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Lecture 7
CAPACITORS AND INDUCTORS

Capacitance
Capacitor
◦ Stores charge
◦ Two conductive plates separated by insulator
◦ Insulating material called dielectric
◦ Conductive plates can become charged with opposite
charges
3

Definition of Capacitance
Amount of charge Q that a capacitor can store depends
on applied voltage
Relationship between charge and voltage given by
Q = CV
C = Q/V (farads, F)
C = farads (F)
Q = coulombs (C)
V = volts (V)
7

Definition of Capacitance
C is capacitance of the capacitor
Unit is the farad (F)
Capacitance of a capacitor
◦ One farad if it stores one coulomb of charge
◦ When the voltage across its terminals is one volt
8

Voltage Breakdown
If voltage is increased enough, dielectric breaks down
This is dielectric strength or breakdown voltage
9

Types of Capacitors
Fixed capacitors
Variable capacitors
10

Energy Stored in a Capacitor
A capacitor does not dissipate power, When power is
transferred to a capacitor
Stored as energy
12
2
2
1
CV

Energy

Capacitor Current and Voltage
13

Capacitor Charging
Capacitor voltage has shape shown:
15

Capacitor Charging Equations
Equation for voltage across the capacitor as a function
of time is
17

The Time Constant
Rate at which a capacitor charges depends on product of R
and C
Product known as time constant
 = RC
 (Greek letter tau) has units of seconds
18

Capacitor Charging – Capacitor
Voltage
19
The transient or charging phase of the capacitor has
Essentially ended after five time constants

Capacitor Charging – Capacitor
Current
20
The current of a capacitor dc network is essentially
Zero amperes after five time constants of the charging
Phase have passed

Capacitor Discharging
Here are the decay waveforms:
22

Capacitor Discharging
Assume capacitor has E volts across it when it
begins to discharge
Current will instantly jump to –E/R
Both voltage and current will decay exponentially
to zero
23

Capacitor Discharging Equations
If a capacitor is charged to voltage V0 and then discharged, the
equations become
24


/
0
/
0
t
C
t
C
e
R
V
i
e
V
v






Capacitor Discharge Equations
Current is negative because it flows opposite
to reference direction
Discharge transients last five time constants
All voltages and currents are at zero when
capacitor has fully discharged
25

Inductors
A magnetic field in space surrounding
current carrying conductor.
A common form of an inductor is a coil of
wire.
Inductor’s effect is to slow the build-up and
collapse of the current and oppose its
changes
26

Applications of Inductors
Radio tuning circuits.
Part of the ballast circuit in fluorescent lights to
limit current while tuning on
In power systems, they are part of the protection
circuitry used to control short-circuit currents
during faults.
28

Inductance and Steady State DC
Voltage across an inductance with constant dc current is
zero
Since it has current but no voltage, it looks like a short
circuit at steady state
For non-ideal inductors Resistance of windings must be
considered
30

Energy Stored by an Inductance
When power flows into an inductor
◦Energy is stored in its magnetic field
When the field collapses
◦Energy returns to the circuit
31

Introduction to Phasors
A phasor is a rotating vector whose projection on a vertical axis can be
used to represent sinusoidally varying quantities.

Phase Difference
Phase difference refers to the angular displacement between different
waveforms of the same frequency.

The terms lead and lag can be understood in terms of phasors.
By definition, the waveform generated by the leading phasor leads the
waveform generated by the lagging phasor and vice versa.
Im leads
phasor Vm;
thus current
i(t) leads
voltage v(t).

Complex Number Review
Complex Numbers in ac Analysis
R, L, and C Circuits with Sinusoidal Excitation
◦Resistance and Sinusoidal ac
◦Inductance and Sinusoidal ac
◦Capacitance and Sinusoidal ac
The Impedance Concept

Complex Number Review
C = a + ib
where a and b are real numbers and i =√ -1.
It has a real part (a) and imaginary part (b).
In circuit theory, j is used to denote the imaginary
component rather than i to avoid confusion with
current i.

Geometrical Representation
Magnitude - length

Conversion between
Rectangular and Polar Forms
Magnitude and angle

Complex Numbers in ac
Analysis
ac voltages and currents can be represented as phasors
Phasors have magnitude and angle
They can be viewed as complex numbers.

Circuit Relationships in the
Phasor Domain
KVL & KCL apply in the time domain and in the phasor domain
Summing ac Voltages and Currents

Representing Phasors as RMS
Values
We used Em, Vm, and Im for simplification
In practice, phasors are always expressed as rms values
because all voltmeters and ammeters display their
measurements in rms.
From now on, a phasor such as V = 240 V∠0o will be taken
to mean 240 V rms at an angle of zero degrees.
To convert this to a time function, first we multiply the rms
value by √2 then we follow the usual procedure.

R, L, and C Circuits with
Sinusoidal Excitation
In general:
◦ Resistance opposes current
◦ inductance opposes changes in current
◦ capacitance opposes changes in voltage
We now investigate these relationships for the case
of sinusoidal ac.

It is important to know:
1. When a circuit consisting of linear circuit
elements R, L, and C is connected to a sinusoidal
source, all currents and voltages in the circuit
will be sinusoidal and of the same frequency as
the source.
2. Because of the rule above, we need only to
determine magnitudes and phase angles to
complete the solution.

In a purely resistive circuit:
◦ Ohm’s Law applies
◦ Current is proportional to the voltage
◦ Voltage and current of a resistor are in phase

Inductance and Sinusoidal AC
Voltage of an inductor
◦ Proportional to rate of change of current
Voltage is greatest when the rate of change (slope)
of the current is greatest
◦ Voltage and current are not in phase
◦ Phase: Voltage leads the current by 90º across an
inductor

Inductive Reactance
Denoted by XL
Represents the opposition inductance presents to current in an ac
circuit

Capacitance and Sinusoidal AC
For capacitance:
◦ Current is proportional to rate of change of
voltage
Current is greatest when rate of change of voltage
is greatest
◦ Voltage and current are out of phase
For a capacitor:
◦ Current leads the voltage by 90º

Capacitive Reactance
Denoted by XC
Represents opposition that capacitance presents to current in an ac
circuit

For capacitance, current always leads voltage by 90o

Impedance
◦ Denoted by Z
◦ The opposition that a circuit element presents to current
◦  is the phase difference between voltage and current

Once impedance is known:

Resistance
For a pure resistance:
◦ Voltage and current are in phase
If the voltage has a phase angle, the current has the same angle
The impedance of a resistor is equal to R0º

Inductance
For an inductor:
Voltage leads current by 90º
If voltage has an angle of 0º:
◦ Current has an angle of -90º
Impedance of an inductor
XL90º

Capacitance
For a capacitor
Current leads the voltage by 90°
If the voltage has an angle of 0°
◦ Current has an angle of 90°
Impedance of a capacitor
XC-90º

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