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CIRCUIT LAWS
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[CHAP. 3
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v iR1 iR2 iR3 i R1 R2 R3 iReq where a single equivalent resistance Req replaces the three series resistors. between i and v will pertain. For any number of resistors in series, we have Req R1 R2 . If the three passive elements are inductances, v L1 di di di L2 L3 dt dt dt di L1 L2 L3 dt di Leq dt The same relationship
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Extending this to any number of inductances in series, we have Leq L1 L2 . If the three circuit elements are capacitances, assuming zero initial charges so that the constants of integration are zero, 1 1 1 v i dt i dt i dt C1 C2 C3   1 1 1 i dt C1 C2 C3 1 i dt Ceq The equivalent capacitance of several capacitances in series is 1=Ceq 1=C1 1=C2 .
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EXAMPLE 3.3. The equivalent resistance of three resistors in series is 750.0
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. Two of the resistors are 40.0 and 410.0
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. What must be the ohmic resistance of the third resistor Req R1 R2 R3 750:0 40:0 410:0 R3
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Find the equivalent
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EXAMPLE 3.4. Two capacitors, C1 2:0 mF and C2 10:0 mF, are connected in series. capacitance. Repeat if C2 is 10.0 pF. Ceq If C2 10:0 pF, Ceq 2:0 10 6 10:0 10 12 20:0 10 18 10:0 pF 2:0 10 6 2:0 10 6 10:0 10 12 C1 C2 2:0 10 6 10:0 10 6 1:67 mF C1 C2 2:0 10 6 10:0 10 6
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where the contribution of 10:0 10 12 to the sum C1 C2 in the denominator is negligible and therefore it can be omitted.
Note: When two capacitors in series di er by a large amount, the equivalent capacitance is essentially equal to the value of the smaller of the two.
CIRCUIT ELEMENTS IN PARALLEL
For three circuit elements connected in parallel as shown in Fig. 3-4, KCL states that the current i entering the principal node is the sum of the three currents leaving the node through the branches.
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CIRCUIT LAWS
Fig. 3-4
i i1 i2 i3 If the three passive circuit elements are resistances,   v v v 1 1 1 1 i v v R1 R2 R3 R1 R2 R3 Req For several resistors in parallel, 1 1 1 Req R1 R2 The case of two resistors in parallel occurs frequently and deserves special mention. resistance of two resistors in parallel is given by the product over the sum. Req R1 R2 R1 R2 The equivalent
EXAMPLE 3.5. Obtain the equivalent resistance of (a) two 60.0-
resistors in parallel and (b) three 60.0-
resistors in parallel. a b 60:0 2 30:0
120:0 1 1 1 1 Req 20:0
Req 60:0 60:0 60:0 Req
Note: For n identical resistors in parallel the equivalent resistance is given by R=n. Combinations of inductances in parallel have similar expressions to those of resistors in parallel: 1 1 1 Leq L1 L2 and, for two inductances, Leq L1 L2 L1 L2
EXAMPLE 3.6.
Two inductances L1 3:0 mH and L2 6:0 mH are connected in parallel. 1 1 1 Leq 3:0 mH 6:0 mH and Leq 2:0 mH
Find Leq .
With three capacitances in parallel, i C1 dv dv dv dv dv C2 C3 C1 C2 C3 Ceq dt dt dt dt dt
For several parallel capacitors, Ceq C1 C2 , which is of the same form as resistors in series.
CIRCUIT LAWS
[CHAP. 3
VOLTAGE DIVISION
A set of series-connected resistors as shown in Fig. 3-5 is referred to as a voltage divider. The concept extends beyond the set of resistors illustrated here and applies equally to impedances in series, as will be shown in 9.
Fig. 3-5
Since v1 iR1 and v i R1 R2 R3 ,
 v1 v
R1 R1 R2 R3
EXAMPLE 3.7. A voltage divider circuit of two resistors is designed with a total resistance of the two resistors equal to 50.0
. If the output voltage is 10 percent of the input voltage, obtain the values of the two resistors in the circuit. v1 0:10 v from which R1 5:0
and R2 45:0
. 0:10 R1 50:0 103
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