EE111 Electrical Circuit Analysis
Lecture 15
Dr. Oliver Faust1
October 22, 2014
1
School of Science & Engineering
Habib University
Motivation
In this lecture we revisit the axiomatic Kirchhoff laws. When we introduced these two laws, we did so with no restrictions as to the types of elements
constituting the network. Therefore, both laws remain valid for circuits with
capacitors and inductors.
Contents
1 Inductance and Capacitance Combinations
1.1 Inductor combinations . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Capacitor combinations . . . . . . . . . . . . . . . . . . . . . . .
1
1
3
2 Consequences of Linearity
5
1
1.1
Inductance and Capacitance Combinations
Inductor combinations
A circuit containing N inductors in series
L1
+
v1
L2
−
v2
+
i
−
+
vs
−
+
vN
−
1
LN
Series equivalent circuit
i
−
+
vs
Leq
Inductors in Series
Applying KVL to the original circuit,
vs
= v1 + v2 + ... + vN
di
di
di
= L1 dt
+ L2 dt
+ ... + LN dt
di
= (L1 + L2 + ... + LN ) dt
(1)
or, written more concisely,
vs =
N
X
N
X
vn =
n=1
Ln
n=1
N
di
di X
=
Ln
dt
dt n=1
(2)
di
dt
(3)
For the equivalent circuit we have
vs = Leq
and thus the equivalent inductance is
Leq = L1 + L2 + ... + LN
or
Leq =
N
X
(4)
Ln
(5)
n=1
A circuit containing N inductors in parallel
+
+
v
is
−
v1
+
L1
L2
− i1
i2
2
vN
LN
− iN
Parallel equivalent circuit
+
Leq
v
is
−
Inductors in Parallel
The combination of N parallel inductors is accomplished by writing the
single nodal equation for the original circuit.
i
PN
PN h R t
is = n=1 in = n=1 L1n t0 v dt0 + in (t0 )
P
R
(6)
PN
t
N
1
=
v dt0 + n=1 in (t0 )
n=1 Ln
t0
and comparing it with the result for the equivalent circuit of
Z t
1
v dt0 + is (t0 )
is =
Leq t0
(7)
Since Kirchhoff’s current law demands that is (t0 ) be equal to the sum of the
branch currents at t0 , the two integral terms must also be equal; hence,
Leq =
1
1/L1 + 1/L2 + ... + 1/LN
(8)
For the special case of two inductors in parallel,
Leq =
L1 L2
L1 + L2
(9)
note that inductors in parallel combine exactly as do resistors in parallel.
1.2
Capacitor combinations
A circuit containing N capacitors in series
C1
−
+
i
−
+
v1
v2
−
+
vs
C2
+
vN
CN
−
3
Series equivalent circuit
i
−
+
vs
Ceq
Capacitors in Series
The combination of N series capacitors is accomplished by writing a single
mesh equation for the original circuit.
i
PN
PN h R t
vs = n=1 vn = n=1 C1n t0 i dt0 + vn (t0 )
R
P
(10)
PN
t
N
1
v dt0 + n=1 vn (t0 )
=
n=1 Cn
t0
and
1
vs =
Ceq
Z
t
i dt0 + vs (t0 )
(11)
t0
Since Kirchhoff’s voltage law establishes that vs (t0 ) is equal the sum of the
capacitor voltages at t0 , thus
Ceq =
1
1/C1 + 1/C2 + ... + 1/CN
(12)
For the special case of two capacitors in series,
Ceq =
C1 C2
C1 + C2
(13)
A circuit containing N capacitors in parallel
+
+
v1
v
is
+
C1
vN
C2
−
−
−
Parallel equivalent circuit
+
Ceq
v
is
CN
−
4
Capacitors in Parallel
We establish the value of the capacitor which is equivalent to N parallel
capacitors as
Ceq = C1 + C2 + ... + CN
(14)
2
Consequences of Linearity
This section focuses on nodal and mesh analysis. We show that the benefits
of linearity apply to RLC circuits as well. In accordance with our previous
definition of a linear circuit, these circuits are also linear, because the voltagecurrent relationships for the inductor and capacitor are linear relationships. For
the inductor, we have
di
(15)
v=L
dt
and multiplication of the current by some constant K leads to a voltage that is
also greater by a factor K. In the integral formulation,
Z
1 t
v dt0 + i(t0 )
(16)
i(t) =
L t0
it can be seen that, if each term is to increase by a factor of K, then the initial
value of the current must also increase by this same factor. A corresponding
investigation of the capacitor shows that it, too, is linear. Thus, a circuit composed of independent sources, linear dependent sources, and linear resistors,
inductors, and capacitors is a linear circuit.
The principle of superposition is a natural consequence of the linear nature
of resistive circuits. The resistive circuits are linear because the voltage-current
relationship for the resistor is linear and Kirchhoff’s laws are linear.
Summary and review
1. The current through a capacitor is given by i = C dv/ dt.
2. The voltage across a capacitor is related to its current by
Z
1 t 0
v(t) =
i(t ) dt0 + v(t0 )
C t0
(17)
3. A capacitor is an open circuit to DC voltages.
4. The voltage across an inductor is given by v = L di/ dt.
5. The current through an inductor is related to its voltage by
Z
1 t 0
i(t) =
v(t ) dt0 + i(t0 )
L t0
6. An inductor is a short circuit to DC currents.
5
(18)
7. The energy presently stored in a capacitor is given by 12 C v 2 , whereas
the energy presently stored in an inductor is given by 12 L i2 ; both are
referenced to a time at which no energy was stored.
8. Series and parallel combinations of inductors can be combined using the
same equations as for resistors.
9. Series and parallel combinations of capacitors work the opposite way as
they do for resistors.
10. Since capacitors and inductors are linear elements, KVL, KCL, superposition, Thévenin’s and Norton’s theorems, and nodal and mesh analysis
apply to their circuits as well.
Outlook
Lecture 16 will cover:
• The Source-Free RL Circuit,
• Properties of the Exponential Response.
6