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 an inductor.
• Know and be able to use the equations for voltage,
current, power, and energy in a capacitor.
• Know that current must be continuous in an inductor
and voltage must be continuous in a capacitor.
• Be able to correctly combine inductors with initial
conditions in series and parallel to form a single
equivalent inductor with an initial condition.
• Be able to correctly combine capacitors with initial
conditions in series and parallel to form a single
equivalent capacitor with an initial condition.
Engr228 - Chapter 6, Nillson 9E
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Real Inductors
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. This
property also is called self inductance to discriminate it from mutual
inductance, describing the voltage induced in one electrical circuit by
the rate of change of the electric current in another 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. To add inductance to a circuit,
electronic components called inductors are used, typically consisting of
coils of wire to concentrate the magnetic field and to collect the
induced voltage.
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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
vL (t ) = L
Engr228 - Chapter 6, Nillson 9E
di (t )
dt
3
DC Characteristics of Inductors
• The inductor acts like a “short circuit” at DC because the
rate of current change is equal to zero.
vL (t ) = L
di (t )
dt
Inductor Example
From the circuit shown
at the right, find i(t).
v(t ) = L
di (t )
dt
1
1
v
(
t
)
dt
=
A sin ωtdt
L∫
L∫
A
A  − cos ωt 
= ∫ sin ωtdt = 

L
L ω 
π
A
A
=
(− cos ωt ) =
(sin ωt − )
ωL
ωL
2
i (t ) =
Engr228 - Chapter 6, Nillson 9E
4
Voltage-Current Relationship in an Inductor
i (t ) =
A
π
(sin ωt − )
ωL
2
Phase shift of -90º
i (t ) =
A
(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.
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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. Do you agree that the voltage waveform
leads the current waveform by 90º?
Power in an Inductor
p = vL iL = LiL
diL
dt
• Instantaneous power is measured in Watts (W).
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6
Energy Storage in an Inductor
• Energy is the integral of power over a time interval.
• Energy is stored in the magnetic field surrounding the
inductor.
• Energy can be recovered by the circuit.
• Energy is measured in Joules (J).
t
wL (t ) = ∫ pdt
t0
wL (t ) =
1
2
LiL
2
Inductor Example
• Find the maximum energy stored in the inductor.
wL =
1 2
πt
Li = 216 sin 2
2
6
p R = i 2 R = 14.4 sin 2
wR =
6
2 πt
0
6
∫ 14.4 sin
πt
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
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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
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Practical Definition of Capacitance
• In electromagnetism and electronics, 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
• The SI unit of capacitance is the farad; 1 farad is 1 coulomb per
volt.
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), microfarad (µF), nano-farad (nF), pico-farad (pF), and femto-farad (fF).
• The capacitance can be calculated if the geometry of the conductors and
the dielectric properties of the insulator between the conductors are
known. For example, 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
×
Engr228 - Chapter 6, Nillson 9E
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Capacitors
• The unit in which capacitance is measured is the Farad (F).
• 1 F = 1 Amp-Second/Volt = 1 Coulomb/Volt.
• Capacitance is related to charge (Q) through the following
equation: Q(t) = CVC(t).
• The governing voltage and current relationship is
iC (t ) = C
dvC (t )
dt
DC Characteristics of a Capacitor
• The capacitor acts like an “open circuit” at DC because
the time rate of change of voltage is zero.
iC (t ) = C
Engr228 - Chapter 6, Nillson 9E
dvC (t )
dt
10
Capacitor Example
From the circuit on the
right, find i(t).
dv(t )
d ( A sin ωt )
=C
dt
dt
= AωC (cos ωt )
i (t ) = C
=
π
A
sin(ωt + )
2
 1 


 ωC 
Phase shift of +90º
Impedance of the capacitor (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. Do you agree that the voltage waveform
lags the current waveform by 90º?
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Power and Energy in a Capacitor
• Power in a capacitor
pC = vC iC = CvC
dvC
dt
• Energy in a capacitor
t
wC (t ) = ∫ pdt
t0
1
2
wC (t ) = CvC
2
Capacitor Example
• Find the maximum energy stored in the capacitor of the
circuit below, and the energy dissipated in the resistor over
the interval 0 < t < 500 ms.
wC =
1 2
Cv = 0.1sin 2 2πt
2
wCmax = 0.1 J
pR =
wR =
∫
0.5
v2
= 0.01sin 2 2πt
R
0.01sin 2 2πt dt = 2.5 mJ
0
Engr228 - Chapter 6, Nillson 9E
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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
=
+
+K+
C eq C1 C 2
CN
Engr228 - Chapter 6, Nillson 9E
Ceq = C1 + C 2 + K + C N
13
Application of Capacitors
Application of Capacitors
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Chapter 6 Summary
• Showed how to calculate voltage, current, power, and
energy in an inductor.
• Showed how to calculate voltage, current, power, and
energy in a capacitor.
• Showed that current must be continuous in an inductor
and voltage must be continuous in a capacitor.
• Showed how to correctly combine inductors with
initial conditions in series and parallel to form a single
equivalent inductor with an initial condition.
• Showed how to correctly combine capacitors with
initial conditions in series and parallel to form a single
equivalent capacitor with an initial condition.
Engr228 - Chapter 6, Nillson 9E
15