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Part 9: Basic AC Theory
9.1 Advantages Of AC Systems
Dealing with alternating current (AC) supplies is on the whole more complicated than dealing with
DC current, However there are certain advantages of AC that have lead to it being the standard for
electrical supplies:
(a)An alternating-current generator (often called an alternator) is more robust, less expensive,
requires less maintenance, and can deliver higher voltages the its DC counterpart.
(b)The power loss in a transmission lines depends on the square of the current carried (P = I
2
R). If
the voltage used is increased, the current is decreased, and losses can be made very small. The
simplest way of stepping up the voltage at the sending end of a line, and stepping it down again
at the receiving end, is to use transformers, which will only operate efficiently from AC supplies.
(c)Three-phase AC induction motors are cheap, robust and easily maintained.
(d)Energy metres, to record the amount of electrical energy used, are much simpler for AC supplies
than for DC supplies.
(e)Discharge lamps (florescent, sodium, mercury vapour etc.) operate more efficiently from AC
supplies, although filament lamps are equally effective on either type of supply.
(f) Direct-current systems are subject to severe corrosion, which is hardly present with AC supplies.
9.2 Waves
As was noted in section 6.2 a coil rotating in a magnetic field will produce an alternating current
(AC) which is made to flow by an alternating EMF. Such generators produce sine wave currents and
voltages (section 8.5) and the equations for a sine wave AC voltage source and the current it
delivers are:
e = E m
sin θ = E m
sin f 2π t = E m
sin ω t
and: i = I m
sin θ = I m
sin f 2π t = I m
sin ω t
where:
e = instantaneous EMF (V)
E
m
= maximum or peak voltage (V)
θ = coil angle relative to magnetic flux (section 6.2)
f = frequency of the supply (Hz)
t = time, (s)
ω = angular velocity of the coil rotating in the magnetic field (rads/s)
i = instantaneous current (A)
I
m
= maximum or peak current (A)
In section 8.5 we considered a stick rotating which produced sine waves but for AC generators we
are considering the coil rotating in a magnetic field. Although nearly all AC supplies are sine waves
other wave shapes are encountered occasionally. Figure 9.1 shows a simple EMF sine wave (a
current sine wave would be similar) and we can see that several values are important:
· Instantaneous values - are values at particular instants in time, and will be different form instant
to instant. Symbols for instantaneous values are lower case symbols, v for voltage and i for
current, e for EMF and so on. These values can be calculated from the above equations if the
maximum values are known.
· Maximum or Peak Values - are the greatest values reached during alternation, usually occurring
once in each half-cycle. Maximum values are indicated by U m
for voltage, I m
for current and so
on.
· Average or Mean Values - are the average value of current or voltage. If an average value is
found over a full cycle, the positive and negative half-cycles will cancel out to give a zero result
if they are identical. In such cases, it is customary to take the average value over a half-cycle.
Mean values have symbols, U av
and I av
etc.
· Root Mean Square Values (RMS) or equivalent Values - RMS values are a method of averaging
sine waves to give a DC equivalent. The heat dissipated in a DC circuit is proportional to the
current squared (P = I
2
R), the equivalent in AC is the RMS current and for a given resistance a
DC current of 1A will dissipate the same heat as an AC RMS current of 1A. The symbols used
fro RMS value are the same as DC symbols, that is U, I etc. Note that from this point on, unless
otherwise stated, all values followed by the symbol V or A (e.g. 240V and 13A) are RMS values.
In figure 9.1 we can see that the voltage becomes alternately positive and negative, meaning that the
current also alternates. In a DC circuit the current flows around the circuit in one direction only and
is always positive. In an AC circuit the current flows first in one direction and then in the other. In a
sine wave supply the current and voltage are constantly changing, this change only ceases for the
very instant that the peak values are reached; at these points the rates of change in current or voltage
are said to be zero. The maximum rate of change occurs as the wave crosses the x -axis (i.e. v or i are
zero).
The concept of AC may seem less intuitive than that of DC. In DC circuits it seems obvious that
electrons can carry energy around a circuit and that this energy can be used to power motors and
light bulbs etc. It may help to consider an analogy with a circular saw and a hand saw. The circular
saw is like DC supplies and its teeth fly around in one direction using the supplied energy to cut
through a plank. A hand saw is like AC supplies and moves the teeth of the saw backwards and
forwards using the supplied energy to cut through a plank. Either way the same amount of energy is
needed to cut through the plank.
Example
Figure 9.1 shows a sine wave for an AC supply with a peak voltage of 155.6V. Find the mean and
RMS values.
By either taking measurements from the graph or using the sin wave equations ( v = 155.6 sin θ) we
can generate values of for v and v
2
for a half-cycle:
θ
0° 15° 30° 45° 60° 75° 90° 105° 120° 135° 150° 165° Total
v (volts) 0 40 78 110 135 150 156 150 141 78 110 78 1226
v
2
0 1600 6084 12 100 18 225 22 500 24 336 22 500 18 225 12 100 6084 1600 145 354
Figure 9.1: A sine wave for a 110V AC supply, showing the maximum, RMS and mean values.
second example in section 8.7.
Example
Two 110V AC supplies, U 1
and U 2
, are 60° out of phase. Add these two supplies together.
The maximum voltage for a 110V supply, U m
= 110 ÷ 0.707 = 155.6V
The parallelogram diagram (also known as a phasor diagram), drawn to scale, is shown in figure
9.2. From it we can measure that the resulting supply, U R
, has a maximum voltage of 269.7V, lags
U
1
by 30° and leads U 2
by 30°. The RMS of U R
is:
RMS = 269.7 ×0.707 = 190.7V
In the above example we used maximum voltage values, although we could equally have use RMS
values all the way through. To do this we can use the same parallelogram (figure 9.2b), but we must
rescale it using U 1
= U
2
= 110V. Now U R
= 190.7V.
9.4 Resistive AC Circuits
Current and voltage stay in-phase in a purely resistive circuit (i.e. a circuit without inductance or
capacitance). The circuit, wave and phase diagrams are shown in figure 9.3. If the alternating
voltage of:
v = U m
sin ω t
is applied to a resistor, the instantaneous current:
t
t v
i I sinω
R
U sin ω
R
m
m
Thus:
R
U
or,usingRMSValues, I
R
U
I
m
m
Example
A 240V AC supply is connected to an 80Ω resistor. Calculate the resulting current flow.
Figure 9.2: The addition of two AC supplies, each with a voltage of 110V but 60° out of phase.
3 A
R
U
I
Note that these Current and voltage values are RMS values.
9.5 Inductive AC Circuit
As we note in section 6.5, any coil of wire carrying a current will set up a magnetic field and
consequently exhibit self inductance. Due to Lenz's law (section 6.3), if the current in the coil is
increasing, the induced EMF will oppose will oppose the supply voltage limiting the rate of
increase. Similarly, if the current in the coil is decreasing the induced EMF will try to keep it
flowing. In a DC circuit the current reaches a steady value at which point the magnetic field
becomes steady and self induction ceases, thus the induced EMF can not prevent a change in current
only slow it down. In an AC circuit the current is constantly changing and matters are more
complex.
Figure 9.3 shows the circuit, wave and phasor diagrams for a purely inductive circuit. The induced
EMF in the coil will always oppose the applied voltage because it is always trying to oppose the
change in current that the supply is causing. Figure 9.3 therefore shows the induced EMF ( e ) and the
supply voltage ( v ) out of phase by 180°. The instantaneous value of the induced EMF depends on
the rate of change of the current (section 6.5):
t
e
L( I I)
2 1
As mentioned in section 8.5, the rate of change of a value will be zero at the maximum and
minimum points on the graph, and have its greatest value when the wave crosses the x -axis (i.e. the
value is zero). Therefore, when i is at a maximum or minimum, the rate of change of i is zero and e
is zero. Also when i = 0, the rate of change of i peaks so that e reaches a maximum. Due to Lenz's
law, when the current is going positive, the EMF must oppose this change and will therefore be
negative. The current wave diagram can therefore be drawn, and figure 9.4b shows that current lags
the applied voltage by 90° and voltage leads current; it is normal the currents relationship to voltage
to be stated, therefore the form description is used. The phasor diagram (figure 9.4c) thus shows the
current under the voltage with an angle of 90° between them.
Figure 9.3: AC resistive circuit; (a) the circuit; (b) the wave diagram, v and i are in phase; (c) the phasor diagram. v is
the instantaneous voltage and i is the instantaneous current. I and U are RMS values. Note that v and i are not plotted on
the same scale.
charged at which point no more current flows (section 7.7). If the capacitor is connected to an AC
supply however, the current constantly changes direction and the capacitor will charge and
discharge accordingly. In AC circuits, although no current flows right through the capacitor, an
alternating current does exist in the circuit.
If an alternating voltage is applied to an uncharged capacitor, as the voltage passes through zero
going positive, the current will immediately reach its maximum value as the capacitor starts to
charge. As the charge increases, charging current will fall, reaching zero when the voltage becomes
steady, which it does for an instant at its maximum value. As the voltage falls, the capacitor will
discharge, and a negative current results. This pattern is shown in figure 9.5, which shows the
circuit, wave and phasor diagrams for a capacitive circuit. It is clear from these diagrams that in a
capacitive circuit current leads voltage by 90° (and so voltage lags current by 90°).
Like inductive circuits it is clear that the current is limited by a property other than resistance. This
property is called capacitive reactance (X C
). It can be shown that:
ωC
2 π C
I
U
X
C
f
where:
Xc = capacitive reactance (Ω)
U = supply voltage (V)
f = supply frequency (Hz)
I = circuit current (A)
C = capacitance (F)
ω = 2π f
With capacitance in microfarads (C'):
2 π C'
X
6
L
f
And (for circuits with capacitance only):
C
X
U
I
Figure 9.5: AC capacitive circuit; (a) the circuit; (b) the wave diagram, v and i are 90° out of phase; (c) the phasor
diagram. v is the instantaneous voltage and i is the instantaneous current. I and U are RMS values. Note that v and i are
not plotted on the same scale.
