Chapter 6
Inductance, Capacitance, and
Mutual Inductance
6.1
6.2
6.3
6.4
6.5
The inductor
The capacitor
Series-parallel combinations of
inductance and capacitance
Mutual inductance
Closer look at mutual inductance
1
Overview

In addition to voltage sources, current sources,
resistors, here we will discuss the remaining 2
types of basic elements: inductors, capacitors.

Inductors and capacitors cannot generate nor
dissipate but store energy.

Their current-voltage (i-v) relations involve with
integral and derivative of time, thus more
complicated than resistors.
2
Key points

di
Why the i-v relation of an inductor is v  L ?
dt

dv
Why the i-v relation of a capacitor is i  C
?
dt

Why the energies stored in an inductor and a
capacitor are:
1 2 1 2
w  Li , Cv , respectively?
2
2
3
Section 6.1
The Inductor
1.
2.
3.
Physics
i-v relation and behaviors
Power and energy
4
Fundamentals

An inductor of inductance L is symbolized by a
solenoidal coil.

Typical inductance L ranges from 10 H to 10
mH.

The i-v relation of an inductor (under the
passive sign convention) is:
di
vL ,
dt
5
Physics of self-inductance (1)

Consider an N1-turn coil C1 carrying current I1.
 
The resulting magnetic field B1 ( r )  N1 I1 (BiotSavart law) will pass through C1 itself, causing a
 
B1 ( r )
flux linkage 1, where
1  N11 ,
1  
S1
  
B1 ( r )  ds  P1 N1 I1 ,
P1 is the permeance.
 1  P1 N I .
2
1 1
6
Physics of self-inductance (2)

The ratio of flux linkage to the driving current is
defined as the self inductance of the loop:
L1 
1
I1
 N12 P1 ,
which describes how easy a coil current can
introduce magnetic flux over the coil itself.
7
Examples

Solenoidal & toroidal coils:
L = 270 H

~2 cm
L = 36 H
RG59/U coaxial cable:
L = 351 nH/m.
8
The i-v relation

Faraday’s law states that the electromotive
force v (emf, in units of volt) induced on a loop
equals the time derivative of the magnetic flux
linkage :
d
d
d
v   ,  v  Li  L i.
dt
dt
dt

Note: The emf of a loop is a non-conservative
force that can drive current flowing along the
loop. In contrast, the current-driving force due
to electric charges is conservative.
9
Behaviors of inductors
di
vL
dt

DC-current: inductor behaves as a short circuit.

Current cannot change instantaneously in an
inductor, otherwise, infinite voltage will arise.

Change of inductor current is the integral of
voltage during the same time interval:
1 t
i (t )  i (t0 )   v ( )d .
L t0
10
Inductive effect is everywhere!

Nearly all electric circuits have currents flowing
through conducting wires. Since it’s difficult to
shield magnetic fields, inductive effect occurs
even we do not purposely add an inductor into
the circuit.
11
Example 6.1: Inductor driven by a current pulse
0, t  0
i (t )  
5t
10
te
, t 0

The inductor voltage is:
di 0, t  0
v(t )  L   5t
dt e (1  5t ), t  0
Inductor
voltage can
jump!
Memory-less
in steady
state.
12
Power & energy (1)

Consider an inductor of inductance L. The
instantaneous power in the inductor is:
di
p  vi  Li .
dt

Assume there is no initial current (i.e. no initial
energy),  i(t = 0) = 0, w(t = 0) = 0. We are
interested in the energy W when the current
increases from zero to I with arbitrary i(t).
13
Power & energy (2)
W
I
di
dw
 Li ,  dw  Li  di,   dw  L  idi,
p
0
0
dt
dt
2 I
1 2
i
1 2
W  L
 LI , i.e. w  Li
2
20 2

How the current changes with time doesn’t
matter. It’s the final current I determining the
final energy.

Inductor stores magnetic energy when there is
nonzero current.
14
Example 6.3: Inductor driven by a current pulse

t < 0.2,  p > 0, w, charging.

t > 0.2,  p < 0, w, discharging.

In steady state (t ), i  0,
v 0, p  0, w  0 (no energy).
15
Example 6.3: Inductor driven by a voltage pulse
1 t
i (t )   v( )d  i (0)
L 0

p > 0, w, always charging.

In steady state (t), i 2 A,
v  0, p  0, w  200 mJ
(sustained current and
constant energy).
With memory in
steady state.
16
Section 6.2
The Capacitor
1.
2.
3.
Physics
i-v relation and behaviors
Power and energy
17
Fundamentals

A capacitor of capacitance C is symbolized by a
parallel-plate.

Typical capacitance C ranges from 10 pF to 470
F.

The i-v relation of an capacitor (under the
passive sign convention) is:
dv
iC .
dt
18
Physics of capacitance (1)

If we apply a voltage V12 between two isolated
conductors, charge Q will be properly
distributed over the conducting surfaces such
 
that the resulting electric field E (r ) satisfies:
2
 
E (r )
1
B1

E
1

2
  
E ( r )  dl  V12 ,
which is valid for any
integral path linking
the two conducting
surfaces.
19
Physics of capacitance (2)

If V12  V12 ,  Q   Q while the spatial
distribution of charge remains such that
 
 

1  
E ( r )  E ( r ), V12   E ( r )  dl  V12 .
2

The ratio of the deposited charge to the bias
voltage is defined as the capacitance of the
conducting pair:
Q
C ,
V
describing how easy a bias voltage can deposit
charge on the conducting pair.
20
Examples

Ceramic disc & electrolytic:

RG59/U coaxial cable:
C = 53 pF/m.
21
The i-v relation

From the definition of capacitance:
Q
d
d
d
q  C v, i  C v.
C  ,  q(t )  Cv(t ),
dt
dt
V
dt

Note: Charge cannot flow through the
dielectric between the conductors. However, a
time-varying voltage causes a time-varying
electric field that can slightly displace the
dielectric bound charge. It is the time-varying
bound charge contributing to the
“displacement current”.
22
Polarization charge
23
Behaviors of capacitors
dv
iC
dt

DC-voltage: capacitor behaves as an open
circuit.

Voltage cannot change instantaneously in an
capacitor, otherwise, infinite current will arise.

Change of capacitor voltage is the integral of
current during the same time interval:
1 t
v(t )  v(t0 )   i ( )d .
C t0
24
Capacitive effect is everywhere!

A Metal-Oxide-Semiconductor (MOS) transistor
has three conducting terminals (Gate, Source,
Drain) separated by a dielectric layer with one
another. Capacitive effect occurs even we do
not purposely add a capacitor into the circuit.
(info.tuwien.ac.at)
charges
25
Power & energy (1)

Consider a capacitor of capacitance C. The
instantaneous power in the capacitor is:
dv
p  vi  Cv .
dt

Assume there is no initial voltage (i.e. no initial
energy),  v(t = 0) = 0, w(t = 0) = 0. We are
interested in the energy W when the voltage
increases from zero to V with arbitrary v(t).
26
Power & energy (2)
W
V
dv
dw
 Cv ,  dw  Cv  dv,   dw  C  vdv,
p
0
0
dt
dt
2 V
v
W  C
2
0
1
 CV 2 , i.e. w  1 Cv 2
2
2

How the voltage increases with time doesn’t
matter. It’s the final voltage V determining the
final energy.

Capacitor stores electric energy when there is
nonzero voltage.
27
Example 6.4: Capacitor driven by a voltage pulse
Capacitor current
can jump!

t < 1,  p > 0, w, charging.

t > 1,  p < 0, w, discharging.

In steady state (t), i 0, v
 0, p  0, w  0 (no energy).
Memory-less in
steady state.
28
Section 6.3
Series-Parallel
Combinations
1.
2.
Inductors in series-parallel
Capacitors in series-parallel
29
Inductors in series
i  i1  i2  i3 ,
v  v1  v2  v3 ,
v j  Lj
di j
,
dt
di
di
di
 v  L1  L2  L3
dt
dt
dt
di
 Leq ,
dt
n
 Leq   L j
j 1
30
Inductors in parallel
1
v  v1  v2  v3 , i  i1  i2  i3 , i j 
Lj

t
t0
 1 t
  3 1
 i     v ( )d  i j (t0 )    

  j 1 L
t
j 1  L j 0
j
 
n
1  t
1
1



 t v ( )d   i (t0 ), 

Leq  0
Leq j 1 L j
3
v ( )d  i j (t0 ),
3
 t


 v ( )d    i j (t0 ) 
 t0
  j 1


31
Capacitors in series
=
v  v1  v2  v3 , i  i1  i2  i3 ,

v  1 t id  v (t ), v  1
j 0
 j C t0
Ceq
j

t
 id  v(t ),
t0
0
n
1
1


Ceq j 1 C j
32
Capacitors in parallel
=
v  v1  v2  v3 , i  i1  i2  i3 ,


dv
dv
i j  C j dt , i  Ceq dt ,
n
 Ceq   C j
j 1
33
Section 6.4, 6.5
Mutual Inductance
1.
2.
3.
Physics
i-v relation and dot convention
Energy
34
Fundamentals

Mutual inductance M is a circuit parameter
between two magnetically coupled coils.

The value of M satisfies M  k L1L2 , where
0  k  1 is the magnetic coupling coefficient.

The emf induced in Coil 2 due to time-varying
current in Coil 1 is proportional to M di1 dt .
35
The i-v relation (1)

Coil 1 of N1 turns is driven by a time-varying
current i1, while Coil 2 of N2 turns is open.

The flux components linking (1) only Coil 1, (2)
both coils, and (3) total flux linking Coil 1 are:
11  
S1
  
  
B1 (r )  ds  P11 N1 I1 , 21   B1 ( r )  ds  P21 N1 I1.
S2
1  11  21 ,  P1 N1 I1  P11  P21 N1 I1 , P1  P11  P21.
36
The i-v relation (2)

Faraday’s law states that the emf induced on
Coil 2 (when i2 remains constant) is:
d
d
d
d
v2  N 221  N 2 ( P21 N1i1 )  N 2 N1 P21 i1  M 21 i1.
dt
dt
dt
dt

One can show that the emf induced on Coil 1
(when i1 remains constant) is:
d
d
v1  N1 N 2 P12 i2  M 12 i2 .
dt
dt

For nonmagnetic media (e.g. air, silicon,
plastic), P21 = P12,  M21 = M12= M= N1N2P21.
37
Mutual inductance in terms of self-inductance

The two self inductances and their product are:
L1  N12 P1 , L2  N 22 P2 ,
L1 L2  N12 N 22 P1 P2  N12 N 22 P11  P21 P22  P12 
 N1 N 2 P21 
2
 P11  P22 

 1
 1, if P21  P12 .
 P21  P21 
1  P11  P22 
M2
 1
 1,  L1 L2  2 , M  k L1 L2 .
Let 2  
k
k
 P21  P21 

The coupling coefficient (1) k = 0, if P21  21 = 0
(i.e. no mutual flux), (2) k = 1, if P11=P22=0 (i.e.
11 = 22 = 0, 1 = 2 = 21, no flux leakage).
38
Dot convention (1)
 Mi2 (t )
leaves the dot of L2,  the “+” polarity of Mi2 (t )
is referred the terminal of L1 without a dot.
 i2

The total voltage across L1 is:
d
d
v1  L1 i1  M i2 .
dt
dt
39
Dot convention (2)
 Mi1(t )
enters the dot of L1,  the “+” polarity of Mi1(t )
is referred the terminal of L2 with a dot.
 i1

The total voltage across L2 is:
d
d
v2  L2 i2  M i1.
dt
dt
40
Example 6.6: Write a mesh current equation
d
d
( 4 H ) i1  ( 20 )(i1  i2 )  (5 )(i1  ig )  (8 H ) (ig  i2 )  0
dt
dt
Self-inductance,
passive sign convention
Mutual-inductance,
igi2 enters the dot of
16-H inductor
41
Example 6.6: Steady-state analysis

In steady state (t  ), inductors are short, 
the 3 resistors are in parallel (Req = 3.75 ).

Let v2 = 0.  (1) v1 = (16A)(3.75 ) = 60 V. (2) i12 =
(60V)/(5) = 12A,  i1 = (16-12) = 4 A (not zero!).
(3) i1'2 = (60V)/(20) = 3 A,  i22' = (12+3) = 15 A.
4A
60 V
1
2
12 A
1'
3A
2'
42
Energy of mutual inductance (1)

Assume i1, i2 = 0 initially. Fix i2 = 0 while
increasing i1 from 0 to some constant I1. The
energy stored in L1 becomes:
W1  L1 
I1
0
1
i1di1  L1 I12 .
2
43
Energy of mutual inductance (2)

Now fix i1 = I1, while increasing i2 from 0 to I2.
During this period, emf’s will be induced in
loops 1 and 2 due to the time-varying i2. The
total power of the two inductors is:
 di2 
 di2 
p (t )  I1   M
  i2 (t )   L2
.
dt 

 dt 

An extra energy of W12+W2 is stored in the pair:
I2
W12  W2  I1M  di2  L2 
0
I2
0
1
i2 di2  MI1 I 2  L2 I 22 .
2
44
Energy of mutual inductance (3)

The entire process contributes to a total energy
1
1
2
Wtot  L1 I1  MI1 I 2  L2 I 22
2
2
for the two-inductor system.

Wtot only depends on the final currents I1, I2
[independent of the time evolution of i1(t), i2(t)].
45
Key points

di
Why the i-v relation of an inductor is v  L ?
dt

dv
Why the i-v relation of a capacitor is i  C
?
dt

Why the energies stored in an inductor and a
capacitor are:
1 2 1 2
w  Li , Cv , respectively?
2
2
46