Parallel (Phase) Resonance

Figure 9. Showing the variations in the resistance, reactances, and impedance of a series-resonant circuit. Note that at resonance the impedance equals the resistance. For a circuit of lower resistance (and higher Q) the impedance would fall to a lower value and the circuit would be more sharply tuned.

Figure 10. Equivalent input impedance of a parallel-resonant circuit. Note that at resonance the impedance rises rapidly and becomes a large value of pure resistance. For a circuit with less loss (and higher Q) the curves rise to higher values and the circuit would be more sharply tuned.

Resonance (phase) occurs in a parallel circuit composed of an inductor and capacitor when the inductive susceptance B_L = X_L/Z_L² equals the capacitive susceptance B_c = X_C/Z_C², and the reactive component of the current through the inductor equals the reactive component of the current through the capacitor. For the practical case of a parallel circuit composed of an inductor having effective resistance R and a capacitor having negligible loss, the inductive susceptance is B_L = X_L/(R_L² + X_L²) and the capacitive susceptance is B_C = 1/X_C. The resonant frequency is

where / is the (phase) resonant frequency in cycles per second when L is the inductance in henrys, C is the capacitance in farads, and R_L is the effective resistance of the inductor in ohms. For many parallel circuits equation 13 is satisfactory. The way the input impedance of a parallel circuit varies with frequency is shown in Fig. 10. Note that the input impedance is high at resonance.

The voltage across a parallel circuit at resonance is E = IR_e, where R_e is the equivalent parallel resistance. From the familiar equation for equivalent impedance of any parallel circuit,

the equivalent parallel impedance of the inductor and capacitor previously considered is approximately

This last expression applies at radio frequencies where R_L is small compared with X_L, where R_L is neglected in the numerator, where

X_L = X_C, and where Q = ωL/R_L. The voltage at resonance across the parallel inductor and capacitor is E = IZ_e = IωLQ. Because the input impedance is greatest at resonance, the voltage across a tuned parallel circuit to which the current input I is constant also is greatest at resonance, and this property is made use of in practice to obtain a high signal voltage. At resonance the current through the inductor and capacitor are I_L = E/Z_L and I_c = E/X_Cj respectively, and may be quite large at resonance; this property also is utilized in practice.