[Prev][Next][Index][Thread]
NEON- AC Circuit Theory - Part 2
Greetings all,
Welcome to part 2 of this very short course in AC circuit theory. Hopefully,
everyone understood the previous DC theory post. If not, do like we did in
engineering school: pretend you did, and press on. Sometimes understanding of
this stuff comes as a sudden flash of insight; sometimes we just drown in the
mud. Pressing on...
I'm trying something I hope will work in my attempts to bypass the agony of
ASCII graphics. I have attached to this post a uuencoded .GIF file of some
waveform plots. I sent a copy to myself and it arrived as a separate post.
Hopefully it will arrive (!) and hopefully, the .GIF format is acceptable to you
all (I could have uuencoded .JPG - it was a toss-up). The original filename was
plots.gif, in case your mail program squirrels it away somewhere in the deep
recesses of your hard drive.
Part 2 - AC theory.
In this session we will be exploring how our resistive, capacitive, and
inductive components react to a time-varying voltage waveform. This is part of
the definition and distinction between DC and AC - the voltage changes with
time. In this discussion, we will concentrate on a very specific type of
voltage changing with time. This is refered to as SINUSOIDAL, otherwise known
as a sine wave. (This is different from a sign wave, which is what some of the
poorer installations I've seen do in a stiff breeze, but I digress).
A sinusiodal waveshape occurs frequently in nature and in mechanical systems.
Consider a flywheel, rotating on a shaft at a constant speed. If we put a paint
dot somewhere on the outer circumference of the flywheel, and then plot the
height of the paint dot vertically above and below an imaginary line passing
horizontally through the center shaft, versus time, you get a sinusoidal
waveform. See the top plot in the attached .gif.
It just so happens that the AC voltage coming out of the wall socket also varies
sinusoidally with time. This makes it possible to analyze circuits connected to
it in a consistent mathematical manner. We will assume, to simplify our
analysis, that when describing resulting waveforms, any applied sinusoidal
voltages have been applied for a long period of time and all possible transient
effects have died out.
a) Resistive Loads. As we know from the previous session, the current passing
through a resistor is a function of the voltage applied across it. This is
expressed in a variation of Ohm's Law: I = E / R. If R is constant (not true
for lightbulbs, by the way), then the current varies timewise as the voltage
does. If we were to plot the current passing through a resistor versus time, on
the same plot with the voltage waveform, and scale one waveform so that it has
the same heigth at the highest point as the other waveform, we would find that
they exactly overlap. See the topmost plot in the .gif file.
b) Capacitive Loads. Now suppose we apply sinusoidally varying voltage to our
capacitor (we assume it's a non-polarized type which can take both positive and
negative voltage.) What happens? We know, from the previous session, that the
current flowing through a capacitor is a function of the RATE OF CHANGE of the
voltage applied across it. This means the maximum current flow would occur at
that point on the voltage waveform where the voltage is varying the fastest.
Where is this point? Why, for a sinusoid, it is where the voltage crosses the
zero axis. The polarity of the current is determined by the direction the
voltage is moving: upwards on the plot results in a positive current:
downwards results in a negative current. If we were to plot the current versus
time, it would result in the middle plot in the .gif file. Mathematically, this
is interesting because it just so happens that the current waveform is also
sinusoidal, but timeshifted from the voltage waveform. As we can see, the
current waveform LEADS the voltage waveform by 25 percent of the waveform
period. Expressed in degrees, we can say that the current waveform leads the
voltage by 90 degrees. By the way, this waveform is also known as a cosine
waveform.
c) Inductive Loads. When we discussed inductors last time, we glossed over a
detail about their operation, which will now become important. When an inductor
experiences a CHANGING magnetic field, a voltage is induced in it's windings
(hence the name 'inductor'). This changing magnetic field can be from an
external source, as occurs in a generator, or internally, from a changing
current flowing through it's windings. If we apply a changing voltage to the
windings of an inductor, a current will start to flow through it's windings.
This (changing) current will induce a voltage across the windings, which will
equal and oppose the previously applied voltage. This is what keeps the
inductor current from becomming incredibly huge (see answers to DC quiz, below).
If we make the applied voltage a sinusoid, the resulting current will also be a
sinusoid. We know, from previous discussion, that the current will be changing
the fastest when the applied voltage is highest. The polarity of the voltage is
determined by the direction the current is moving. Refer to the lower plot in
the .gif. We see that, like the capacitor, the current and voltage waveforms
are out of phase - this time, however, the current LAGS the voltage by 90
degrees.
End of part 2. Part 3 (putting the pieces together) tomorrow.
-------------------------------
Answers to previous quiz:
Q1) Assuming that the characteristics of a capacitor are as previously stated,
what happens if we connect it to the 'constant current source', as described in
the inductor example? What would the ammeter and voltmeter readings show?
ANS: applying a constant current to a capacitor will result in a linearly
increasing voltage. The ammeter will read a steady 1 amp; the voltmeter reading
will continually increase. The voltage will increase until either the capacitor
breaks down or the current source can't increase it's voltage any further. In a
race like this, don't bet on the capacitor.
Q2) Assuming that the characteristics of an inductor are as previously stated,
what happens if we connect it to the 12 volt car battery, as described in the
capacitor example? What would the ammeter and voltmeter readings show?
ANS: Applying a constant voltage to an inductor will result in a linearly
increasing current. The voltmeter will read a steady 12 volts; the ammeter
reading will continuously increase. The current will increase until either
something melts, burns out, or the battery goes dead. In this race, don't bet
on the inductor.
3) Would it be prudent to try the above tests (Yes or No)? Why?
ANS: No, unless you like fireworks. These modes of operation generally result
in failure of some kind. For instance, in a solid state neon 'transformer',
failure of the driver circuitry may apply a constant voltage across the
transformer winding (an inductor). Now you understand what happens, and why.
By the way, the results to the quiz answers are theoretical - in the real world,
there is resistance in both circuits which will change the actual result a bit,
especially in the inductor question.
-------------------------------
New quiz: Suppose we have an inductor of such a size that, when 120 volts AC is
applied to it, it draws a current of 1 amp. Suppose we also have a capacitor of
such a size that, when 120 volts AC is applied to it, it also draws a current of
1 amp. What would we measure on an ammeter if we connected both of them, in
parallel, to the 120 VAC source at once? (The ammeter is on one of the wires
going from the wall socket to the paralleled inductor and capacitor.)
-------------------------------
Regards,
Telford Dorr
Answer to be posted later.