Voltage division is a circuit analysis tool that is used to find the voltage drop across a single resistance from a collection of series-connected resistances when the volt-age drop acr
Trang 1Summary
• Series resistors can be combined to obtain a single
equivalent resistance according to the equation
#eq = 2 * * = *1 + R 2 + • ' + **'
/ = 1
(See page 58.)
Parallel resistors can be combined to obtain a single
equivalent resistance according to the equation
— = 2 — = — + — + ••• +—•
^eq (=1 Ri Rl Rl Rk
When just two resistors are in parallel, the equation for
equivalent resistance can be simplified to give
Rp-n —
R[Rj
eq /?! + R 2
(See pages 59-60.)
• When voltage is divided between series resistors, as
shown in the figure, the voltage across each resistor can
be found according to the equations
v 2 =
(See page 61.)
Ri
Ri
Ri + R 2 s ' <
)
+
+
v 2 :
Ui
\Ri
When current is divided between parallel resistors, as
shown in the figure, the current through each resistor
can be found according to the equations
R-,
'2
Ri + R 2 V
Ri + Ri
(See page 63.)
Voltage division is a circuit analysis tool that is used to
find the voltage drop across a single resistance from a
collection of series-connected resistances when the volt-age drop across the collection is known:
Ri
R eq where Vj is the voltage drop across the resistance Rj
and v is the voltage drop across the series-connected
resistances whose equivalent resistance is i?eq (See page 65.)
Current division is a circuit analysis tool that is used to
find the current through a single resistance from a col-lection of parallel-connected resistances when the cur-rent into the collection is known:
Rcq
where /,- is the current through the resistance Rj and i is
the current into the parallel-connected resistances
whose equivalent resistance is R cq (See page 65.)
A voltmeter measures voltage and must be placed in
par-allel with the voltage being measured An ideal voltmeter has infinite internal resistance and thus does not alter the voltage being measured (See page 66.)
An ammeter measures current and must be placed in
series with the current being measured An ideal amme-ter has zero inamme-ternal resistance and thus does not alamme-ter the current being measured (See page 66.)
Digital meters and analog meters have internal
resist-ance, which influences the value of the circuit variable being measured Meters based on the d'Arsonval meter movement deliberately include internal resistance as a way to limit the current in the movement's coil (See page 67.)
The Wheatstone bridge circuit is used to make precise
measurements of a resistor's value using four resistors, a dc voltage source, and a galvanometer A Wheatstone bridge
is balanced when the resistors obey Eq 3.33, resulting in
a galvanometer reading of 0 A (See page 69.)
A circuit with three resistors connected in a A
configu-ration (or a IT configuconfigu-ration) can be transformed into an
equivalent circuit in which the three resistors are Y con-nected (or T concon-nected) The A-to-Y transformation is given by Eqs 3.44-3.46; the Y-to-A transformation is given by Eqs 3.47-3.49 (See page 72.)
Trang 2Problems
Sections 3.1-3.2
3.1 For each of the circuits shown,
a) identify the resistors connected in series,
b) simplify the circuit by replacing the
series-connected resistors with equivalent resistors
3.2 For each of the circuits shown in Fig P3.2,
a) identify the resistors connected in parallel,
b) simplify the circuit by replacing the
parallel-connected resistors with equivalent resistors
3.3 Find the equivalent resistance seen by the source in
each of the circuits of Problem 3.1
3.4 Find the equivalent resistance seen by the source in each of the circuits of Problem 3.2
3.5 Find the equivalent resistance R a ^ for each of the
PSPICE circuits in Fig P3.5
MULTISIM
3.6 Find the equivalent resistance #at, for each of the
PSPICE c i r c u i t s in Fig P3.6
MULTISIM °
Figure P3.1
10V
6 0
>vw- 120
^ W v <
4 a:
(a)
9 0
7 0 : 3mA( f
200 mV
300 O
W V
500 O
Figure P3.2
60 V 1000¾ 25 0.¾ 22 O
(a)
©
2kO
50 mA t 10 kO
6kO
- ^ W v — i
9kO% 18 kO:
(b)
250 O
/ VW—r
(c)
Figure P3.5
10 O
:5 O f 2 0 O
6 0
b»—"vW-(a)
30 kO i 60 kO 1200 kO \ 50 kO
(b)
Figure P3.6
15 0
25 0
12 0
24 0
->vw-(a)
12()0 | 6 0 O | 2 0 O
7 0
- V W 5 0
b • 'vw
(b)
50 0
40 O
140
24 0
(c)
Trang 33.7 a) In the circuits in Fig P3.7(a)-(c), find the
equiv-alent resistance /?., h
MULTISIM u
b) For each circuit find the p o w e r delivered by the
source
3.8 a) Find t h e p o w e r dissipated in each resistor in the
circuit shown in Fie 3.9
MULTISIM °
b) Find the p o w e r delivered by the 120 V source
c) Show that the p o w e r delivered equals the p o w e r
dissipated
3.9 a) Show that the solution of the circuit in Fig 3.9
(see E x a m p l e 3.1) satisfies Kirchhoffs current
law at junctions x a n d y
b) Show that the solution of the circuit in Fig 3.9
satisfies Kirchhoffs voltage law a r o u n d every
closed loop
Sections 3.3-3.4
3.10 Find the p o w e r dissipated in the 5 ft resistor in the
PSPICE circuit in Fig P3.10
MULTISIM ^
Figure P3.10
PSPICE
MULTISIM
3.11 For the circuit in Fig P3.11 calculate
PSPICE
MULTISIM a ) V (> nd l a
b) the p o w e r dissipated in the 6 ft resistor
c) the p o w e r d e v e l o p e d by the current source
Figure P3.ll
3.12 a) Find an expression for the equivalent resistance
of two resistors of value R in series
b) Find an expression for the equivalent resistance
of n resistors of value R in series
c) Using the results of (a), design a resistive net-work with an equivalent resistance of 3 kft using two resistors with the same value from A p p e n d i x
H d) Using the results of (b), design a resistive
net-w o r k net-with an equivalent resistance of 4 kft using
a m i n i m u m n u m b e r of identical resistors from
A p p e n d i x H
3.13 a) Find an expression for the equivalent resistance
of two resistors of value R in parallel
b) Find an expression for the equivalent resistance
of n resistors of value R in parallel
c) Using the results of (a), design a resistive
net-w o r k net-with an equivalent resistance of 5 kft using two resistors with the s a m e value from
A p p e n d i x H d) Using the results of (b), design a resistive net-work with an equivalent resistance of 4 kft using
a m i n i m u m n u m b e r of identical resistors from
A p p e n d i x H
3.14 In the voltage-divider circuit shown in Fig P3.14, the
PSPICE no - l o a d value of vn is 4 V W h e n the load resistance
MULTISIM , , , , , , ,
R L is attached across the terminals a and b, v() drops
t o 3 V Find RL
Figure P3.14
20 V
40 ft
Figure P3.7
15 V
6ft
b 2ft
i5 A 60 ft
5.6 ft
A/W-12 ft
(c)
Trang 4DESIGN
PROBLEM
PSPICE
HULTISIM
3.15 a) Calculate the no-load voltage v„ for the
voltage-divider circuit shown in Fig P3.15
b) Calculate the power dissipated in Rx and R2
c) Assume that only 0.5 W resistors are available
The no-load voltage is to be the same as in (a)
Specify the smallest ohmic values of R] and R2
Figure P3.15
DE5IGN
PROBLEM
PSPICE
MULTISIM
/?i|4.7kfi
160 V ©
/?2<3.3kfl v„
3.16 The no-load voltage in the voltage-divider circuit
shown in Fig P3.16 is 8 V The smallest load resistor
that is ever connected to the divider is 3.6 kfl When
the divider is loaded, v() is not to drop below 7.5 V
a) Design the divider circuit to meet the
specifica-tions just mentioned Specify the numerical values
of /?, and R2
b) Assume the power ratings of commercially
available resistors are 1/16,1/8,1/4,1, and 2 W
What power rating would you specify?
Figure P3.16
40 V
3.17 Assume the voltage divider in Fig P3.16 has been
constructed from 1 W resistors What is the smallest
resistor from Appendix H that can be used as R L
before one of the resistors in the divider is
operat-ing at its dissipation limit?
3.18 Specify the resistors in the circuit in Fig P3.18 to
PROBLEM meet the following design criteria:
i H = 1 mA; vg = 1 V; iY = 2i 2 ;
i 2 = 2i3; and i3 = 2iA
Figure P3.18
3.19
PSPICE
a) The voltage divider in Fig P3.19(a) is loaded
with the voltage divider shown in Fig P3.19(b);
that is, a is connected to a', and b is connected to
b' Find vlt
b) Now assume the voltage divider in Fig P3.19(b)
is connected to the voltage divider in Fig P3.19(a) by means of a current-controlled
voltage source as shown in Fig P3.19(c) Find va
c) What effect does adding the dependent-voltage source have on the operation of the voltage divider that is connected to the 380 V source?
Figure P3.19
75 kn
380 V 25 kO
- • b
40 kO a'o vw f •
60kft:
b'<
(a)
75 kil
(b)
40 kn
^vw—
380 V 25 kH > 25,000/ 60 kn:
3.20 There is often a need to produce more than one
PROBLEM voltage using a voltage divider For example, the memory components of many personal computers require voltages of —12 V, 5 V, and +12 V, all with respect to a common reference terminal Select the
values of R],R 2 , and /?3 in the circuit in Fig P3.20 to meet the following design requirements:
a) The total power supplied to the divider circuit
by the 24 V source is 80 W when the divider is unloaded
b) The three voltages, all measured with respect to
the common reference terminal, are V\ = 12 V,
v 2 = 5 V, and v$ ~ - 1 2 V
Figure P3.20
24 V ©
'ih
/?,;
-• Common
* , :
3.21
PSPICE
MULTISIM
»3
a) Show that the current in the kth branch of the
circuit in Fig P3.21(a) is equal to the source current
i s times the conductance of the kth branch divided
by the sum of the conductances, that is,
h
ipk
G t + G 2 + G 3 + • • • + G k + • • • + G>
Trang 5b) Use the result derived in (a) to calculate the
cur-rent in the 5 0 resistor in the circuit in
Fig.P3.21(b)
Figure P3.21
(a)
0.5 a ^5 o f 8 a f io ft ^20 a ^ 40 a
L
(b)
3.22 A voltage divider like that in Fig 3.13 is to be
PROBLEM designed so that v0 = kv s at no load (RL = oo) and
v 0 = avs at full load (RL = Ra ) Note that by
defini-tion a < k < 1
a) Show that
and
_ k - a
R\ - — ; K
ak
b) Specify the numerical values of R[ and R2 if
k = 0.85, a = 0.80, and R 0 = 34 kO
c) If v s = 60 V, specify the maximum power that
will be dissipated in R\ and R 2
d) Assume the load resistor is accidentally short
circuited How much power is dissipated in Rx
and /?2?
3.24 Look at the circuit in Fig P3.2(b)
a) Use current division to find the current flowing from top to bottom in the 10 kfi resistor b) Using your result from (a), find the voltage drop across the 10 k l l resistor, positive at the top c) Starting with your result from (b), use voltage division to find the voltage drop across the 2 kfl resistor, positive at the top
d) Using your result from part (c), find the current through the 2 kH resistor from top to bottom e) Starting with your result from part (d), use cur-rent division to find the curcur-rent through the
18 kft resistor from top to bottom
3.25 Find vx and v2 in the circuit in Fig P3.25
PSPICE MULTISIM
Figure P3.25
t'2130 a
40 a
3.26 Find va in the circuit in Fig P3.26
PSPICE MULTISIM
Figure P3.26
18 mA
12 k a
3.23 Look at the circuit in Fig P3.2(a)
a) Use voltage division to find the voltage drop
across the 18 II resistor, positive at the left
b) Using your result from (a), find the current
flow-ing in the 18 il resistor from left to right
c) Starting with your result from (b), use current
division to find the current in the 25 fi resistor
from top to bottom
d) Using your result from part (c), find the voltage
drop across the 25 Q resistor, positive at the top
e) Starting with your result from (d), use voltage
division to find the voltage drop across the 10 fl
resistor, positive on the left
3.27 a) Find the voltage v x in the circuit in Fig P3.27
PSPICE
MULTISIM b) Replace the 18 V source with a general voltage
source equal to Vs Assume V s is positive at the upper terminal Find v x as a function of V y
Figure P3.27
18V
Trang 63.28 Find i a and i g in the circuit in Fig P3.28
'5P1CE _ „ _ _
Fiqure P3.28
1 2 f t
i3 n
3.32 Suppose the d'Arsonval voltmeter described in Problem 3.31 is used to measure the voltage across the 45 ft resistor in Fig P3.32
a) What will the voltmeter read?
b) Find the percentage of error in the voltmeter reading if
( measured value
% error = - 1 I X 100
\ true value Figure P3.32
3.29 For the circuit in Fig P3.29, calculate (a) ig and
PSPKE (b) the power dissipated in the 30 ft resistor
4ULTISIM
Figure P3.29
3.30 The current in the 12 ft resistor in the circuit in
PSPICE Fig P3.30 is 1 A, as shown
WLTISIM
a) Find vg
b) Find the power dissipated in the 20 ft resistor
Figure P3.30
Section 3.5
3.31 A d'Arsonval voltmeter is shown in Fig P3.31 Find
the value of Rv for each of the following full-scale
readings: (a) 50 V, (b) 5 V, (c) 250 mV, and (d) 25 mV
3.33 The ammeter in the circuit in Fig P3.33 has a resist-ance of 0.1 ft Using the definition of the percent-age error in a meter reading found in Problem 3.32, what is the percentage of error in the reading of this ammeter?
Figure P3.33
60 ft
'VW-3.34 The ammeter described in Problem 3.33 is used to
measure the current i0 in the circuit in Fig P3.32 What
is the percentage of error in the measured value?
3.35 a) Show for the ammeter circuit in Fig P3.35 that
the current in the d'Arsonval movement is always 1/25th of the current being measured b) What would the fraction be if the 100 mV, 2 m A movement were used in a 5 A ammeter?
c) Would you expect a uniform scale on a dc d'Arsonval ammeter?
100 mV, 2 raA
-AAA *
(25/12) ft
Trang 7PSPICE
MULTISIM
3.36 A shunt resistor and a 50 mV, 1 mA d'Arsonval
movement are used to build a 5 A ammeter A
resistance of 20 mO is placed across the terminals
of the ammeter What is the new full-scale range of
the ammeter?
3.37 The elements in the circuit in Fig 2.24 have the
follow-ing values: flj = 20 kO,, R2 = 80 kft, Rc = 0.82 kfl,
R E = 0.2 kO, Vcc = 7.5 V, V() = 0.6 V, and j3 = 39
a) Calculate the value of i B in microamperes
b) Assume that a digital multimeter, when used as a
dc ammeter, has a resistance of 1 kfl If the
meter is inserted between terminals b and 2 to
measure the current i Br what will the meter read?
c) Using the calculated value of i R in (a) as the
cor-rect value, what is the percentage of error in the
measurement?
3.38
DESIGN
PROBLEM
A d'Arsonval ammeter is shown in Fig P3.38
Design a set of d'Arsonval ammeters to read the
fol-lowing full-scale current readings: (a) 10 A, (b) 1 A,
(c) 50 mA, and (d) 2 mA Specify the shunt resistor
for each ammeter
Figure P3.38
3.39 A d'Arsonval movement is rated at 1 mA and
PROBLEM 50 m V- Assume 0.5 W precision resistors are
avail-able to use as shunts What is the largest
full-scale-reading ammeter that can be designed using a
single resistor? Explain
3.40 The voltmeter shown in Fig P3.40(a) has a
full-scale reading of 750 V The meter movement is
rated 75 mV and 1.5 mA What is the percentage of
error in the meter reading if it is used to measure
the voltage v in the circuit of Fig P3.40(b)?
Figure P3.40
750 V
30 m A M ) 25 kfR 125 kO f v
Common
3.41 You have been told that the dc voltage of a power supply is about 350 V When you go to the instrument room to get a dc voltmeter to measure the power supply voltage, you find that there are only two dc voltmeters available One voltmeter is rated 300 V
full scale and has a sensitivity of 900 fl/V The other
voltmeter is rated 150 V full scale and has a
sensitiv-ity of 1200 fl/V {Hint: you can find the effective
resistance of a voltmeter by multiplying its rated full-scale voltage and its sensitivity.)
a) How can you use the two voltmeters to check the power supply voltage?
b) What is the maximum voltage that can be measured?
c) If the power supply voltage is 320 V, what will each voltmeter read?
3.42 Assume that in addition to the two voltmeters described in Problem 3.41, a 50 k(l precision
resis-tor is also available The 50 kft resisresis-tor is
con-nected in series with the series-concon-nected voltmeters This circuit is then connected across the terminals of the power supply The reading on the 300 V meter is 205.2 V and the reading on the
150 V meter is 136.8 V What is the voltage of the power supply?
3.43 The voltage-divider circuit shown in Fig P3.43 is designed so that the no-load output voltage is 7/9ths of the input voltage A d'Arsonval volt-meter having a sensitivity of 100 fl/V and a full-scale rating of 200 V is used to check the operation
of the circuit
a) What will the voltmeter read if it is placed across the 180 V source?
b) What will the voltmeter read if it is placed across the 70 kO resistor?
c) What will the voltmeter read if it is placed across the 20 kil resistor?
d) Will the voltmeter readings obtained in parts (b) and (c) add to the reading recorded in part (a)? Explain why or why not
Figure P3.43
180 V
:20 Ml
:70kfi i\,
(bj
Trang 83.44 The circuit model of a dc voltage source is shown in
Fig P3.44 The following voltage measurements are
made at the terminals of the source: (1) With the
terminals of the source open, the voltage is
meas-ured at 50 raV, and (2) with a 15 Mfi resistor
con-nected to the terminals, the voltage is measured at
48.75 mV All measurements are made with a digital
voltmeter that has a meter resistance of 10 MH
a) What is the internal voltage of the source (v s ) in
millivolts?
b) What is the internal resistance of the source (Rs )
in kilo-ohms?
Figure P3.44
Terminals of ' the source
Figure P3.46
3.45 Assume in designing the multirange voltmeter
PROBLEM shown in Fig P3.45 that you ignore the resistance of
the meter movement
a) Specify the values of R iy R2, and R$
b) For each of the three ranges, calculate the
percent-age of error that this design strategy produces
Figure P3.45
100 V i • A W
-10
V»-I V '
* 2 -AA/V
*3
0 50 m V
2 m A
DESIGN
PROBLEM
Common
3.46 Design a d'Arsonval voltmeter that will have the
three voltage ranges shown in Fig P3.46
a) Specify the values of Rh R 2 , and 7?3
b) Assume that a 750 kil resistor is connected
between the 150 V terminal and the common
terminal The voltmeter is then connected to an
unknown voltage using the common terminal
and the 300 V terminal The voltmeter reads
288 V What is the unknown voltage?
c) What is the maximum voltage the voltmeter in (b)
can measure?
• 300 V
- • 1 5 0 V
•30 V
y x 5 0 m V
/J 1mA
1 Common
3.47 A 600 kH resistor is connected from the 200 V
ter-minal to the common terter-minal of a dual-scale volt-meter, as shown in Fig P3.47(a) This modified voltmeter is then used to measure the voltage across the 360 kO resistor in the circuit in Fig P3.47(b) a) What is the reading on the 500 V scale of the meter?
b) What is the percentage of error in the measured voltage?
Figure P3.47
r
500 V
600 k Q
40 k O
• ,
- • 5 0 0 VI
600 V © 360 m
Modified | voltmeter I
I
I .Common
— • I
J (b)
Trang 9Sections 3.6-3.7 Figure P3.53
3.48 Assume the ideal voltage source in Fig 3.26 is
replaced by an ideal current source Show that
Eq 3.33 is still valid
3.49 Find the power dissipated in the 3 kQ, resistor in the
PSPICE circuit in Fig P3.49
Figure P3.49
192 V
750 n
A W
25 kfl
3.50 Find the detector current id in the unbalanced
SPICE bridge in Fig P3.50 if the voltage drop across the
detector is negligible
Figure P3.50
75 V
20 kn
3.51 The bridge circuit shown in Fig 3.26 is energized
PSPICE from a 24 V dc source The bridge is balanced when
MULT1SIM _ „
R l = 500 H, /?2 = 1000 n , and R3 = 750 IX
a) What is the value of Rx t
b) How much current (in milliamperes) does the dc
source supply?
c) Which resistor in the circuit absorbs the most
power? How much power does it absorb?
d) Which resistor absorbs the least power? How
much power does it absorb?
3.52 In the Wheatstone bridge circuit shown in Fig 3.26,
PSPICE fa & r a tjQ RJR c a n be s et to the following values:
MULT I SIM
0.001, 0.01,0.1,1,10,100, and 1000 The resistor R3
can be varied from 1 to 11,110 ft, in increments of
1 ft An unknown resistor is known to lie between
4 and 5 ft What should be the setting of the R 2 /R\
ratio so that the unknown resistor can be measured
to four significant figures?
3.53 Use a A-to-Y transformation to find the voltages V\
and v-> in the circuit in Fig P3.53
MUITISIM
50 n
3.54 Use a Y-to-A transformation to find (a) i 0 ; (b) i\, (c) i< and (d) the power delivered by the ideal
cur-JLTISIM x v ' « « * * ! •
rent source in the circuit in Fig P3.54
Figure P3.54
320 a
/ „T^ 6 0 o a
3.55 Find i?ab in the circuit in Fig P3.55
PSPICE MULTISIM
Figure P3.55
PSPICE MULTISIH
3.56 a) Find the equivalent resistance Rah in the circuit
in Fig P3.56 by using a A-to-Y transformation
involving the resistors R 2 , R$, and R 4
b) Repeat (a) using a Y-to-A transformation
involving resistors R2 , R4, and R 5
c) Give two additional A-to-Y or Y-to-A
transfor-mations that could be used to find R db
Figure P3.56
^ 2 1 i o n 50 a
40 n
R*
Rsisn R, 4X1
in
Ry
Trang 103.57 a) Find the resistance seen by the ideal voltage 3.61 In the circuit in Fig P3.61(a) the device labeled D
PSPICE
MULTISIM source in the circuit in Fig P3.57
b) If vah equals 400 V, how much power is
dissi-pated in the 31 Cl resistor?
Figure P3.57
a
PSPICE MULTISIM
W ab ©
1.5 n
^ v w
-50 n
7i a
60 a :
20 a
100 a
so a
—vw-40 a
3 0 a
3i a
20 a
represents a component that has the equivalent cir-cuit shown in Fig P3.61(b).The labels on the termi-nals of D show how the device is connected to the
circuit Find v x and the power absorbed by the device Figure P3.61
3.58 Find the equivalent resistance R ah in the circuit in
PSPICE F i g p 3 5 8 i
MULTISIM °
32 a
20 a
3.62 Derive Eqs 3.44-3.49 from Eqs 3.41-3.43 The
fol-lowing two hints should help you get started in the right direction:
1) To find Ri as a function of R a , R f} , and R c , first
subtract Eq 3.42 from Eq 3.43 and then add this result to Eq 3.41 Use similar manipulations to
find R2 and R3 as functions of R(l , R b , and R c 2) To find R b as a function of R^, R 2 , and R 3 , take
advantage of the derivations obtained by hint (1), namely, Eqs 3.44-3.46 Note that these equa-tions can be divided to obtain
3.59 Find iQ and the power dissipated in the 140 ft
resis-'SPICE t o r jn t^e cjr c uit |n pig P359,
Figure P3.59
240 V
22 a
3.60 For the circuit shown in Fig P3.60, find (a) i h (b) v,
(c) i2 , and (d) the power supplied by the voltage
JLTISIM
source
Figure P3.60
120 a
or R,
R, Rh,
and
R$ Rb
~7T = ~TT, or R, = —/?»,
Now use these ratios in Eq 3.43 to eliminate R a and R c Use similar manipulations to find R a and
R c as functions of Ri, R2 , and i?3,
3.63 Show that the expressions for A conductances as
functions of the three Y conductances are
n
G,
G2G3
G1G3
+ G 2 + G 3 '
Gi + G2 + G3' where
43 a C - l r - l etc