Parallel resistor-capacitor circuits
Using the same value components in our
series example circuit, we will connect them in parallel and see what
happens:
Because the power source has the same
frequency as the series example circuit, and the resistor and capacitor
both have the same values of resistance and capacitance, respectively,
they must also have the same values of impedance. So, we can begin our
analysis table with the same "given" values:
This being a parallel circuit now, we
know that voltage is shared equally by all components, so we can place
the figure for total voltage (10 volts ∠ 0o) in all the
columns:
Now we can apply Ohm's Law (I=E/Z)
vertically to two columns in the table, calculating current through the
resistor and current through the capacitor:
Just as with DC circuits, branch currents
in a parallel AC circuit add up to form the total current (Kirchhoff's
Current Law again):
Finally, total impedance can be
calculated by using Ohm's Law (Z=E/I) vertically in the "Total" column.
As we saw in the AC inductance chapter, parallel impedance can also be
calculated by using a reciprocal formula identical to that used in
calculating parallel resistances. It is noteworthy to mention that this
parallel impedance rule holds true regardless of the kind of impedances
placed in parallel. In other words, it doesn't matter if we're
calculating a circuit composed of parallel resistors, parallel
inductors, parallel capacitors, or some combination thereof: in the form
of impedances (Z), all the terms are common and can be applied uniformly
to the same formula. Once again, the parallel impedance formula looks
like this:
The only drawback to using this equation
is the significant amount of work required to work it out, especially
without the assistance of a calculator capable of manipulating complex
quantities. Regardless of how we calculate total impedance for our
parallel circuit (either Ohm's Law or the reciprocal formula), we will
arrive at the same figure:
- REVIEW:
- Impedances (Z) are managed just like
resistances (R) in parallel circuit analysis: parallel impedances
diminish to form the total impedance, using the reciprocal formula.
Just be sure to perform all calculations in complex (not scalar) form!
ZTotal = 1/(1/Z1 + 1/Z2 + . . . 1/Zn)
- Ohm's Law for AC circuits: E = IZ ; I
= E/Z ; Z = E/I
- When resistors and capacitors are
mixed together in parallel circuits (just as in series circuits), the
total impedance will have a phase angle somewhere between 0o
and -90o. The circuit current will have a phase angle
somewhere between 0o and +90o.
- Parallel AC circuits exhibit the same
fundamental properties as parallel DC circuits: voltage is uniform
throughout the circuit, branch currents add to form the total current,
and impedances diminish (through the reciprocal formula) to form the
total impedance.
|
