Chapter 6

Engr228
Circuit Analysis

Dr Curtis Nelson

Chapter 6 Objectives

• Know and be able to use the equations for voltage,

current, power, and energy in inductors and capacitors;
• Know that current must be continuous in an inductor
and voltage must be continuous in a capacitor;
• Be able to correctly combine inductors in series and
parallel to form a single equivalent inductor;
• Be able to correctly combine capacitors in series and
parallel to form a single equivalent capacitor.

Engr228 - Chapter 6, Nillson 9E 1

 Practical Definition of Inductance

• Inductance is the property of an electrical circuit causing

voltage to be generated proportional to the rate of change
in current in a circuit.
• The term 'inductance' was coined by Oliver Heaviside in
February 1886. It is customary to use the symbol L for
inductance, possibly in honor of the physicist Heinrich
Lenz. The SI unit of inductance is the henry (H), named
after American scientist and magnetic researcher Joseph
Henry.
• Inductance is caused by the magnetic field generated by
electric currents according to Ampere's law. Inductors
typically consist of coils of wire to concentrate the
magnetic field and to collect the induced voltage.

Real Inductors

Engr228 - Chapter 6, Nillson 9E 2

 Inductors

• Faraday’s law states that a voltage is induced in an ideal

conductor if a changing current passes through it.
• The induced voltage is proportional to the time rate of
change (derivative) of the current.
• The physics behind this is governed by Maxwell’s
equations.
• As a result, energy is stored in the magnetic field
surrounding the wire.

Inductors

• The inductor is often called a coil because physically coiling

a wire greatly increases its inductance.
• The governing voltage and current relationship is

di (t )
vL (t ) = L
dt

Engr228 - Chapter 6, Nillson 9E 3

 DC Characteristics of Inductors

The inductor acts like a “short circuit” at DC because the time

rate of current change is equal to zero.

di (t )
vL (t ) = L
dt

Inductor Example
From the circuit shown
at the right, find i(t).

di (t )
v(t ) = L
dt
1 1
L∫ L∫
i (t ) = v (t ) dt = A sin ωtdt

A A  − cos ωt 
= ∫ sin ωtdt =  
L L ω 
A A π
= (− cos ωt ) = (sin ωt − )
ωL ωL 2

Engr228 - Chapter 6, Nillson 9E 4

 Voltage-Current Relationship in an Inductor

A π
i (t ) = (sin ωt − )
ωL 2

Phase shift of -90º A

i (t ) = (sin ωt )
R

ωL is known as the inductive impedance

Voltage-Current Relationship in a Resistor

Voltage and current in a resistor are in phase as shown below.

The amplitudes may vary due to Ohm’s Law, but the phase is
the same for the current and the voltage.

Engr228 - Chapter 6, Nillson 9E 5

 Voltage-Current Relationship in an Inductor
Current and voltage in an inductor are not in phase with each
other. For sinusoidal waves, the voltage across an inductor
leads the current through it by 90º. (In other words, the
current lags the voltage by 90º.) For instance, in the diagram
below, the tall blue waveform represents the voltage across an
inductor, and the short purple waveform represents the current
through the inductor.

Power in an Inductor

diL
p = vL iL = LiL
dt

Instantaneous power is measured in Watts (W).

Engr228 - Chapter 6, Nillson 9E 6

 Energy Storage in an Inductor

• Energy is the integral of power over a time interval and is

measured in Joules (J).
• Energy is stored in the magnetic field surrounding the
inductor and can be recovered by the circuit.

t
wL (t ) = ∫ pdt
t0

1 2
wL (t ) = LiL
2

Inductor Example
• Find the maximum energy stored in the inductor.

1 2 πt
wL = Li = 216 sin 2
2 6
πt
p R = i 2 R = 14.4 sin 2
6
6
2 πt
wR = ∫ 14.4 sin
0 6
= 43.2 J

• Energy is stored in the inductor from 0 to 3 seconds where it

reaches its peak value, and then begins to leave the inductor.
• The energy dissipated as heat in the resistor from 0 to 6 sec is
about 20% of the peak value.

Engr228 - Chapter 6, Nillson 9E 7

 Capacitors

• Like the inductor, the capacitor is an energy storing device.

• The capacitance (C) is a measure of the capacitor’s
potential to store energy in an electric field.
• Physically, a capacitor is constructed of two conducting
plates separated by an insulator.

Real Capacitors

Engr228 - Chapter 6, Nillson 9E 8

 Practical Definition of Capacitance

• Capacitance is the ability of a body to hold an electrical

charge. Capacitance is also a measure of the amount of
electrical energy stored for a given electric potential. A
common form of an energy storage device is a parallel-plate
capacitor. In a parallel plate capacitor, capacitance is directly
proportional to the surface area of the conductor plates and
inversely proportional to the separation distance between the
plates. If the charges on the plates are +Q and −Q, and V
gives the voltage between the plates, then the capacitance is
given by

C = Q/V

Capacitors

• The unit in which capacitance is measured is the Farad (F).

• 1 F = 1 Amp-Second/Volt = 1 Coulomb/Volt.
• The governing voltage and current relationship is

dvC (t )
iC (t ) = C
dt

Engr228 - Chapter 6, Nillson 9E 9

 More on Capacitance
• The capacitance of the majority of capacitors used in
electronic circuits is several orders of magnitude smaller
than the farad. The most common subunits of capacitance in
use today are the milli-farad (mF), micro-farad (µF), nano-
farad (nF), pico-farad (pF), and femto-farad (fF).
• The capacitance of a parallel-plate capacitor constructed of
two parallel plates both of area S separated by a distance d is
approximately equal to the following:
C = εRε0(S/d)
• C is the capacitance;
• S is the area of overlap of the two plates;
• εr is the relative static permittivity (sometimes called the dielectric
constant) of the material between the plates (for a vacuum, εr = 1);
×
• ε0 is the electric constant (ε0 ≈ 8.854 10−12 Fm–1);
• d is the separation between the plates.

DC Characteristics of a Capacitor

The capacitor acts like an “open circuit” at DC because

the time rate of change of voltage is zero so no current
can flow through it.
dvC (t )
iC (t ) = C
dt

Engr228 - Chapter 6, Nillson 9E 10

 Capacitor Example

From the circuit on the

right, find i(t).

dv(t ) d ( A sin ωt )
i (t ) = C =C
dt dt
= AωC (cos ωt )
A π
= sin(ωt + )
 1  2
 
 ωC 
Phase shift of +90º

Impedance of the capacitor (also called reactance).

Voltage-Current Relationship in a Capacitor

Current and voltage in a capacitor are not in phase with each
other. For sinusoidal waves, the voltage across an capacitor
lags the current through it by 90º. (In other words, the
current leads the voltage by 90º.) For instance, in the diagram
below, the tall purple waveform represents the current through
a capacitor, and the short blue waveform represents the voltage
across a capacitor.

Engr228 - Chapter 6, Nillson 9E 11

 Power and Energy in a Capacitor

Power in a capacitor

dvC
pC = vC iC = CvC
dt

Energy is stored in the electric field in a capacitor and can be

recovered by the circuit
t
wC (t ) = ∫ pdt
t0

1 2
wC (t ) = CvC
2

Capacitor Example
Find the maximum energy stored in the capacitor in the
circuit below, and the energy dissipated in the resistor over
the interval 0 < t < 500 ms.

1 2
wC = Cv = 0.1sin 2 2πt
2
wCmax = 0.1 J

v2
pR = = 0.01sin 2 2πt
R
0.5
wR = ∫
0
0.01sin 2 2πt dt = 2.5 mJ

Engr228 - Chapter 6, Nillson 9E 12

 Inductor Combinations

Leq = L1 + L2 + K + LN

(d)
(c)
1 1 1 1
= + +K+
Leq L1 L2 LN

Capacitor Combinations

(c)

(d)

1 1 1 1 Ceq = C1 + C 2 + K + C N
= + +K+
C eq C1 C 2 CN

Engr228 - Chapter 6, Nillson 9E 13

 Application of Capacitors

Application of Capacitors

Engr228 - Chapter 6, Nillson 9E 14

 Chapter 6 Summary

• Showed how to calculate voltage, current, power, and

energy in inductors and capacitors;
• Showed that current must be continuous in an inductor
and voltage must be continuous in a capacitor;
• Showed how to correctly combine inductors in series
and parallel to form a single equivalent inductor;
• Showed how to correctly combine capacitors in series
and parallel to form a single equivalent capacitor.

Engr228 - Chapter 6, Nillson 9E 15