Some examples with AC circuits
Let's connect three AC voltage sources in
series and use complex numbers to determine additive
voltages. All the rules and laws learned in the study of DC
circuits apply to AC circuits as well (Ohm's Law,
Kirchhoff's Laws, network analysis methods), with the
exception of power calculations (Joule's Law). The only
qualification is that all variables must be expressed
in complex form, taking into account phase as well as
magnitude, and all voltages and currents must be of the same
frequency (in order that their phase relationships remain
constant).
The polarity marks for all three voltage
sources are oriented in such a way that their stated
voltages should add to make the total voltage across the
load resistor. Notice that although magnitude and phase
angle is given for each AC voltage source, no frequency
value is specified. If this is the case, it is assumed that
all frequencies are equal, thus meeting our qualifications
for applying DC rules to an AC circuit (all figures given in
complex form, all of the same frequency). The setup of our
equation to find total voltage appears as such:
Graphically, the vectors add up in this
manner:
The sum of these vectors will be a resultant
vector originating at the starting point for the 22 volt
vector (dot at upper-left of diagram) and terminating at the
ending point for the 15 volt vector (arrow tip at the
middle-right of the diagram):
In order to determine what the resultant
vector's magnitude and angle are without resorting to
graphic images, we can convert each one of these polar-form
complex numbers into rectangular form and add. Remember,
we're adding these figures together because the
polarity marks for the three voltage sources are oriented in
an additive manner:
In polar form, this equates to 36.8052 volts
∠ -20.5018o. What this means in real terms is
that the voltage measured across these three voltage sources
will be 36.8052 volts, lagging the 15 volt (0o
phase reference) by 20.5018o. A voltmeter
connected across these points in a real circuit would only
indicate the polar magnitude of the voltage (36.8052 volts),
not the angle. An oscilloscope could be used to display two
voltage waveforms and thus provide a phase shift
measurement, but not a voltmeter. The same principle holds
true for AC ammeters: they indicate the polar magnitude of
the current, not the phase angle.
This is extremely important in relating
calculated figures of voltage and current to real circuits.
Although rectangular notation is convenient for addition and
subtraction, and was indeed the final step in our sample
problem here, it is not very applicable to practical
measurements. Rectangular figures must be converted to polar
figures (specifically polar magnitude) before they
can be related to actual circuit measurements.
We can use SPICE to verify the accuracy of
our results. In this test circuit, the 10 kΩ resistor value
is quite arbitrary. It's there so that SPICE does not
declare an open-circuit error and abort analysis. Also, the
choice of frequencies for the simulation (60 Hz) is quite
arbitrary, because resistors respond uniformly for all
frequencies of AC voltage and current. There are other
components (notably capacitors and inductors) which do not
respond uniformly to different frequencies, but that is
another subject!
ac voltage addition
v1 1 0 ac 15 0 sin
v2 2 1 ac 12 35 sin
v3 3 2 ac 22 -64 sin
r1 3 0 10k
.ac lin 1 60 60
I'm using a frequency of 60 Hz
.print ac v(3,0) vp(3,0) as a default value
.end
freq v(3) vp(3)
6.000E+01 3.681E+01 -2.050E+01
Sure enough, we get a total voltage of 36.81
volts ∠ -20.5o (with reference to the 15 volt
source, whose phase angle was arbitrarily stated at zero
degrees so as to be the "reference" waveform).
At first glance, this is counter-intuitive.
How is it possible to obtain a total voltage of just over 36
volts with 15 volt, 12 volt, and 22 volt supplies connected
in series? With DC, this would be impossible, as voltage
figures will either directly add or subtract, depending on
polarity. But with AC, our "polarity" (phase shift) can vary
anywhere in between full-aiding and full-opposing, and this
allows for such paradoxical summing.
What if we took the same circuit and
reversed one of the supply's connections? Its contribution
to the total voltage would then be the opposite of what it
was before:
Note how the 12 volt supply's phase angle is
still referred to as 35o, even though the leads
have been reversed. Remember that the phase angle of any
voltage drop is stated in reference to its noted polarity.
Even though the angle is still written as 35o,
the vector will be drawn 180o opposite of what it
was before:
The resultant (sum) vector should begin at
the upper-left point (origin of the 22 volt vector) and
terminate at the right arrow tip of the 15 volt vector:
The connection reversal on the 12 volt
supply can be represented in two different ways in polar
form: by an addition of 180o to its vector angle
(making it 12 volts ∠ 215o), or a reversal of
sign on the magnitude (making it -12 volts ∠ 35o).
Either way, conversion to rectangular form yields the same
result:
The resulting addition of voltages in
rectangular form, then:
In polar form, this equates to 30.4964 V ∠
-60.9368o. Once again, we will use SPICE to
verify the results of our calculations:
ac voltage addition
v1 1 0 ac 15 0 sin
v2 1 2 ac 12 35 sin Note the reversal of node numbers 2 and 1
v3 3 2 ac 22 -64 sin to simulate the swapping of connections
r1 3 0 10k
.ac lin 1 60 60
.print ac v(3,0) vp(3,0)
.end
freq v(3) vp(3)
6.000E+01 3.050E+01 -6.094E+01
-
REVIEW:
-
All the laws and rules of DC circuits
apply to AC circuits, with the exception of power
calculations (Joule's Law), so long as all values are
expressed and manipulated in complex form, and all
voltages and currents are at the same frequency.
-
When reversing the direction of a vector
(equivalent to reversing the polarity of an AC voltage
source in relation to other voltage sources), it can be
expressed in either of two different ways: adding 180o
to the angle, or reversing the sign of the magnitude.
-
Meter measurements in an AC circuit
correspond to the polar magnitudes of calculated
values. Rectangular expressions of complex quantities in
an AC circuit have no direct, empirical equivalent,
although they are convenient for performing addition and
subtraction, as Kirchhoff's Voltage and Current Laws
require.
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