0195136047.pdf

60 CIRCUIT CONCEPTS

10

20

30

− 1 1

(20t + 10)

(− 10 t + 40)

10 t

30

10(t + 1)

0

L = 100 pH

23

+

−

i(t), A

i(t) v(t)

t, μs

45

Figure P1.2.26

1.2.28The inductance per unit length in H/m for parallel-
plate infinitely long conductors in air is given by
L=μ 0 d/w= 4 π× 10 −^7 d/w, wheredandware
in meters. ComputeL(per unit length) ford=1m
andw= 0 .113 m. See Problem 1.2.23 and show
that the product of inductance (per unit length) and
capacitance (per unit length) isμ 0 ε 0.

1.2.29Determine the duals for the circuit configurations
of Problem 1.2.24 and determine the equivalent
inductance at terminalsA–Bfor each case.

1.2.30Consider a pair of coupled coils as shown in Figure
1.2.10 of the text, with currents, voltages, and
polarity dots as indicated. Show that the mutual
inductance isL 12 =L 21 = Mby following
these steps:
(a) Starting at timet 0 withi 1 (t 0 )=i 2 (t 0 )=0,
maintaini 2 =0 and increasei 1 until, at time
t 1 ,i 1 (t 1 )= I 1 andi 2 (t 1 ) =0. Determine
the energy accumulated during this time. Now
maintainingi 1 =I 1 , increasei 2 until at time
t 2 ,i 2 (t 2 )=I 2. Find the corresponding energy
accumulated and the total energy stored at
timet 2.
(b) Repeat the process in the reverse order, al-
lowing the currents to reach their final values.
Compare the expressions obtained for the total
energy stored and obtain the desired result.

1.2.31For the coupled coils shown in Figure P1.2.31, a
dot has been arbitrarily assigned to a terminal as

indicated. Following the dot convention presented

in the text, place the other dot in the remaining

coil and justify your answer with an explanation.

Comment on whether the polarities are consistent

with Lenz’s law.

*1.2.32For the configurations of the coupled coils shown

in Figure P1.2.32, obtain the voltage equations for

v 1 andv 2.

1.2.33The self-inductances of two coupled coils areL 11

andL 22 , and the mutual inductance between them

isM. Show that the effective inductance of the two

coils in series is given by

Lseries=L 11 +L 22 ± 2 M

and the effective inductance of the two coils in

parallel is given by

Lparallel=

L 11 L 22 −M^2

L 11 ∓ 2 M+L 22

Specify the conditions corresponding to different

signs of the term 2M.

1.2.34For the coupled inductors shown in Figure

P1.2.34, neglecting the coil resistances, write the

volt-ampere relations.

1.2.35Consider an amplifier as a voltage source with an

internal resistance of 72. Find the turns ratio of

the ideal transformer such that maximum power

is delivered when the amplifier is connected to an

8-speaker through anN 1 :N 2 transformer.

1.2.36For the circuit shown in Figure P1.2.36, determine

vout(t).

Figure P1.2.31