Putting It All Together

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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:

  1. Find the conductance G = 1/R for the resistor. (It will be positive or zero.)
  2. Find the susceptance BL of the inductor using the appropriate formula. (It will be negative or zero.)
  3. Find the susceptance BC of the capacitor using the appropriate formula. (It will be positive or zero.)
  4. Find the net susceptance B = BL + BC. (It might be positive, negative, or zero.)
  5. Compute R and X in terms of G and B using the appropriate formulas.
  6. Assemble the complex impedance R + jX.

Problem 1

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:

  1. Calculate G = 1/R = 1/10.0 = 0.100.
  2. Calculate BL = −1/(6.28fL) = −1/(6.28 × 1.00 × 10.0) = −0.0159.
  3. Calculate BC = 6.28fC = 6.28 × 1.00 × 0.000820 = 0.00515. (Remember to first convert the capacitance to microfarads, to go with megahertz.)
  4. Calculate B = BL + BC = −0.0159 + 0.00515 = −0.0108.
  5. 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.
  6. The complex impedance is R + jX = 9.88 + j1.07.

Problem 2

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:

  1. Calculate G = 1/R = 1/47.0 = 0.021277.
  2. Calculate BL = −1/(6.28fL) = −1/(6.28 × 2.252 × 10.0) = −0.00707.
  3. Calculate BC = 6.28fC = 6.28 × 2.252 × 0.000500 = 0.00707. (Remember to first convert the capacitance to microfarads, to go with megahertz.)
  4. Calculate B = BL + BC = −0.00707 + 0.00707 = 0.00000.
  5. 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.
  6. 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.