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35
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 Thévenin 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 Thévenin 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
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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
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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 ]
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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
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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
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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 Thévenin’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 Thévenin 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 Thévenin’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 Thévenin 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
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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
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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
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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,
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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
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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]
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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]
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