Dr.
Modar Shbat
Division of Engineering
modar.shbat@smu.ca
2
This section introduces two more circuit elements, the capacitor and the inductor. The
constitutive equations for the devices involve either integration or differentiation.
Consequently:
Electric circuits that contain capacitors and/or inductors are represented by
differential equations. Circuits that do not contain capacitors or inductors are
represented by algebraic equations. We say that circuits containing capacitors and/or
inductors are dynamic circuits, whereas circuits that do not contain capacitors or
inductors are static circuits.
Circuits that contain capacitors and/or inductors are able to store energy.
Circuits that contain capacitors and/or inductors have memory. The voltages and
currents at a particular time depend not only on other voltages at currents at that
same instant of time but also on previous values of those currents and voltages.
In a DC circuit, capacitors act like open circuits, and inductors act like short circuits.
Series or parallel capacitors can be reduced to an equivalent capacitor. Series or
parallel inductors can be reduced to an equivalent inductor.
Electric Circuit I 3
A capacitor voltage v(t) deposits a charge +q(t) on one plate and a charge -q(t) on the
other plate. We say that the charge q(t) is stored by the capacitor. The charge stored by
a capacitor is proportional to the capacitor voltage v(t). Thus, we write:
In general, the capacitor voltage v(t) varies as a function of time. Consequently, q(t), the
charge stored by the capacitor, also varies as a function of time. The variation of the
capacitor charge with respect to time implies a capacitor current i(t), given by:
Electric Circuit I 4
Charge on a Capacitor:
Electric Circuit I 5
Charge on a Capacitor:
Charging & Discharging of a
Capacitor
Electric Circuit I 6
Example:
The figure shows a circuit together with two plots. The plots represent the current and
voltage of the capacitor in the circuit. Determine the value of the capacitance of the
capacitor.
Solution:
Electric Circuit I 7
Let us consider the parallel connection of N
capacitors as shown in the figure:
The equivalent capacitance of a set of N parallel capacitors is simply the sum of the
individual capacitances. It must be noted that all the parallel capacitors will have the
same initial condition v(0).
Electric Circuit I 8
Now let us determine the equivalent capacitance
Cs of a set of N series-connected capacitances:
For the case of two series capacitors:
Electric Circuit I 9
Electric Circuit I 10
Capacitors that are connected to a sinusoidal supply (AC)
produce reactance from the effects of supply frequency
and capacitor size.
If we apply an alternating current or AC supply, the capacitor will alternately charge and
discharge at a rate determined by the frequency of the supply. Then the Capacitance in
AC circuits varies with frequency as the capacitor is being constantly charged and
discharged. The least voltage rate-of-change occurs when
the AC sine wave crosses over at its maximum
positive peak ( +VMAX ) and its minimum
negative peak, ( -VMAX ). At these two positions
within the cycle, the sinusoidal voltage is
constant, therefore its rate-of-change is zero, so
dv/dt is zero, resulting in zero current change
within the capacitor. Thus when dv/dt = 0, the
capacitor acts as an open circuit, so i = 0.
Electric Circuit I 11
Capacitive Reactance in a purely capacitive circuit is the opposition to current flow in
AC circuits. Like resistance, reactance is also measured in Ohm’s but is given the
symbol X to distinguish it from a purely resistive value. As reactance is a quantity that
can also be applied to Inductors as well as Capacitors, when used with capacitors it is
more commonly known as Capacitive Reactance.
Where: f is in Hertz and C is in Farads, 2πƒ (Omega, ω) is used
to denote an angular frequency (radial frequency).
The capacitive reactance of the capacitor decreases as the
frequency across it increases therefore capacitive reactance is
inversely proportional to frequency.
As the frequency increases the current flowing through the
capacitor increases in value because the rate of voltage
change across its plates increases. 12
Then we can see that at DC a capacitor has infinite reactance (open-circuit), at very high
frequencies a capacitor has zero reactance (short-circuit).
Example:
Find the reactance of a capacitor in an AC capacitive circuit when C= 4μF and f= 60Hz
Electric Circuit I 13
An Inductor is a passive electrical component consisting
of a coil of wire which is designed to take advantage of
the relationship between magnetism and electricity as a
result of an electric current passing through the coil.
An inductor is a circuit element that stores energy
in a magnetic field.
Inductors are represented by a parameter called the inductance. The inductance of the
inductor is given by:
where N is the number of turns, A is the cross-sectional area of the
core in square meters; l the length of the winding in meters; and μ
is a property of the magnetic core known as the permeability. The
unit of inductance is called henry (H) in honor of the American
physicist Joseph Henry.
Electric Circuit I 14
A current source is used to cause a coil current i(t). We
find that the voltage v(t) across the coil is proportional
to the rate of change of the coil current. That is,
Integrating both sides of the last form gives us the
current:
The time t0 is called
the initial time, and the
inductor current i(t0) is
called initial condition.
Frequently, we select
t0 = 0 as the initial time.
Electric Circuit I 15
Example:
Solution:
Electric Circuit I 16
Example:
Solution:
Electric Circuit I 17
Energy Stored in an Inductor
Note that for all i(t), so the inductor is a passive element. The inductor does not
generate or dissipate energy but only stores energy.
Series Inductors A series and parallel connection of inductors can
be reduced to an equivalent simple inductor.
Consider a series connection of N inductors as
shown:
Electric Circuit I 18
Now, consider the set of N inductors in parallel, the
current i is equal to the sum of the currents in the N
inductors:
Example:
Electric Circuit I 19
Inductive Reactance of a coil depends on the
frequency of the applied voltage as reactance is
directly proportional to frequency.
Just like resistance, the value of reactance is also measured in Ohm’s but is given the
symbol X, to distinguish it from a purely resistive value.
Example
A coil of inductance 150mH is connected across 50Hz
power supply. Calculate the inductive reactance of the coil.
Electric Circuit I 20
Electric Circuit I 21
