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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