What is the effect of adding a capacitor in parallel with a resistor on the combined impedance and phase at a given frequency?

• A. The parallel impedance is lower than either branch; the overall impedance has a phase between 0 and -90°, approaching -90° as frequency increases.
• B. The parallel impedance is higher than either branch; the phase is always 0°.
• C. The parallel impedance equals the larger of the two branches; phase is +90°.
• D. The parallel impedance is unchanged; phase is +0°.

Answer: (A)

Adding a capacitor in parallel with a resistor changes the total impedance because the two branches add their conductances, not their impedances. The resistor contributes a real conductance G = 1/R and the capacitor contributes an imaginary conductance jωC, so the total admittance is Y = 1/R + jωC. The combined impedance is Z = 1/Y, which is a complex number with a negative imaginary part. Its magnitude is |Z| = 1 / sqrt((1/R)^2 + (ωC)^2), which is smaller than either branch’s impedance magnitude. The phase of Z is φ = -arctan(ωCR), meaning the phase sits between 0° and -90° and moves toward -90° as frequency increases (the capacitor conducts more as frequency rises). So at a given frequency, the parallel impedance is lower than either branch, and the phase lies between 0 and -90°, approaching -90° as frequency grows.