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4.4: Parallel Resistor-Capacitor Circuits

  • Page ID
    1404
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    Using the same value components in our series example circuit, we will connect them in parallel and see what happens: (Figure below)

    02263.png

    Parallel R-C circuit.

    Resistor and Capacitor in Parallel

    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:

    12051.webp

    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:

    12052.png

    Calculation Using Ohm’s Law

    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:

    12053.png

    Just as with DC circuits, branch currents in a parallel AC circuit add up to form the total current (Kirchhoff’s Current Law again):

    12054.webp

    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:

    12036 (1).png

    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:

    12055.webp

    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.

    This page titled 4.4: Parallel Resistor-Capacitor Circuits is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Tony R. Kuphaldt (All About Circuits) via source content that was edited to the style and standards of the LibreTexts platform.