Series resistor-capacitor circuits
In the last section, we learned what would
happen in simple resistor-only and capacitor-only AC
circuits. Now we will combine the two components together in
series form and investigate the effects.
Take this circuit as an example to analyze:
The resistor will offer 5 Ω of resistance to
AC current regardless of frequency, while the capacitor will
offer 26.5258 Ω of reactance to AC current at 60 Hz. Because
the resistor's resistance is a real number (5 Ω ∠ 0o,
or 5 + j0 Ω), and the capacitor's reactance is an imaginary
number (26.5258 Ω ∠ -90o, or 0 - j26.5258 Ω), the
combined effect of the two components will be an opposition
to current equal to the complex sum of the two numbers. The
term for this complex opposition to current is impedance,
its symbol is Z, and it is also expressed in the unit of
ohms, just like resistance and reactance. In the above
example, the total circuit impedance is:
Impedance is related to voltage and current
just as you might expect, in a manner similar to resistance
in Ohm's Law:
In fact, this is a far more comprehensive
form of Ohm's Law than what was taught in DC electronics (E=IR),
just as impedance is a far more comprehensive expression of
opposition to the flow of electrons than simple resistance
is. Any resistance and any reactance, separately or in
combination (series/parallel), can be and should be
represented as a single impedance.
To calculate current in the above circuit,
we first need to give a phase angle reference for the
voltage source, which is generally assumed to be zero. (The
phase angles of resistive and capacitive impedance are
always 0o and -90o, respectively,
regardless of the given phase angles for voltage or
current).
As with the purely capacitive circuit, the
current wave is leading the voltage wave (of the source),
although this time the difference is 79.325o
instead of a full 90o.
As we learned in the AC inductance chapter,
the "table" method of organizing circuit quantities is a
very useful tool for AC analysis just as it is for DC
analysis. Let's place out known figures for this series
circuit into a table and continue the analysis using this
tool:
Current in a series circuit is shared
equally by all components, so the figures placed in the
"Total" column for current can be distributed to all other
columns as well:
Continuing with our analysis, we can apply
Ohm's Law (E=IR) vertically to determine voltage across the
resistor and capacitor:
Notice how the voltage across the resistor
has the exact same phase angle as the current through it,
telling us that E and I are in phase (for the resistor
only). The voltage across the capacitor has a phase angle of
-10.675o, exactly 90o less than
the phase angle of the circuit current. This tells us that
the capacitor's voltage and current are still 90o
out of phase with each other.
Let's check our calculations with SPICE:
ac r-c circuit
v1 1 0 ac 10 sin
r1 1 2 5
c1 2 0 100u
.ac lin 1 60 60
.print ac v(1,2) v(2,0) i(v1)
.print ac vp(1,2) vp(2,0) ip(v1)
.end
freq v(1,2) v(2) i(v1)
6.000E+01 1.852E+00 9.827E+00 3.705E-01
freq vp(1,2) vp(2) ip(v1)
6.000E+01 7.933E+01 -1.067E+01 -1.007E+02
Once again, SPICE confusingly prints the
current phase angle at a value equal to the real phase angle
plus 180o (or minus 180o). However,
it's a simple matter to correct this figure and check to see
if our work is correct. In this case, the -100.7o
output by SPICE for current phase angle equates to a
positive 79.3o, which does correspond to our
previously calculated figure of 79.325o.
Again, it must be emphasized that the
calculated figures corresponding to real-life voltage and
current measurements are those in polar form, not
rectangular form! For example, if we were to actually build
this series resistor-capacitor circuit and measure voltage
across the resistor, our voltmeter would indicate 1.8523
volts, not 343.11 millivolts (real rectangular) or 1.8203
volts (imaginary rectangular). Real instruments connected to
real circuits provide indications corresponding to the
vector length (magnitude) of the calculated figures. While
the rectangular form of complex number notation is useful
for performing addition and subtraction, it is a more
abstract form of notation than polar, which alone has direct
correspondence to true measurements.
-
REVIEW:
-
Impedance is the total measure of
opposition to electric current and is the complex (vector)
sum of ("real") resistance and ("imaginary") reactance.
-
Impedances (Z) are managed just like
resistances (R) in series circuit analysis: series
impedances add to form the total impedance. Just be sure
to perform all calculations in complex (not scalar) form!
ZTotal = Z1 + Z2 + . . .
Zn
-
Please note that impedances always add in
series, regardless of what type of components comprise the
impedances. That is, resistive impedance, inductive
impedance, and capacitive impedance are to be treated the
same way mathematically.
-
A purely resistive impedance will always
have a phase angle of exactly 0o (ZR
= R Ω ∠ 0o).
-
A purely capacitive impedance will always
have a phase angle of exactly -90o (ZC
= XC Ω ∠ -90o).
-
Ohm's Law for AC circuits: E = IZ ; I =
E/Z ; Z = E/I
-
When resistors and capacitors are mixed
together in circuits, the total impedance will have a
phase angle somewhere between 0o and -90o.
-
Series AC circuits exhibit the same
fundamental properties as series DC circuits: current is
uniform throughout the circuit, voltage drops add to form
the total voltage, and impedances add to form the total
impedance.
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