Page 38 - Schaum's Outline of Theory and Problems of Electric Circuits

CIRCUIT LAWS
CHAP. 3]

Fig. 3-4 27

i ¼ i 1 þ i 2 þ i 3
If the three passive circuit elements are resistances,

v v v 1 1 1 1
i ¼ þ þ ¼ þ þ v ¼ v
R 1 R 2 R 3 R 1 R 2 R 3 R eq

For several resistors in parallel,
1 1 1
¼ þ þ
R eq R 1 R 2

The case of two resistors in parallel occurs frequently and deserves special mention. The equivalent
resistance of two resistors in parallel is given by the product over the sum.

R 1 R 2
R eq ¼
R 1 þ R 2

EXAMPLE 3.5. Obtain the equivalent resistance of (a) two 60.0-
resistors in parallel and (b) three 60.0-
resistors in parallel.

ð60:0Þ 2
ðaÞ R eq ¼ ¼ 30:0
120:0
1 1 1 1
ðbÞ ¼ þ þ R eq ¼ 20:0
R eq 60:0 60:0 60:0
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 L 1 L 2
¼ þ þ and, for two inductances, L eq ¼
L eq L 1 L 2 L 1 þ L 2

EXAMPLE 3.6. Two inductances L 1 ¼ 3:0 mH and L 2 ¼ 6:0 mH are connected in parallel. Find L eq .
1 1 1
¼ þ and L eq ¼ 2:0mH
L eq 3:0mH 6:0mH
With three capacitances in parallel,

dv dv dv dv dv
i ¼ C 1 þ C 2 þ C 3 ¼ðC 1 þ C 2 þ C 3 Þ ¼ C eq
dt dt dt dt dt
For several parallel capacitors, C eq ¼ C 1 þ C 2 þ , which is of the same form as resistors in series.