Alternating Current (AC) Circuits
Dr Miguel Cavero
August 19, 2014
AC Circuits
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AC Versus DC
Many electrical appliances use DC current (e.g. computers). The
current always flows in the same direction from the terminals of the
battery.
Current from a power plant is alternating current (AC).
Mains electricity in South Africa is given by 230 V, 50 Hz AC power.
The main reasons for the choice of AC are due to the transmission
over long distances.
Transmission is more efficient when the current is low. (Recall that
power in a conductor is I 2 R.)
It is easier and more efficient to change the voltage in AC power
with the use of a transformer, thereby using high voltages to
transmit power over long distances.
AC Circuits
AC Circuits
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Power Transmission
Consider a power station that can generate a million watts (1 MW) or
power.
The power could be transmitted as follows: (a) 1 MA at 1 V, or (b) 1 A at
1 MV.
Which wire would need to be bigger, the wire in (a) or the wire in (b)?
The wire in (a) would be very small compared to that in (b). The power
loss through the wire would also be small, since the current is less
than in (b).
Power companies use transformers to raise the voltage for
transmission and then drop the voltage again for distribution to a
house or business.
AC Circuits
AC Circuits
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AC Sources
A source of alternating emf or voltage is needed to supply an
alternating current in a circuit.
A simple example is coild of wire rotating at a constant rate in a
magnetic field. This produces an emf that varies sinusoidally.
This is a basic form of an alternating-current generator, or alternator.
The symbol for the AC source (or alternating voltage/emf source) in a
circuit is
AC Circuits
AC Circuits
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Expression For An Alternating Current
Since the voltage source in an AC circuit varies sinusoidally, the
alternating current can be written as
I = I0 cos ωt
This graph represents a sine curve. It does not matter whether the
current is expressed in terms of a sine or cosine function.
AC Circuits
AC Circuits
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Expression For An Alternating Current
I = I0 cos ωt
I is the instantaneous current at time t, and I0 is the peak current.
ω is the angular frequency, where ω = 2πf .
AC Circuits
AC Circuits
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Expression For An Alternating Voltage
The alternating voltage is written as
V = V0 cos (ωt + α)
where V0 is the peak voltage, and α is the phase angle, between the
current and the voltage.
Depending on the sign of α, the the voltage in a component is either
ahead or behind the current.
For example, when V = V0 cos (ωt + 53◦ ), the voltage is ahead of (or
leads) the current by 53◦ .
When V = V0 cos (ωt − 53◦ ), the voltage is behind (or lags) the current
by 53◦ .
AC Circuits
AC Circuits
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Phase Angle
Consider a sine and a cosine function:
Depending on where the initial point is taken, the sine curve is ahead
of the cosine curve by 90◦ , or by one quarter-cycle.
AC Circuits
AC Circuits
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Resistors In AC Circuits
The voltage across a resistor in an AC circuit is
VR = (I0 cos ωt) R
= V0R cos ωt
where V0R = I0 R is the peak voltage across the resistor.
Both I in the resistor and VR across it are proportional to cos ωt.
Therefore they are in phase with one another.
AC Circuits
Resistance and Reactance
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Capacitors In AC Circuits
The voltage across a capacitor in an AC circuit can be shown to be
VC = V0C cos (ωt − π/2)
where V0C is the peak voltage across the capacitor.
The voltage across the capacitor lags the current through it by
π/2 = 90◦ .
AC Circuits
Resistance and Reactance
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Inductance
An inductor is generally a coil of wire, containing many closely-spaced
turns. Many inductors have a magnetic core (such as iron) inside the
coil.
Inductors are widely used in AC circuits, for example as filters to
separate signals at different frequencies.
The symbol for an inductor is
An inductor has a property called inductance, which is the ratio of
voltage to the rate of change of current.
AC Circuits
Resistance and Reactance
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Inductance
When there is a current flowing in an inductor, energy is stored in the
magnetic field in the coil.
When the current changes, the magnetic field induces an emf in the
coil, which opposes the change in the current.
(This is a consequence of Faraday’s Law of electromagnetic induction.)
The induced emf is sometimes called a ”back emf” because of its
opposition to the change in the current.
When a current increases in the coil, the induced emf opposes the
original current and slows down the current increase. When the current
decreases, the induced emf slows down the decrease in the current.
The inductance L is given by
E L=
dI/dt The SI unit of inductance is the henry, H (1 H = 1 V s A−1 ).
AC Circuits
Resistance and Reactance
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Inductors In AC Circuits
The voltage across a inductor in an AC circuit can be shown to be
VL = V0L cos (ωt + π/2)
where V0L is the peak voltage across the inductor.
The voltage across the inductor leads the current through it by
π/2 = 90◦ .
AC Circuits
Resistance and Reactance
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Peak Voltages
The peak voltages across each component are
V0R = I0 R
1
V0C = I0
ωC
V0L = I0 ωL
The ratio of voltage and current for each component is a constant,
given by
V0
I0
V0
I0
V0
I0
AC Circuits
= R
=
resistance
1
= XC
ωC
= ωL = XL
capacitative reactance
inductive reactance
Resistance and Reactance
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Resistance And Reactance
Note that the capacitative reactance XC and the inductive reactance
XL both have units [V]/[A] = [Ω].
The capacitative reactance XC is the opposition to the flow of charge
through it as a capacitor charges/discharges.
The inductive reactance XL is the opposition to any change in current
due to the induced emf in the coil (inductor).
The combined opposition to current in an AC circuit (composed of
resistors, capacitors and inductors) is called the impedance Z.
q
Z = R2 + (XL − XC )2
AC Circuits
Resistance and Reactance
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