When you’re confronted with a parallel circuit containing resistance, inductance, and capacitance, and you want to determine the complex impedance of the combination, do these things:
- Find the conductance G = 1/R for the resistor. (It will be positive or zero.)
- Find the susceptance BL of the inductor using the appropriate formula. (It will be negative or zero.)
- Find the susceptance BC of the capacitor using the appropriate formula. (It will be positive or zero.)
- Find the net susceptance B = BL + BC. (It might be positive, negative, or zero.)
- Compute R and X in terms of G and B using the appropriate formulas.
- Assemble the complex impedance R + jX.
Suppose a resistor of 10.0 Ω, a capacitor of 820 pF, and a coil of 10.0 μH are in parallel. The frequency is 1.00 MHz. What is the complex impedance?
Proceed according to the above steps, as follows:
- Calculate G = 1/R = 1/10.0 = 0.100.
- Calculate BL = −1/(6.28fL) = −1/(6.28 × 1.00 × 10.0) = −0.0159.
- Calculate BC = 6.28fC = 6.28 × 1.00 × 0.000820 = 0.00515. (Remember to first convert the capacitance to microfarads, to go with megahertz.)
- Calculate B = BL + BC = −0.0159 + 0.00515 = −0.0108.
- Define G2 + B2 = 0.1002 + (−0.0108)2 = 0.010117. Then R = G/0.010117 = 0.100/0.010117 = 9.88 Ω, and X = −B/0.010117 = 0.0108/0.010117 = 1.07 Ω.
- The complex impedance is R + jX = 9.88 + j1.07.
Suppose a resistor of 47.0 Ω, a capacitor of 500 pF, and a coil of 10.0 μH are in parallel. What is their complex impedance at a frequency of 2.252 MHz?
Proceed as before:
- Calculate G = 1/R = 1/47.0 = 0.021277.
- Calculate BL = −1/(6.28fL) = −1/(6.28 × 2.252 × 10.0) = −0.00707.
- Calculate BC = 6.28fC = 6.28 × 2.252 × 0.000500 = 0.00707. (Remember to first convert the capacitance to microfarads, to go with megahertz.)
- Calculate B = BL + BC = −0.00707 + 0.00707 = 0.00000.
- Define G2 + B2 = 0.0212772 + 0.000002 = 0.00045271. Then R = G/0.00045271 = 0.021277/0.00045271 = 46.999 Ω, and X = −B/0.00045271 = 0.00000/0.00045271 = 0.00000.
- The complex impedance is R + jX = 46.9999 + j0.00000. When we round it off to three significant figures, we get 47.0 + j 0.00. This a pure resistance equal to the value of the resistor in the circuit.