power factor of the circuit is given by:**

Power consumed in the circuit, P = I^{2} R or V I cos ɸ.

**Reactance:**

Inductive reactance, X_{L} is directly proportional to frequency being equal to ωL or 2 π f L and capacitive reactance, X_{C} is inversely proportional to frequency being equal to 1/ω C or 1/2 π fC.

Inductive reactance causes the current to lag behind the applied voltage, while the capacitive reactance causes the current to lead the voltage. So when inductance and capacitance are connected in series, their effects neutralize each other and their combined effect is then their difference.

**The combined effect of inductive reactance and capacitive reactance is called the reactance and is found by subtracting the capacitive reactance from the inductive reactance or according to equation: **

X = X_{L} – X_{C}

When X_{L} > X_{C} i.e. X_{L} – X_{C} is positive, the circuit is inductive and phase angle is ɸ positive.

When X_{L} < X_{C} i.e. X_{L} – X_{C} is negative, the circuit is capacitive and phase angle is ɸ negative.

When X_{L} = X_{C} i.e. X_{L} – X_{C} = 0, the circuit is purely resistive and phase angle ɸ is zero.

**If the expression for applied voltage is taken as:**

v = V_{max} Sin^{ }ωt

**Then expression for the current will be:**

The value of ɸ will be positive when current leads i.e., when X_{C} > X_{L} and negative when current lags i.e., when X_{L} > X_{C}.

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