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 +90o), while a
capacitive reactance translates into a negative imaginary impedance
(impedance at -90o). Resistance, of course, is still regarded
as a purely "real" impedance (polar angle of 0o):
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 0o. 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 -90o, 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 r-l-c 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 7.773E-02
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
hand-calculated 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:
ZTotal = Z1 + Z2 + . . . Zn
- 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.
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