Chapter 29: Parallel Resonance and Q-factor

29.1 Introduction to Parallel Resonance

A parallel resonant circuit is usually used to establish a condition of stable frequency or current designed to produce oscillation.[cite: 181, 182, 183] In this circuit, the capacitor and inductor are connected directly together, exchanging energy with each other.[cite: 183, 184] They work in much the same way the electrical frequency of a pendulum stabilizes an oscillator circuit.[cite: 185, 186, 188, 189]

Another use is where greatly reduced or increased impedance at a certain frequency is desired.[cite: 187, 190, 191] They can be used to block or tune a certain frequency range.[cite: 191, 192] These circuits are called filters.[cite: 195]

29.2 LR-C Parallel Network

This is a practical network where the coil has inductance L and resistance R, and next to it (in parallel) is pure capacitance C.[cite: 9, 11, 12, 199, 200]

Again we sum the admittances and make the imaginary parts sum to zero in order to start to derive an equation for resonance.[cite: 13, 14, 15, 16, 17]

The equation to find the resonant frequency is given by:[cite: 18]

When R² is very small, then f_r = 1 / (2π√LC).[cite: 21, 22, 23]

29.3 Dynamic Resistance

Since the current at resonance is in phase with the voltage, the impedance of the network acts as a pure resistance.[cite: 34, 35, 38] The dynamic resistance R_D is given by:[cite: 38]

Where I_r is the current at resonance.[cite: 40] Substituting Y_r = R / (R² + ω_r²L²) and following a few steps leads to:[cite: 41, 42]

29.4 LR-CR Parallel Network

R_C and R_L are the resistances of the capacitor and coil respectively.[cite: 45, 46, 48, 49] The total network admittance is Y_T = Y_C + Y_L.[cite: 52]

The resonant frequency is given by:[cite: 53]

29.5 Q-factor in a Parallel Network

The Q-factor in a series R-L-C circuit is a measure of the voltage magnification.[cite: 55] In a parallel network, it is a measure of current magnification.[cite: 56, 58, 59] Energy flows between the inductor and capacitor. It leaves the capacitor and establishes a magnetic field in the inductor; this field then collapses and charges the capacitor, and so on.[cite: 57]

Circulating currents (i.e., I_C or I_L) may be several times greater than the supply current at resonance for the parallel network.[cite: 61, 62, 63]

Given the capacitor current I_C = V / X_C = Vω_rC and current at resonance I_r = V / R_D = V / (L/CR) = VCR / L:[cite: 71, 72]

Which is the same equation for series resonance and can also be expressed as:[cite: 75, 78]

Natural Frequency vs. Forced Resonant Frequency

f_n is the natural frequency at which energy would naturally resonate in a closed loop of a capacitor and inductor.[cite: 93, 100, 101]

f_r is called the forced resonant frequency seen at the terminals. For a series circuit, the forced and natural frequencies coincide.[cite: 102, 105, 106]

Since the smallest Q we may come across is 10, we can see that even with a small Q, the difference between f_n and f_r tends to be small.[cite: 118, 120, 121]

Worked Problems