b Test your solution for the branch currents by showing that the total power dissipated equals the total power developed.. Figure P4.13 4.14 Use the node-voltage method to find the tota
Trang 1126 Techniques of Circuit Analysis
dv 2 = R 3 R 4 {R y R,I, 2 - [R 2 (R 3 + R 4 ) + R 3 R 4 ]I gl }
dR, " [(R l + R 2 )(R 3 + R 4 ) + R 3 R 4 ] 2 (4.98)
Figure 4.69 A Circuit used to introduce sensitivity analysis
We now consider an example with actual component values to illustrate the use of Eqs 4.97 and 4.98
EXAMPLE
Assume the nominal values of the components in the circuit in Fig 4.69 are:
R l = 25 H ; R 2 = 5 ft; R 3 = 50 ft; R 4 = 75 ft; I g] = 12 A and
I g2 = 16 A Use sensitivity analysis to predict the values of v x and v 2 if
the value of R^ is different by 10% from its nominal value
Solution
From Eqs 4.95 and 4.96 we find the nominal values of v v and v 2 Thus
25(3750(16) - [5(125) + 3750112}
and
30(125) + 3750
3750[30(16) - 25(12)]
30(125) + 3750 (4.100)
Now from Eqs 4.97 and 4.98 we can find the sensitivity of V\ and v 2 to
changes in R t Hence
dv { [3750 + 5(125)] - (3750(16) - [3750 + 5(125)]12}
dRi [(30)(125) + 3750]'
and
dv 2 _ 3750(3750(16) - [5(125) + 3750J12}]
Trang 2Ai?! = - 2 5 ft and Eq 4.101 predicts Av x will be
A vi = ( ^ V2-5) = -1-4583 V
Therefore, if R^ is 10% less than its nominal value, our analysis predicts
that v\ will be
Vi = 25 - 1.4583 = 23.5417 V (4.103)
Similarly for Eq 4.102 we have
Av 2 = 0.5(-2.5) = -1.25 V,
v 2 = 90 - 1.25 = 88.75 V (4.104)
We attempt to confirm the results in Eqs 4.103 and 4.104 by substituting
the value R^ = 22.5 ft into Eqs 4.95 and 4.96 When we do, the results are
vx = 23.4780 V, (4.105)
Why is there a difference between the values predicted from the sensitivity
analysis and the exact values computed by substituting for R^ in the
equa-tions for V\ and v 2l We can see from Eqs 4.97 and 4.98 that the sensitivity
of Vi and v 2 with respect to R^ is a function of R lt because R^ appears in
the denominator of both Eqs 4.97 and 4.98 This means that as Ri
changes, the sensitivities change and hence we cannot expect Eqs 4.97 and
4.98 to give exact results for large changes in /¾ Note that for a 10%
change in R u the percent error between the predicted and exact values of
vx and v 2 is small Specifically, the percent error in v { = 0.2713% and the
percent error in v 2 = 0.0676%
From this example, we can see that a tremendous amount of work is
involved if we are to determine the sensitivity of v^ and v 2 to changes in
the remaining component values, namely R 2f R 3t R 4f I gU and I g2
Fortunately, PSpice has a sensitivity function that will perform sensitivity
analysis for us The sensitivity function in PSpice calculates two types of
sensitivity The first is known as the one-unit sensitivity, and the second
is known as the 1% sensitivity In the example circuit, a one-unit change
in a resistor would change its value by 1 ft and a one-unit change in a
current source would change its value by 1 A In contrast, 1%
sensitiv-ity analysis determines the effect of changing resistors or sources by
1% of their nominal values
The result of PSpice sensitivity analysis of the circuit in Fig 4.69 is
shown in Table 4.2 Because we are analyzing a linear circuit, we can use
superposition to predict values of v\ and v 2 if more than one component's
value changes For example, let us assume /?, decreases to 24 ft and R 2
decreases to 4 ft From Table 4.2 we can combine the unit sensitivity of V\
to changes in R { and R 2 to get
Auj Av\
A#[ + 1R~ 2 = 0.5833 - 5.417 = -4.8337 V/ft,
Trang 3Techniques of Circuit Analysis
Similarly,
At?? Av 2
+
AS, ' AR 2 0.5+ 6.5 = 7.0 V / a
Thus if both R { and R 2 decreased by 1II we would predict
Vi = 25 + 4.8227 = 29.8337 V,
v 2 = 90 - 7 = 83 V
TABLE 4.2 PSpice Sensitivity Analysis Results
Element Element Name Value
Element! Sensitivity (Volts/Unit)
(a) DC Sensitivities of Node Voltage VI
Rl 25 R2 5 R3 50 R4 75
IG1 12
IG2 16
(b) Sensitivities of Output V2
Rl 25
R2 5
R3 50
R4 75
IG1 12 IG2 16
0.5833 -5.417 0.45 0.2 -14.58 12.5
0.5 6.5 0.54 0.24 -12.5
15
Normalized Sensitivity (Volts/Percent)
0.1458 -0.2708 0.225 0.15 -1.75
2
0.125 0.325 0.27 0.18 -1.5 2.4
If we substitute R { = 24 fl and R 2 = 4 ft into Eqs 4.95 and 4.96 we get
vi = 29.793 V,
v 2 = 82.759 V
In both cases our predictions are within a fraction of a volt of the actual node voltage values
Circuit designers use the results of sensitivity analysis to determine which component value variation has the greatest impact on the output of the circuit As we can see from the PSpice sensitivity analysis in Table 4.2,
the node voltages Dj and v 2 are much more sensitive to changes in R 2 than
to changes in R u Specifically, V\ is (5.417/0.5833) or approximately
9 times more sensitive to changes in R 2 than to changes in R x and v 2 is
(6.5/0.5) or 13 times more sensitive to changes in R 2 than to changes in
Ri, Hence in the example circuit, the tolerance on R 2 must be more
strin-gent than the tolerance on R^ if i t is important to keep V\ and v 2 close to their nominal values
NOTE: Assess your understanding of this Practical Perspective by trying Chapter Problems 4.105-4.107
Trang 4Summary
• For the topics in this chapter, mastery of some basic terms,
and the concepts they represent, is necessary Those terms
are node, essential node, path, branch, essential branch,
mesh, and planar circuit Table 4.1 provides definitions
and examples of these terms (See page 91.)
• Two new circuit analysis techniques were introduced in
this chapter:
• The node-voltage method works with both planar
and nonplanar circuits A reference node is chosen
from among the essential nodes Voltage variables
are assigned at the remaining essential nodes, and
Kirchhoff s current law is used to write one equation
per voltage variable The number of equations is
n e — 1, where n e is the number of essential nodes
(See page 93.)
• The mesh-current method works only with planar
circuits Mesh currents are assigned to each mesh,
and Kirchhoff's voltage law is used to write one
equation per mesh The number of equations is
b — (n — 1), where b is the number of branches in
which the current is unknown, and n is the number of
nodes The mesh currents are used to find the branch
currents (See page 99.)
• Several new circuit simplification techniques were
introduced in this chapter:
• Source transformations allow us to exchange a
volt-age source (v s) and a series resistor (R) for a current
source (i s) and a parallel resistor (R) and vice versa
The combinations must be equivalent in terms of
their terminal voltage and current Terminal
equiva-lence holds provided that
(See page 109.)
• Thevenin equivalents and Norton equivalents allow
us to simplify a circuit comprised of sources and resis-tors into an equivalent circuit consisting of a voltage source and a series resistor (Thevenin) or a current source and a parallel resistor (Norton) The simplified circuit and the original circuit must be equivalent in terms of their terminal voltage and current Thus keep in mind that (1) the Thevenin voltage (Kiii) is the open-circuit voltage across the terminals of the original circuit, (2) the Thevenin resistance (i?Th) is the ratio of the Thevenin voltage to the short-circuit current across the terminals of the original circuit; and (3) the Norton equivalent is obtained by per-forming a source transformation on a Thevenin equivalent (See page 113.)
• Maximum power transfer is a technique for calculating
the maximum value of p that can be delivered to a load,
RL Maximum power transfer occurs when Ri = Rjh,
the Thevenin resistance as seen from the resistor R L
The equation for the maximum power transferred is
(See page 120.)
• In a circuit with multiple independent sources,
superposition allows us to activate one source at a time
and sum the resulting voltages and currents to deter-mine the voltages and currents that exist when all inde-pendent sources are active Deinde-pendent sources are never deactivated when applying superposition (See page 122.)
Trang 5130 Techniques of Circuit Analysis
Problems
Section 4.1
4.1 For the circuit shown in Fig P4.1, state the numerical
value of the number of (a) branches, (b) branches
where the current is unknown, (c) essential branches,
(d) essential branches where the current is unknown,
(e) nodes, (f) essential nodes, and (g) meshes
Figure P4.1
4.4 Assume the current i g in the circuit in Fig P4.4 is
known The resistors R^ - R 5 are also known a) How many unknown currents are there?
b) How many independent equations can be writ-ten using Kirchhoff s current law (KCL)?
c) Write an independent set of KCL equations d) How many independent equations can be derived from Kirchhoff s voltage law (KVL)? e) Write a set of independent KVL equations
Figure P4.4
V L
4.2 a) If only the essential nodes and branches are
identified in the circuit in Fig P4.1, how many
simultaneous equations are needed to describe
the circuit?
b) How many of these equations can be derived
using Kirchhoff s current law?
c) How many must be derived using Kirchhoffs
voltage law?
d) What two meshes should be avoided in applying
the voltage law?
4.3 a) How many separate parts does the circuit in
Fig P4.3 have?
b) How many nodes?
c) How many branches are there?
d) Assume that the lower node in each part of the
circuit is joined by a single conductor Repeat
the calculations in (a)-(c)
Figure P4.3
4.5 A current leaving a node is defined as positive a) Sum the currents at each node in the circuit shown in Fig P4.4
b) Show that any one of the equations in (a) can be derived from the remaining three equations
Section 4.2
4.6 Use the node-voltage method to find Uj and v 2 in PSPICE the circuit in Fig P4.6
Figure P4.6
144 V
son
4.7 Use the node-voltage method to find how much
PSPICE power the 2 A source extracts from the circuit in
HULTISIM p j g p 4 J
Figure P4.7
Trang 64.8 Use the node-voltage method to find v x and v 2 in
PSPICE the circuit shown in Fig P4.8
MULTISIM
Figure P4.8
8 f t
-VvV-<>i|40ft kmVL u 2f 120ft C\ J 1 A
4.9 Use the node-voltage method to find v () in the
cir-PSPICE cuit in Fig P4.9
Figure P4.9
2011
24 V
8012
40 mA
4.10 a) Find the power developed by the 40 mA current
PSPKE source in the circuit in Fig P4.9
b) Find the power developed by the 24 V voltage
source in the circuit in Fig P4.9
c) Verify that the total power developed equals the
total power dissipated
4.11 A 50 O resistor is connected in series with the
PSPICE 40 mA current source in the circuit in Fig P4.9
MULTISIM N ^ ,
a) Find v a
b) Find the power developed by the 40 mA current
source
c) Find the power developed by the 24 V voltage
source
d) Verify that the total power developed equals the
total power dissipated
e) What effect will any finite resistance connected
in series with the 40 mA current source have on
the value of v a l
4.12 The circuit shown in Fig P4.12 is a dc model of a
PSPICE residential power distribution circuit
a) Use the node-voltage method to find the branch
currents i { — /6
b) Test your solution for the branch currents by showing that the total power dissipated equals the total power developed
Figure P4.12
125 V
125 V
':,
4.13 a) Use the node-voltage method to find the
PSPKE branch currents /a - /e in the circuit shown in
MULTISIM F i g p 4 > 1 3
b) Find the total power developed in the circuit
Figure P4.13
4.14 Use the node-voltage method to find the total power
PSPICE dissipated in the circuit in Fig P4.14
MULTISIM
Figure P4.14
40 V
Trang 7132 Techniques of Circuit Analysis
4.15 a) Use the node-voltage method to find v h v 2 , and
! ™ v 3 in the circuit in Fig P4.15
MULTISIM
b) How much power does the 40 V voltage source
deliver to the circuit?
Figure P4.15
40 V
4.16 a) Use the node-voltage method to show that the
PSPICE output voltage v 0 in the circuit in Fig P4.16 is
equal to the average value of the source voltages
b) Find v 0 if v { = 100 V, v 2 = 80 V, and
v 3 = - 6 0 V
Figure P4.16
4.19 Use the node-voltage method to calculate the
PSPICE power delivered by the dependent voltage source in
WLTISIM , ^ „ , -. n
the circuit in Fig P4.19
Figure P4.19
160 V
('„ I loo a 150 L
20 a
^vw-Section 4.3
4.17 a) Use the node-voltage method to find the total
power developed in the circuit in Fig P4.17
MULTISIM
b) Check your answer by finding the total power
absorbed in the circuit
Figure P4.17
84/A
PSPICE
MULTISIM
4.18 a) Use the node-voltage method to find v„ in the
circuit in Fig P4.18
b) Find the power absorbed by the dependent source
c) Find the total power developed by the
independ-ent sources
Figure P4.18
20 a
4.20 a) Find the node voltages V\, «2, and v$ in the
cir-PSPICE c ui t in Fig P4.20
•IULTISIM
b) Find the total power dissipated in the circuit
Figure P4.20
5 n 10 a
m V A f
VvV-f \ 5 /(, iv*20ft y2?40O y y k (^/) 96 V
Section 4.4
4.21 Use the node-voltage method to find i 0 in the
cir-PSPICE cuit in Fig P4.21
MULTISIM
Figure P4.21
20 V 6
30kll lkft
PSPICE MULTISIM
4.22 a) Use the node-voltage method to find the
branch currents ij, i 2, and /3 in the circuit in
Fig P4.22
b) Check your solution for ij, i 2, and i3 by showing that the power dissipated in the circuit equals the power developed
Figure P4.22
Trang 84.23 a) Use the node-voltage method to find the power
dissipated in the 2 (2 resistor in the circuit in
Fig P4.23
b) Find the power supplied by the 230 V source
Figure P4.23
230 vC-y
i n
i n
i n :
la-i n
l n
i n
s n 2n :5 n
4.24 Use the node-voltage method to find the value of v 0
KM« in the circuit in Fig P4.24
MUITISIM
Figure P4.26
25 V
55 n
4.27 Use the node-voltage method to find v 0 in the cir-PSPICE cuit in Fig P4.27
MULTISIM
Figure P4.27
15 V
Figure P4.24
50 V
4.25 Use the node-voltage method to find the value of v a
PSPICE in the circuit in Fig P4.25
«!ULTISIM
Figure P4.25
4.26 a) Use the node-voltage method to find v 0 and
in the circuit in Fig P4.26 Use node a as the
reference node
b) Repeat part (a), but use node b as the
refer-ence node
c) Compare the choice of reference node in (a)
and (b) Which is better, and why?
4.28 Use the node-voltage method to find the power
devel-oped by the 20 V source in the circuit in Fig P4.28
PSPICE MULTISIM
Figure P4.28
8 0 n (1)3.125 v A
4.29 Assume you are a project engineer and one of your PSPICE
MULTISIM staff is assigned to analyze the circuit shown in Fig P4.29 The reference node and node numbers given on the figure were assigned by the analyst
Her solution gives the values of v 3 and v 4 as 108 V and 81.6 V, respectively
Test these values by checking the total power developed in the circuit against the total power dis-sipated Do you agree with the solution submitted
by the analyst?
Figure P4.29
Trang 9134 Techniques of Circuit Analysis
4.30 Show that when Eqs 4.16,4.17, and 4.19 are solved
for i B, the result is identical to Eq 2.25
Section 4.5
4.31 Solve Problem 4.12 using the mesh-current method
4.32 Solve Problem 4.13 using the mesh-current method
4.33 a) Use the mesh-current method to find the branch
currents L, //„ and L in the circuit in Fig P4.33
b) Repeat (a) if the polarity of the 60 V source is
reversed
Figure P4.33
4.34 a) Use the mesh-current method to find the total
power developed in the circuit in Fig P4.34
MULTISIM
b) Check your answer by showing that the total
power developed equals the total power
dissipated
Figure P4.34
460 V
4.35 Solve Problem 4.21 using the mesh-current method
4.36 Solve Problem 4.23 using the mesh-current method
Section 4.6
4.37 a) Use the mesh-current method to find v 0 in the
PSPICE circuit in Fig P4.37
MULTISIM
b) Find the power delivered by the dependent source
Figure P4.37
io a
7 a
4.38 Use the mesh-current method to find the power
dissi-PSPICE p a t e (j in the 20 fl resistor in the circuit in Fig P4.38
MULTISIM
Figure P4.38
5 0
3 a
• A W
135 V ©
4 f t
2 a l a
-"VW
4.39 Use the mesh-current method to find the power delivered by the dependent voltage source in the
circuit seen in Fig P4.39
Figure P4.39
660 V
25 a
20 /,,
4.40 Use the mesh-current method to find the power
>SPICE developed in the dependent voltage source in the
JLTISIM _ _ _
circuit in Fig P4.40
Figure P4.40
30 V
Section 4.7 4.41 Solve Problem 4.8 using the mesh-current method 4.42 a) Use the mesh-currcnt method to find how much
power the 4 A current source delivers to the cir-cuit in Fig P4.42
b) Find the total power delivered to the circuit c) Check your calculations by showing that the total power developed in the circuit equals the total power dissipated
Trang 10Figure P4.42
120 V
5 0
80 V
4.43 Solve Problem 4.20 using the mesh-current method
4.44 a) Use the mesh-current method to solve for i± in
PSPICE the circuit in Fig P4.44
MULTISIM
b) Find the power delivered by the independent
current source
c) Find the power delivered by the dependent
volt-age source
4.47 Solve Problem 4.22 using the mesh-current method 4.48 Use the mesh-current method to find the total
PSPICE power dissipated in the circuit in Fig P4.48
MULTISIM
Figure P4.48
4.49 a) Assume the 20 V source in the circuit in
Fig P4.48 is changed to 60 V Find the total power dissipated in the circuit
b) Repeat (a) if the 6 A current source is replaced
by a short circuit
c) Explain why the answers to (a) and (b) are the same
Figure P4.44
i k n
150 k
4.45 Use the mesh-current method to find the total power
PSPICE developed in the circuit in Fig P4.45
MULTISIM
Figure P4.45
4.46 a) Use the mesh-current method to determine
PSPICE which sources in the circuit in Fig P4.46 are
gen-IULTISIM crating power
b) Find the total power dissipated in the circuit
Figure P4.46
2 f t
4.50 a) Find the branch currents /a - i c for the circuit
nna shown in Fig P4.50
MULTISIM
b) Check your answers by showing that the total power generated equals the total power dissipated
Figure P4.50
4/d - ic
19 A
4.51 a) Use the mesh-current method to find the branch
PSPICE MULTISIM currents in i a — ie in the circuit in Fig P4.51 b) Check your solution by showing that the total power developed in the circuit equals the total power dissipated
Figure P4.51
100ft