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Maximum power transfer theorems and impedance matching

At the end of this chapter you should be able to: ž appreciate the conditions for maximum power transfer in a.c. networks ž apply the maximum power transfer theorems to a.c. networks ž appreciate advantages of impedance matching in a.c. networks ž perform calculations involving matching transformers for impedance matching in a.c. networks

35.1 Maximum power transfer theorems

Figure 35.1

A network that contains linear impedances and one or more voltage or current sources can be reduced to a Th´evenin equivalent circuit as shown in Chapter 33. When a load is connected to the terminals of this equivalent circuit, power is transferred from the source to the load. A Th´evenin equivalent circuit is shown in Figure 35.1 with source internal impedance, z D r C jx and complex load Z D R C jX. The maximum power transferred from the source to the load depends on the following four conditions. Condition 1. Let the load consist of a pure variable resistance R (i.e. let X D 0). Then current I in the load is given by: ID

E r C R C jx

and the magnitude of current, jIj D


E [r C R2 C x 2 ]

The active power P delivered to load R is given by P D jIj2 R D

E2 R r C R2 C x 2

To determine the value of R for maximum power transferred to the load, P is differentiated with respect to R and then equated to zero (this being the normal procedure for finding maximum or minimum values using 1

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calculus). Using the quotient rule of differentiation, dP D E2 dR



[r C R2 C x 2 ]1  R2r C R [r C R2 C x 2 ]2



D 0 for a maximum (or minimum) value. For

dP to be zero, the numerator of the fraction must be zero. dR r C R2 C x 2  2Rr C R D 0

Hence i.e.,

r 2 C 2rR C R2 C x 2  2Rr  2R2 D 0

from which, r 2 C x 2 D R2 R=

or

35.1




.r 2 Y x 2 / = jz j

Thus, with a variable purely resistive load, the maximum power is delivered to the load if the load resistance R is made equal to the magnitude of the source impedance. Condition 2. Let both the load and the source impedance be purely resistive (i.e., let x D X D 0). From equation (35.1) it may be seen that the maximum power is transferred when

R=r

(this is, in fact, the d.c.

condition explained in Chapter 13, page 187) Condition 3. Let the load Z have both variable resistance R and variable reactance X. From Figure 35.1, current I D

E E and jIj D
2 r C R C jx C x [r C R C x C X2 ]

The active power P delivered to the load is given by P D jIj2 R (since power can only be dissipated in a resistance) i.e., PD

r C

E2 R C x C X2

R2

If X is adjusted such that X D x then the value of power is a maximum. E2 R If X D x then P D r C R2 dP D E2 dR Hence i.e., from which,



r C R2 1  R2r C R r C R4



D 0 for a maximum value

r C R2  2Rr C R D 0 r 2 C 2rR C R2  2Rr  2R2 D 0 r 2  R2 D 0 and R D r 2

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Thus with the load impedance Z consisting of variable resistance R and variable reactance X, maximum power is delivered to the load when X = −x and R = r,

i.e., when R C jX D r  jx. Hence maximum

power is delivered to the load when the load impedance is the complex conjugate of the source impedance. Condition 4. Let the load impedance Z have variable resistance R and fixed reactance X. From Figure 35.1, the magnitude of current, jIj D


E [r C R2 C x C X2 ]

and the power dissipated in the load, PD

r C

dP D E2 dR



D0

E2 R C x C X2

R2

[r C R2 C x C X2 ]1  R2r C R [r C R2 C x C X2 ]2



for a maximum value r C R2 C x C X2  2Rr C R D 0

Hence

r 2 C 2rR C R2 C x C X2  2Rr  2R2 D 0 from which, R2 D r 2 C x C X2 and R =




[r 2 Y .x Y X /2 ]

Summary With reference to Figure 35.1: 1

When the load is purely resistive (i.e., X D 0) and adjustable, maximum power transfer is achieved when

2

R = jz j =




.r 2 Y x 2 /

When both the load and the source impedance are purely resistive (i.e., X D x D 0), maximum power transfer is achieved when

3

R=r

When the load resistance R and reactance X are both independently adjustable, maximum power transfer is achieved when X = −x and R = r

4

When the load resistance R is adjustable with reactance X fixed, maximum power transfer is achieved when R=




[r 2 Y .x Y X /2 ] 3

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The maximum power transfer theorems are primarily important where a small source of power is involved — such as, for example, the output from a telephone system (see Section 35.2) Problem 1. For the circuit shown in Figure 35.2 the load impedance Z is a pure resistance. Determine (a) the value of R for maximum power to be transferred from the source to the load, and (b) the value of the maximum power delivered to R. (a)

From condition 1, maximum power transfer occurs when R D jzj, i.e., when R D j15 C j20j D

Figure 35.2

(b)




152 C 202  D 25 Z

Current I flowing in the load is given by I D E/ZT , where the total circuit impedance ZT D z C R D 15 C j20 C 25 D 40 C j20 or 44.726 26.57°  Hence

ID

1206 0° D 2.6836 26.57° A 44.726 26.57°

Thus maximum power delivered, P D I2 R D 2.6832 25 D 180 W Problem 2. If the load impedance Z in Figure 35.2 of problem 1 consists of variable resistance R and variable reactance X, determine (a) the value of Z that results in maximum power transfer, and (b) the value of the maximum power. (a)

From condition 3, maximum power transfer occurs when X D x and R D r. Thus if z D r C jx D 15 C j20 then Z = .15 − j20/Z or 25 6 −53.13° Z

(b)

Total circuit impedance at maximum power transfer condition, ZT D z C Z, i.e., ZT D 15 C j20 C 15  j20 D 30  Hence current in load, I D

1206 0° E D D 4 6 0° A ZT 30

and maximum power transfer in the load, P D I2 R D 42 15 D 240 W Problem 3. For the network shown in Figure 35.3, determine (a) the value of the load resistance R required for maximum power transfer, and (b) the value of the maximum power transferred. Figure 35.3 4

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(a)

This problem is an example of condition 1, where maximum power transfer is achieved when R D jzj. Source impedance z is composed of a 100  resistance in parallel with a 1 µF capacitor. 1 1 D 2fC 210001 ð 106  D 159.15 

Capacitive reactance, XC D

Hence source impedance, zD

159156 90° 100j159.15 D 100  j159.15 187.966  57.86°

D 84.676 32.14°  or 71.69  j45.04 Thus the value of load resistance for maximum power transfer is 84.67 Z (i.e., jzj) (b)

With z D 71.69  j45.04 and R D 84.67  for maximum power transfer, the total circuit impedance, ZT D 71.69 C 84.67  j45.04 D 156.36  j45.04 or 162.726 16.07°  Current flowing in the load, I D

2006 0° V D ZT 162.726 16.07° D 1.236 16.07° A

Thus the maximum power transferred, P D I2 R D 1.232 84.67 D 128 W Problem 4. In the network shown in Figure 35.4 the load consists of a fixed capacitive reactance of 7  and a variable resistance R. Determine (a) the value of R for which the power transferred to the load is a maximum, and (b) the value of the maximum power. (a)

From condition (4), maximum power transfer is achieved when RD




[r 2 C x C X2 ] D D

(b)

Current I D D




[42 C 10  72 ]




42 C 32  D 5 Z

606 0° 606 0° D 4 C j10 C 5  j7 9 C j3 606 0° D 6.3246 18.43° A 9.4876 18.43°

Thus the maximum power transferred, P D I2 R D 6.3242 5 Figure 35.4

D 200 W 5

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Problem 5. Determine the value of the load resistance R shown in Figure 35.5 that gives maximum power dissipation and calculate the value of this power. Using the procedure of Th´evenin’s theorem (see page 576):

Figure 35.5

(i) (ii) (iii) (iv)

R is removed from the network as shown in Figure 35.6 P.d. across AB, E D 15/15 C 520 D 15 V Impedance ‘looking-in’ at terminals AB with the 20 V source removed is given by r D 5 ð 15/5 C 15 D 3.75  The equivalent Th´evenin circuit supplying terminals AB is shown in Figure 35.7. From condition (2), for maximum power transfer, R D r, i.e., R = 3.75 Z Current I D

15 E D D2A RCr 3.75 C 3.75

Thus the maximum power dissipated in the load,

Figure 35.6

P D I2 R D 22 3.75 D 15 W Problem 6. Determine, for the network shown in Figure 35.8, (a) the values of R and X that will result in maximum power being transferred across terminals AB, and (b) the value of the maximum power.

Figure 35.7

(a)

Using the procedure for Th´evenin’s theorem: (i) Resistance R and reactance X are removed from the network as shown in Figure 35.9 (ii) P.d. across AB, 

ED



5 C j10 11.186 63.43° 1006 30°  1006 30°  D 5 C j10 C 5 14.146 45° D 79.076 48.43° V

Figure 35.8

(iii) With the 1006 30° V source removed the impedance, z, ‘looking in’ at terminals AB is given by: zD

55 C j10 511.186 63.43°  D 5 C 5 C j10 14.146 45° 

D 3.9536 18.43°  or 3.75 C j1.25

Figure 35.9

(iv) The equivalent Th´evenin circuit is shown in Figure 35.10. From condition 3, maximum power transfer is achieved when X D x and R D r, i.e., in this case when X = −1.25 Z and R = 3.75 Z 6

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(b)

Current I D

E 79.076 48.43° D zCZ 3.75 C j1.25 C 3.75  j1.25 D

79.076 48.43° D 10.5436 48.43° A 7.5

Thus the maximum power transferred, P D I2 R D 10.5432 3.75 D 417 W Figure 35.10

35.2

Further problems on the maximum power transfer theorems may be found in Section 35.3, problems 1 to 10, page 626.

Impedance matching

It is seen from Section 35.1 that when it is necessary to obtain the maximum possible amount of power from a source, it is advantageous if the circuit components can be adjusted to give equality of impedances. This adjustment is called ‘impedance matching’ and is an important consideration in electronic and communications devices which normally involve small amounts of power. Examples where matching is important include coupling an aerial to a transmitter or receiver, or coupling a loudspeaker to an amplifier. The mains power supply is considered as infinitely large compared with the demand upon it, and under such conditions it is unnecessary to consider the conditions for maximum power transfer. With transmission lines (see Chapter 44), the lines are ‘matched’, ideally, i.e., terminated in their characteristic impedance. With d.c. generators, motors or secondary cells, the internal impedance is usually very small and in such cases, if an attempt is made to make the load impedance as small as the source internal impedance, overloading of the source results. A method of achieving maximum power transfer between a source and a load is to adjust the value of the load impedance to match the source impedance, which can be done using a ‘matching-transformer’. A transformer is represented in Figure 35.11 supplying a load impedance ZL .

Figure 35.11

Matching impedance by means of a transformer

Small transformers used in low power networks are usually regarded as ideal (i.e., losses are negligible), such that 7

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N1 I2 V1 D D V2 N2 I1 From Figure 35.11, the primary input impedance jzj is given by jzj D

N1 /N2 V2 V1 D D I1 N2 /N1 I2



N1 N2

2

V2 I2

Since the load impedance jZL j D V2 /I2 , 

jz j =

N1 N2

2

jZL j

35.2

If the input impedance of Figure 35.11 is purely resistive (say, r) and the load impedance is purely resistive (say, RL ) then equation (35.2) becomes 

r=

N1 N2

2

RL

35.3

(This is the case introduced in Section 20.10, page 334). Thus by varying the value of the transformer turns ratio, the equivalent input impedance of the transformer can be ‘matched’ to the impedance of a source to achieve maximum power transfer. Problem 7. Determine the optimum value of load resistance for maximum power transfer if the load is connected to an amplifier of output resistance 448  through a transformer with a turns ratio of 8:1 The equivalent input resistance r of the transformer must be 448  for maximum power transfer. From equation (35.3), r D N1 /N2 2 RL , from which, load resistance RL D rN2 /N1 2 D 4481/82 D 7 Z Problem 8. A generator has an output impedance of 450 C j60. Determine the turns ratio of an ideal transformer necessary to match the generator to a load of 40 C j19 for maximum transfer of power. Let the output impedance of the generator be z, where z D 450 C j60 or 453.986 7.59° , and the load impedance be ZL , where ZL D 40 C j19 or 44.286 25.41° . From Figure 35.11 and equation (35.2), z D N1 /N2 2 ZL . Hence     N1 z 453.98 p transformer turns ratio D D D 10.25 D 3.20 N2 ZL 44.28 8

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Problem 9. A single-phase, 240 V/1920 V ideal transformer is supplied from a 240 V source through a cable of resistance 5 . If the load across the secondary winding is 1.60 k determine (a) the primary current flowing, and (b) the power dissipated in the load resistance. The network is shown in Figure 35.12. (a)

Turns ratio,

V1 240 1 N1 D D D N2 V2 1920 8

Figure 35.12 Equivalent input resistance of the transformer, 

rD

N1 N2

2

 2

1 8

RL D

1600 D 25 

Total input resistance, RIN D R1 C r D 5 C 25 D 30 . Hence the primary current, I1 D V1 /RIN D 240/30 D 8 A (b)

For an ideal transformer, 

from which, I2 D I1

V1 V2

I2 V1 D V2 I1



D 8



240 1920



D1A

Power dissipated in the load resistance, P D I22 RL D 12 1600 D 1.6 kW Problem 10. An ac. source of 306 0° V and internal resistance 20 k is matched to a load by a 20:1 ideal transformer. Determine for maximum power transfer (a) the value of the load resistance, and (b) the power dissipated in the load. The network diagram is shown in Figure 35.13. (a) Figure 35.13

For maximum power transfer, r1 must be equal to 20 k. From equation (35.3), r1 D N1 /N2 2 RL from which, 9

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load resistance RL D r1 (b)

N2 N1

2



D 20 000

1 20

2

D 50 Z

The total input resistance when the source is connected to the matching transformer is r C r1 , i.e., 20 k C 20 k D 40 k. Primary current, I1 D V/40 000 D 30/40 000 D 0.75 mA I2 N1 D from which, I2 D I1 N2 I1



N1 N2



D 0.75 ð 103 



20 1



D 15 mA Power dissipated in load resistance RL is given by P D I22 RL D 15 ð 103 2 50 D 0.01125 W or 11.25 mW Further problems on impedance matching may be found in Section 35.3 following, problems 11 to 15, page 627.

35.3 Further problems on maximum power transfer theorems and impedance matching

Maximum power transfer theorems 1

For the circuit shown in Figure 35.14 determine the value of the source resistance r if the maximum power is to he dissipated in the 15  load. Determine the value of this maximum power. [r D 9 , P D 208.4 W]

2

In the circuit shown in Figure 35.15 the load impedance ZL is a pure resistance R. Determine (a) the value of R for maximum power to be transferred from the source to the load, and (b) the value of the maximum power delivered to R. [(a) 11.18  (b) 151.1 W]

3

If the load impedance ZL in Figure 35.15 of problem 2 consists of a variable resistance R and variable reactance X, determine (a) the value of ZL which results in maximum power transfer, and (b) the value of the maximum power. [(a) 10 C j5 (b) 160 W]

4

For the network shown in Figure 35.16 determine (a) the value of the load resistance RL required for maximum power transfer, and (b) the value of the maximum power. [(a) 26.83  (b) 35.4 W]

5

Find the value of the load resistance RL shown in Figure 35.17 that gives maximum power dissipation, and calculate the value of this power. [RL D 2.1 , P D 23.3 W]

6

For the circuit shown in Figure 35.18 determine (a) the value of load resistance RL which results in maximum power transfer, and (b) the value of the maximum power. [(a) 16  (b) 48 W]

Figure 35.14

Figure 35.15

10

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Figure 35.16

Figure 35.17

Figure 35.18

7

Determine, for the network shown in Figure 35.19, (a) the values of R and X which result in maximum power being transferred across terminals AB, and (b) the value of the maximum power. [(a) R D 1.706 , X D 0.177  (b) 269 W]

8

A source of 1206 0° V and impedance 5 C j3 supplies a load consisting of a variable resistor R in series with a fixed capacitive reactance of 8 . Determine (a) the value of R to give maximum power transfer, and (b) the value of the maximum power. [(a) 7.07  (b) 596.5 W]

9

If the load ZL between terminals A and B of Figure 35.20 is variable in both resistance and reactance determine the value of ZL such that it will receive maximum power. Calculate the value of the maximum power. [R D 3.47 , X D 0.93 , 13.6 W]

10

For the circuit of Figure 35.21, determine the value of load impedance ZL for maximum load power if (a) ZL comprises a variable resistance R and variable reactance X, and (b) ZL is a pure resistance R. Determine the values of load power in each case [(a) R D 0.80 , X D 1.40 , P D 225 W (b) R D 1.61 , P D 149.2 W]

Figure 35.19

Figure 35.20 Impedance matching

Figure 35.21

11

The output stage of an amplifier has an output resistance of 144 . Determine the optimum turns ratio of a transformer that would match a load resistance of 9  to the output resistance of the amplifier for maximum power transfer. [4:1]

12

Find the optimum value of load resistance for maximum power transfer if a load is connected to an amplifier of output resistance 252  through a transformer with a turns ratio of 6:1 [7 ]

13

A generator has an output impedance of 300 C j45. Determine the turns ratio of an ideal transformer necessary to match the generator to a load of 37 C j19 for maximum power transfer. [2.70] 11

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14

A single-phase, 240 V/2880 V ideal transformer is supplied from a 240 V source through a cable of resistance 3.5 . If the load across the secondary winding is 1.8 k, determine (a) the primary current flowing, and (b) the power dissipated in the load resistance. [(a) 15 A (b) 2.81 kW]

15

An a.c. source of 206 0° V and internal resistance 10.24 k is matched to a load for maximum power transfer by a 16:1 ideal transformer. Determine (a) the value of the load resistance, and (b) the power dissipated in the load. [(a) 40  (b) 9.77 mW]

12