WEEK 14: Parallel Resonance

The pure parallel resonant circuit is shown in the figure below.

If we neglect the loss resistance of the inductor, R represents the leakage resistance of the capacitor. However, as we will see below the loss resistance of the inductor can be transformed into this resistor.

The total admittance:

The admittances (called susceptances) of the capacitor and inductor are imaginary and have opposite sign. At the frequency w₀C = 1/w₀L the total imaginary part is zero, so the total admittance is 1/R�its minimum value�and the total impedance has its maximum value. This frequency is called the parallel resonant frequency.

The total impedance characteristic of the pure parallel resonant circuit is shown in the figure below:

Note that the impedance changes very rapidly around the resonance frequency, even though we used a logarithmic impedance axis for better resolution. The same curve with a linear impedance axis is shown below. Note that viewed with this axis, the impedance appears to be changing even more rapidly near resonance.

The susceptances of the inductance and capacitance are equal but of opposite sign at resonance: B_L = B_C, 1/w₀L = w₀C, hence the angular frequency of the parallel resonance:

determined again by the Thomson formula.

Solving for the resonant frequency in Hz:

At this frequency the admittance Y = 1/R = G and is at its minimum (i.e., the impedance is maximum). The currents through the inductance and capacitance can be much higher then the current of the total circuit. If R is relatively large, the voltage and admittance changes sharply around the resonant frequency. In this case we say the circuit has goodselectivity.

Selectivity can be measured by the quality factor Q

When the angular frequency equals the angular frequency of resonance, we get the resonant quality factor:

There is also a more general definition of the quality factor:

Another important property of the parallel resonant circuit is itsbandwidth. The bandwidth is the difference between the two cutoff frequencies, where the impedance drops from its maximum value to

the maximum.

It can be shown that the Δf bandwidth is determined by the following simple formula:

This formula is also applicable for series resonant circuits.