Series R, L, and C
Let's take the following example circuit and
analyze it:
The first step is to determine the
reactances (in ohms) for the inductor and the capacitor.
The next step is to express all resistances
and reactances in a mathematically common form: impedance.
Remember that an inductive reactance translates into a
positive imaginary impedance (or an impedance at +90^{o}),
while a capacitive reactance translates into a negative
imaginary impedance (impedance at 90^{o}).
Resistance, of course, is still regarded as a purely "real"
impedance (polar angle of 0^{o}):
Now, with all quantities of opposition to
electric current expressed in a common, complex number
format (as impedances, and not as resistances or reactances),
they can be handled in the same way as plain resistances in
a DC circuit. This is an ideal time to draw up an analysis
table for this circuit and insert all the "given" figures
(total voltage, and the impedances of the resistor,
inductor, and capacitor).
Unless otherwise specified, the source
voltage will be our reference for phase shift, and so will
be written at an angle of 0^{o}. Remember that there
is no such thing as an "absolute" angle of phase shift for a
voltage or current, since it's always a quantity relative to
another waveform. Phase angles for impedance, however (like
those of the resistor, inductor, and capacitor), are known
absolutely, because the phase relationships between voltage
and current at each component are absolutely defined.
Notice that I'm assuming a perfectly
reactive inductor and capacitor, with impedance phase angles
of exactly +90 and 90^{o}, respectively. Although
real components won't be perfect in this regard, they should
be fairly close. For simplicity, I'll assume perfectly
reactive inductors and capacitors from now on in my example
calculations except where noted otherwise.
Since the above example circuit is a series
circuit, we know that the total circuit impedance is equal
to the sum of the individuals, so:
Inserting this figure for total impedance
into our table:
We can now apply Ohm's Law (I=E/R)
vertically in the "Total" column to find total current for
this series circuit:
Being a series circuit, current must be
equal through all components. Thus, we can take the figure
obtained for total current and distribute it to each of the
other columns:
Now we're prepared to apply Ohm's Law (E=IZ)
to each of the individual component columns in the table, to
determine voltage drops:
Notice something strange here: although our
supply voltage is only 120 volts, the voltage across the
capacitor is 137.46 volts! How can this be? The answer lies
in the interaction between the inductive and capacitive
reactances. Expressed as impedances, we can see that the
inductor opposes current in a manner precisely opposite that
of the capacitor. Expressed in rectangular form, the
inductor's impedance has a positive imaginary term and the
capacitor has a negative imaginary term. When these two
contrary impedances are added (in series), they tend to
cancel each other out! Although they're still added
together to produce a sum, that sum is actually less
than either of the individual (capacitive or inductive)
impedances alone. It is analogous to adding together a
positive and a negative (scalar) number: the sum is a
quantity less than either one's individual absolute value.
If the total impedance in a series circuit
with both inductive and capacitive elements is less than the
impedance of either element separately, then the total
current in that circuit must be greater than what it
would be with only the inductive or only the capacitive
elements there. With this abnormally high current through
each of the components, voltages greater than the source
voltage may be obtained across some of the individual
components! Further consequences of inductors' and
capacitors' opposite reactances in the same circuit will be
explored in the next chapter.
Once you've mastered the technique of
reducing all component values to impedances (Z), analyzing
any AC circuit is only about as difficult as analyzing any
DC circuit, except that the quantities dealt with are vector
instead of scalar. With the exception of equations dealing
with power (P), equations in AC circuits are the same as
those in DC circuits, using impedances (Z) instead of
resistances (R). Ohm's Law (E=IZ) still holds true, and so
do Kirchhoff's Voltage and Current Laws.
To demonstrate Kirchhoff's Voltage Law in an
AC circuit, we can look at the answers we derived for
component voltage drops in the last circuit. KVL tells us
that the algebraic sum of the voltage drops across the
resistor, inductor, and capacitor should equal the applied
voltage from the source. Even though this may not look like
it is true at first sight, a bit of complex number addition
proves otherwise:
Aside from a bit of rounding error, the sum
of these voltage drops does equal 120 volts. Performed on a
calculator (preserving all digits), the answer you will
receive should be exactly 120 + j0 volts.
We can also use SPICE to verify our figures
for this circuit:
ac rlc circuit
v1 1 0 ac 120 sin
r1 1 2 250
l1 2 3 650m
c1 3 0 1.5u
.ac lin 1 60 60
.print ac v(1,2) v(2,3) v(3,0) i(v1)
.print ac vp(1,2) vp(2,3) vp(3,0) ip(v1)
.end
freq v(1,2) v(2,3) v(3) i(v1)
6.000E+01 1.943E+01 1.905E+01 1.375E+02 7.773E02
freq vp(1,2) vp(2,3) vp(3) ip(v1)
6.000E+01 8.068E+01 1.707E+02 9.320E+00 9.932E+01
The SPICE simulation shows our
handcalculated results to be accurate.
As you can see, there is little difference
between AC circuit analysis and DC circuit analysis, except
that all quantities of voltage, current, and resistance
(actually, impedance) must be handled in complex
rather than scalar form so as to account for phase angle.
This is good, since it means all you've learned about DC
electric circuits applies to what you're learning here. The
only exception to this consistency is the calculation of
power, which is so unique that it deserves a chapter devoted
to that subject alone.

REVIEW:

Impedances of any kind add in series: Z_{Total}
= Z_{1} + Z_{2} + . . . Z_{n}

Although impedances add in series, the
total impedance for a circuit containing both inductance
and capacitance may be less than one or more of the
individual impedances, because series inductive and
capacitive impedances tend to cancel each other out. This
may lead to voltage drops across components exceeding the
supply voltage!

All rules and laws of DC circuits apply to
AC circuits, so long as values are expressed in complex
form rather than scalar. The only exception to this
principle is the calculation of power, which is
very different for AC.
