Chapter 29: Parallel Resonance and Q factor

Date: 9/7/19

A parallel resonant circuit is usually used to establish a condition of stable frequency or current designed to produce an oscillation. In this circuit, a capacitor and inductor are connected directly together, exchanging energy with each other. They work in much the same way as a pendulum stabilizes the mechanical frequency of an oscillator circuit.

Another use is where a greatly increased impedance is desired at a certain frequency. They can be used to block a frequency range or to strain certain frequencies out of others. These circuits are called filters.

29.1 Parallel Network with Resistance, Pure Inductance, and Pure Capacitance

This is a parallel network containing resistance R, pure inductance L, and pure capacitance C connected in parallel. In a parallel circuit, we sum the admittances to get the overall admittance.

• Admittance of the Resistor R is 1/R
• Admittance of the inductor L is 1/(jX_L) = -j/(ωL)
• Admittance of the capacitor C is jωC

The total admittance Y is given by:

The circuit is resonant when ω_rC – 1/(ω_rL) = 0.

29.2 LR-C Parallel Network

This is a more practical network where the coil has inductance L and resistance R, and next to it (in parallel) is pure capacitance C. The admittances are:

• Y_coil = 1/(R + jX_L) = (R – jX_L) / (R² + X_L²)
• Y_c = 1/(-jX_c) = jωC

By summing the admittances and setting the imaginary part to zero, the equation to find the resonant frequency is derived as:

When R² / L² is much less than 1/LC, then f_r ≈ 1 / (2π√(LC)). The real part at resonance is a combination of resistance and inductance.

29.3 Dynamic Resistance

Since the current at resonance is in phase with the voltage, the impedance of the network acts as a resistance. This is known as the dynamic resistance (R_D):

where I_r is the current at resonance.

29.4 LR-CR Parallel Network

In this network, R_c and R_L are the resistances of the capacitor and coil respectively. Summing the admittances of the inductive and capacitive branches gives the total network admittance Y_T = Y_c + Y_L. The resonant frequency equation emerges as:

29.5 Q-factor in a Parallel Network

The Q-factor in a series R-L-C circuit is a measure of voltage magnification, whereas in a parallel network it is a measure of current magnification. As energy flows between the inductor and capacitor, circulating currents (I_c or I_L) may be several times greater than the supply current at resonance.

The Q-factor can be expressed as:

The natural frequency f_n is the frequency the loop would naturally resonate at, given by f_n = 1 / (2π√(LC)). The relationship between the resonant (forced) frequency and natural frequency is:

Since Q is usually 10 or greater, the difference between f_r and f_n tends to be small. A high Q tends to make parallel and series resonant frequencies have the same value.

Bandwidth and Resonant Frequency

The expressions for Q-factor, bandwidth, and resonant frequency are similar to those in the series R-L-C circuit:

• Q_r = f_r / (f₂ – f₁)
• Q_T = (Q_LQ_c) / (Q_L + Q_c)
• f_r = √(f₁f₂)

At the half-power frequencies, the impedance is Z = Z_r / √2. The admittance relationship is Y / Y_r = 1 + j2δQ, where δ is the fractional deviation from the resonant frequency.

29.6 Problems Re-started

Evaluating total admittance Y_T = Y_c + Y_L:

Given Y_c = 1 / (4 – j10) and Y_L = 1 / (3 + jX_L)

Rationalizing the admittances:

• Y_c = (4 + j10) / (4² + 10²) = (4 + j10) / 116
• Y_L = (3 – jX_L) / (3² + X_L²)

At resonance, the imaginary parts of the admittances sum to zero:

Cross multiplying gives:

This quadratic equation can be solved using the standard formula X_L = (-b ± √(b² – 4ac)) / 2a.