Bird J. Electrical Circuit Theory and Technology
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778 Electrical Circuit Theory and Technology
R
OC
D
20004000
2000 C 4000
D 1333
and the impedance at port 2 with port 1 short-circuited,
R
SC
D
20003000
2000 C 3000
D 1200
Hence R
OB
D
p
[13331200] D 1265 Z
Thus the image impedances are 790 Z and 1265 Z.
Figure 41.29
(b) The iterative impedance at port 1 in Figure 41.29 is shown as R
1
.
From circuit theory,
R
1
D
1000[3000 C2000R
1
/2000 C R
1
]
1000 C 3000 C 2000R
1
/2000 C R
1
i.e., R
1
D
3 ð 10
6
C 2 ð10
6
R
1
/2000 C R
1
4000 C 2000R
1
/2000 C R
1
4000R
1
C
2000R
2
1
2000 C R
1
D 3 ð 10
6
C
2 ð 10
6
R
1
2000 C R
1
8 ð 10
6
R
1
C 4000R
2
1
C 2000R
2
1
D 6 ð 10
9
C 3 ð 10
6
R
1
C 2 ð 10
6
R
1
6000R
2
1
C 3 ð 10
6
R
1
6 ð 10
9
D 0
2R
2
1
C 1000R
1
2 ð 10
6
D 0
Using the quadratic formula gives
R
1
D
1000 š
[1000
2
422 ð 10
6
]
4
D
1000 š 4123
4
D 781 Z
(neglecting the negative value).
Figure 41.30
The iterative impedance at port 2 in Figure 41.30 is shown as R
2
.
R
2
D
2000[3000 C1000R
2
/1000 C R
2
]
2000 C 3000 C 1000R
2
/1000 C R
2
D
6 ð 10
6
C 2 ð10
6
R
2
/1000 C R
2
5000 C 1000R
2
/1000 C R
2
Hence
5000R
2
C
1000R
2
2
1000 C R
2
D 6 ð 10
6
C
2 ð 10
6
R
2
1000 C R
2
5 ð 10
6
R
2
C 5000R
2
2
C 1000R
2
2
D 6 ð 10
9
C 6 ð 10
6
R
2
C 2 ð 10
6
R
2
Attenuators 779
6000R
2
2
3 ð 10
6
R
2
6 ð 10
9
D 0
2R
2
2
1000R
2
2 ð 10
6
D 0
from which
R
2
D
1000 š
[1000
2
422 ð 10
6
]
4
D
1000 š 4123
4
D 1281 Z
Thus the iterative impedances of the section shown in
Figure 41.28 are 781 Z and 1281 Z.
Further problems on asymmetrical T—and -sections may be found in
Section 41.9, problems 19 to 21, page 787.
41.7 The L-section
attenuator
A typical L-section attenuator pad is shown in Figure 41.31. Such a pad
is used for matching purposes only, the design being such that the atten-
uation introduced is a minimum. In order to derive values for R
1
and R
2
,
consider the resistances seen from either end of the section.
Looking in at port 1,
R
OA
D R
1
C
R
2
R
OB
R
2
C R
OB
from which
R
OA
R
2
C R
OA
R
OB
D R
1
R
2
C R
1
R
OB
C R
2
R
OB
41.22
Figure 41.31 L-section
attenuator pad
Looking in at port 2,
R
OB
D
R
2
R
1
C R
OA
R
1
C R
OA
C R
2
from which
R
OB
R
1
C R
OA
R
OB
C R
OB
R
2
D R
1
R
2
C R
2
R
OA
41.23
Adding equations (41.22) and (41.23) gives
R
OA
R
2
C 2R
OA
R
OB
C R
OB
R
1
C R
OB
R
2
D 2R
1
R
2
C R
1
R
OB
C R
2
R
OB
C R
2
R
OA
i.e., 2R
OA
R
OB
D 2R
1
R
2
and R
1
D
R
OA
R
OB
R
2
41.24
780 Electrical Circuit Theory and Technology
Substituting this expression for R
1
into equation (41.22) gives
R
OA
R
2
C R
OA
R
OB
D
R
OA
R
OB
R
2
R
2
C
R
OA
R
OB
R
2
R
OB
C R
2
R
OB
i.e., R
OA
R
2
C R
OA
R
OB
D R
OA
R
OB
C
R
OA
R
2
OB
R
2
C R
2
R
OB
from which R
2
R
OA
R
OB
D
R
OA
R
2
OB
R
2
R
2
2
R
OA
R
OB
D R
OA
R
2
OB
and resistance,
R
2
=
R
OA
R
2
OB
R
OA
− R
OB
41.25
Thus, from equation (41.24),
R
1
D
R
OA
R
OB
R
OA
R
2
OB
/R
OA
R
OB
D
R
OA
R
OB
R
OB
p
R
OA
/R
OA
R
OB
D
R
OA
p
R
OA
p
R
OA
R
OB
Hence resistance,
R
1
=
p
[R
OA
.R
OA
− R
OB
/]
41.26
Figure 41.32 shows an L-section attenuator pad with its resistances
expressed in terms of the input and output resistances, R
OA
and R
OB
.
Figure 41.32
Problem 15. A generator having an internal resistance of 500
is connected to a 100 load via an impedance-matching resis-
tance pad as shown in Figure 41.33. Determine (a) the values of
resistance R
1
and R
2
, (b) the attenuation of the pad in decibels, and
(c) its insertion loss.
Attenuators 781
Figure 41.33
(a) From equation (41.26), R
1
D
p
[500500 100] D 447.2 Z
From equation (41.25), R
2
D
500100
2
500 100
D 111.8 Z
(b) From section 41.3, the attenuation is given by 10lgP
1
/P
2
dB. Note
that, for an asymmetrical section such as that shown in Figure 41.33,
the expression 20 lgV
1
/V
2
or 20 lgI
1
/I
2
may not be used for
attenuation since the terminals of the pad are not matched to equal
impedances. In Figure 41.34,
current I
1
D
E
500 C 447.2 C 111.8 ð100/111.8 C 100
D
E
1000
Figure 41.34
and current
I
2
D
111.8
111.8 C 100
I
1
D
111.8
211.8
E
1000
D
E
1894.5
Thus input power,
P
1
D I
2
1
500 D
E
1000
2
500
and output power,
P
2
D I
2
2
100 D
E
1894.5
2
100
Hence
attenuation D 10 lg
P
1
P
2
D 10lg
[E/1000]
2
500
[E/1894.5]
2
100
D 10lg
1894.5
1000
2
5
dB
i.e., attenuation = 12.54 dB
(c) Insertion loss A
L
is defined as
voltage across load when connected directly to the generator
voltage across load when the two-port network is connected
Figure 41.35 shows the generator connected directly to the load.
Load current, I
L
D
E
500 C 100
D
E
600
and voltage, V
L
D I
L
100 D
E
600
100 D
E
6
Figure 41.35
782 Electrical Circuit Theory and Technology
From Figure 41.34 voltage, V
1
D E I
1
500 D E E/1000500 from
part (b)
i.e., V
1
D 0.5E
voltage, V
2
D V
1
I
1
R
1
D 0.5E
E
1000
447.2 D 0.0528 E
insertion loss, A
L
D
V
L
V
2
D
E/6
0.0528E
D 3.157
In decibels, the insertion loss D 20 lg
V
L
V
2
D 20lg3.157 D 9.99 dB
Further problems on L-section attenuators may be found in Section 41.9,
problems 22 and 23, page 787.
41.8 Two-port networks
in cascade
Often two-port networks are connected in cascade, i.e., the output from
the first network becomes the input to the second network, and so on,
as shown in Figure 41.36. Thus an attenuator may consist of several
cascaded sections so as to achieve a particular desired overall performance.
Figure 41.36 Two-port networks connected in cascade
If the cascade is arranged so that the impedance measured at one port
and the impedance with which the other port is terminated have the same
value, then each section (assuming they are symmetrical) will have the
same characteristic impedance Z
0
and the last network will be terminated
in Z
0
. Thus each network will have a matched termination and hence
the attenuation in decibels of section 1 in Figure 41.36 is given by a
1
D
20lgV
1
/V
2
. Similarly, the attenuation of section 2 is given by a
2
D
20lgV
2
/V
3
, and so on.
The overall attenuation is given by
a D 20
V
1
V
n
D 20lg
V
1
V
2
ð
V
2
V
3
ð
V
3
V
4
ðÐÐÐð
V
n1
V
n
D 20lg
V
1
V
2
C 20lg
V
2
V
3
CÐÐÐC20lg
V
n1
V
n
Attenuators 783
by the laws of logarithms, i.e.,
overall attenuation, a
= a
1
Y a
2
Y ···Y a
n−1
41.27
Thus the overall attenuation is the sum of the attenuations (in decibels)
of the matched sections.
Problem 16. Five identical attenuator sections are connected in
cascade. The overall attenuation is 70 dB and the voltage input to
the first section is 20 mV. Determine (a) the attenuation of each
individual attenuation section, (b) the voltage output of the final
stage, and (c) the voltage output of the third stage.
(a) From equation (41.27), the overall attenuation is equal to the sum
of the attenuations of the individual sections and, since in this case
each section is identical, the attenuation of each section D 70/5 D
14 dB.
(b) If V
1
D the input voltage to the first stage and V
0
D the output of
the final stage, then the overall attenuation D 20lgV
1
/V
0
, i.e.,
70 D 20lg
20
V
0
where V
0
is in millivolts
3.5 D lg
20
V
0
10
3.5
D
20
V
0
from which
output voltage of final stage, V
0
D
20
10
3.5
D 6.32 ð 10
3
mV
D 6.32 mV
(c) The overall attenuation of three identical stages is 3 ð 14 D 42 dB.
Hence 42 D 20lgV
1
/V
3
, where V
3
is the voltage output of the
third stage. Thus
42
20
D lg
20
V
3
, 10
42/20
D
20
V
3
from which the voltage output of the third stage, V
3
D 20/10
2.1
D
0.159 mV
784 Electrical Circuit Theory and Technology
Problem 17. A d.c. generator has an internal resistance of 450
and supplies a 450 load.
(a) Design a T-network attenuator pad having a characteristic
impedance of 450 which, when connected between the
generator and the load, will reduce the load current to
1
8
of its
initial value.
(b) If two such networks as designed in (a) were connected in
series between the generator and the load, determine the frac-
tion of the initial current that would now flow in the load.
(c) Determine the attenuation in decibels given by four such
sections as designed in (a).
The T-network attenuator is shown in Figure 41.37 connected between
the generator and the load. Since it is matching equal impedances, the
network is symmetrical.
Figure 41.37
(a) Since the load current is to be reduced to
1
8
of its initial value, the
attenuation N D 8. From equation (41.13),
resistance, R
1
D
R
0
N 1
N C 1
D 450
8 1
8 C 1
D 350 Z
and from equation (41.14),
resistance, R
2
D R
0
2N
N
2
1
D 450
2 ð 8
8
2
1
D 114 Z
(b) When two such networks are connected in series, as shown in
Figure 41.38, current I
1
flows into the first stage and
1
8
I
1
flows
out of the first stage into the second.
Again,
1
8
of this current flows out of the second stage, i.e.,
1
8
ð
1
8
I
1
, i.e.,
1
64
of I
1
flows into the load.
Thus
1
64
of the original current flows in the load.
Attenuators 785
Figure 41.38
Figure 41.39
(c) The attenuation of a single stage is 8. Expressed in decibels,
the attenuation is 20 lgI
1
/I
2
D 20lg8 D 18.06 dB. From equa-
tion (41.27), the overall attenuation of four identical stages is given
by 18.06 C 18.06 C 18.06 C 18.06, i.e., 72.24 dB.
Further problems on cascading two-port networks may be found in
Section 41.9 following, problems 24 to 26, page 787.
41.9 Further problems
on attenuators
Logarithmic ratios
1 The ratio of two powers is (a) 3, (b) 10, (c) 30, (d) 10000. Determine
the decibel power ratio for each.
[(a) 4.77 dB (b) 10 dB (c) 14.8 dB (d) 40 dB]
2 The ratio of two powers is (a)
1
10
,(b)
1
2
, (c)
1
40
,(d)
1
1000
. Determine
the decibel power ratio for each.
[(a) 10 dB (b) 3 dB (c) 16 dB (d) 30 dB]
3 An amplifier has (a) a gain of 25 dB, (b) an attenuation of 25 dB. If
the input power is 12 mW, determine the output power in each case.
[(a) 3795 mW (b) 37.9
µW]
4 7.5% of the power supplied to a cable appears at the output terminals.
Determine the attenuation in decibels. [11.25 dB]
5 The current input of a system is 250 mA. If the current ratio of the
system is (i) 15 dB, (ii) 8 dB, determine (a) the current output and
(b) the current ratio expressed in nepers.
[(i) (a) 1.406 A (b) 1.727 Np
(ii) (a) 99.53 mA (b) 0.921 Np]
Symmetrical T—and p-attenuators
6 Determine the characteristic impedances of the T-network attenuator
sections shown in Figure 41.39.
[(a) 26.46 (b) 244.9 (c) 1.342 k]
786 Electrical Circuit Theory and Technology
Figure 41.40
7 Determine the characteristic impedances of the -network attenuator
pads shown in Figure 41.40. [(a) 7.45 (b) 353.6 (c) 189.7 ]
8 A T-section attenuator is to provide 18 dB voltage attenuation per
section and is to match a 1.5 k line. Determine the resistance values
necessary per section. [R
1
D 1165 , R
2
D 384 ]
9A-section attenuator has a series resistance of 500 and shunt
resistances of 2 k. Determine (a) the characteristic impedance, and
(b) the attenuation produced by the network. [(a) 667 (b) 6 dB]
10 For each of the attenuator pads shown in Figure 41.41 determine
(a) the input resistance when the output port is open-circuited, (b) the
input resistance when the output port is short-circuited, and (c) the
characteristic impedance.
[(i) (a) 50 (b) 42 (c) 45.83
(ii) (a) 285.7 (b) 240 (c) 261.9 ]
11 A television signal received from an aerial through a length of coaxial
cable of characteristic impedance 100 has to be attenuated by
15 dB before entering the receiver. If the input impedance of the
receiver is also 100 , design a suitable T-attenuator network to
give the necessary reduction. [R
1
D 69.8 , R
2
D 36.7 ]
12 Design (a) a T-section symmetrical attenuator pad, and (b) a -
section symmetrical attenuator pad, to provide a voltage attenuation
of 15 dB and having a characteristic impedance of 500 .
[(a) R
1
D 349 , R
2
D 184
(b) R
1
D 1.36 k, R
2
D 716 ]
13 Determine the values of the shunt and series resistances for
T-pad attenuators of characteristic impedance 400 to provide the
following voltage attenuations: (a) 12 dB (b) 25 dB (c) 36 dB
[(a) R
1
D 239.4 , R
2
D 214.5
(b) R
1
D 357.4 , R
2
D 45.13
(c) R
1
D 387.5 , R
2
D 12.68 ]
14 Design a -section symmetrical attenuator network to provide a
voltage attenuation of 24 dB and having a characteristic impedance
of 600 .[R
1
D 4.736 k, R
2
D 680.8 ]
15 A d.c. generator has an internal resistance of 600 and supplies a
600 load. Design a symmetrical (a) T-network and (b) -network
attenuator pad, having a characteristic impedance of 600 which
when connected between the generator and load will reduce the load
current to
1
4
its initial value.
[(a) R
1
D 360 , R
2
D 320
(b) R
1
D 1125 , R
2
D 1000 ]
Insertion loss
16 The attenuator section shown in Figure 41.42 feeds a matched load.
Determine (a) the characteristic impedance R
0
and (b) the insertion
loss. [(a) 282.8 (b) 15.31 dB]
Figure 41.41
Attenuators 787
Figure 41.42
17 A 0–10 k variable resistor is connected across the output of
a generator of internal resistance 500 . If a load of 1500 is
connected across the variable resistor, determine the insertion loss
in decibels at a tapping of (a) 7.5 k, (b) 2.5 k
[(a) 8.13 dB (b) 17.09 dB]
18 A symmetrical attenuator pad has a series arm resistance of 800
and shunt arms each of 250 . Determine (a) the characteristic
impedance of the section, and (b) the insertion loss when feeding
a matched load. [(a) 196.1 (b) 18.36 dB]
Asymmetric T—and p-attenuators
19 An asymmetric section is shown in Figure 41.43. Determine for the
section (a) the image impedances, and (b) the iterative impedances.
[(a) 144.9 , 241.5 (b) 143.6 , 243.6 ]
Figure 41.43
20 An asymmetric -section is shown in Figure 41.44. Determine
for the section (a) the image impedances, and (b) the iterative
impedances.
[(a) 329.5 , 285.6 (b) 331.2 , 284.2 ]
Figure 41.44
21 Distinguish between image and iterative impedances of a network.
An asymmetric T-attenuator section has series arms of resistance
200 and 400 respectively, and a shunt arm of resistance 300 .
Determine the image and iterative impedances of the section.
[(a) 430.9 , 603.3 ; 419.6 , 619.6 ]
L-section attenuators
22 Figure 41.45 shows an L-section attenuator. The resistance across
the input terminals is 250 and the resistance across the output
terminals is 100 . Determine the values R
1
and R
2
.
[R
1
D 193.6 , R
2
D 129.1 ]
Figure 41.45
23 A generator having an internal resistance of 600 is connected to
a 200 load via an impedance-matching resistive pad as shown
in Figure 41.46. Determine (a) the values of resistances R
1
and R
2
,
(b) the attenuation of the matching pad, and (c) its insertion loss.
[(a) R
1
D 489.9 , R
2
D 249.9
(b) 9.96 dB (c) 8.71 dB]
Cascading two-port networks
24 The input to an attenuator is 24 V and the output is 4 V. Deter-
mine the attenuation in decibels. If five such identical attenuators
are cascaded, determine the overall attenuation.
[(a) 15.56 dB, 77.80 dB]
25 Four identical attenuator sections are connected in cascade. The
overall attenuation is 60 dB. The input to the first section is 50 mV.
Figure 41.46