Parallel Resonant Circuit

The basic format of the series resonant circuit is a series RLC combination in series with an applied voltage source. The parallel resonant circuit has the basic configuration of Fig. 1, a parallel RLC combination in parallel with an applied current source.
Ideal parallel resonant network.
Fig. 1: Ideal parallel resonant network.
For the series circuit, the impedance was a minimum at resonance, producing a significant current that resulted in a high output voltage for $V_C$ and $V_L$. For the parallel resonant circuit, the impedance is relatively high at resonance, producing a significant voltage for $V_C$ and $V_L$ through the Ohm's law relationship ($V_C = IZ_T$). For the network of Fig. 1, resonance will occur when $X_L = X_C$, and the resonant frequency will have the same format obtained for series resonance. If the practical equivalent of Fig. 1 had the format of Fig. 1, the analysis would be as direct and lucid as that experienced for series resonance. However, in the practical world, the internal resistance of the coil must be placed in series with the inductor, as shown in Fig. 2.
Practical parallel L-C network
Fig. 2: Practical parallel L-C network.
The resistance $R_l$ can no longer be included in a simple series or parallel combination with the source resistance and any other resistance added for design purposes. Even though $R_l$ is usually relatively small in magnitude compared with other resistance and reactance levels of the network, it does have an important impact on the parallel resonant condition, as will be demonstrated in the sections to follow. In other words, the network of Fig. 1 is an ideal situation that can be assumed only for specific network conditions.
Our first effort will be to find a parallel network equivalent (at the terminals) for the series $RL$ branch of Fig. 2 using the technique introduced in Frequency Response of the Parallel RL Network (Section 14.9). That is,
$$Z_{R-L} = R_l + j X_L$$
$$ \begin{split} Y_{R-L} &= { 1 \over R_l + j X_L}\\ &={ R_l \over R_l^2 + X^2_L} - j { X_L \over R_l^2 + X^2_L}\\ &={ 1 \over { R_l^2 + X^2_L \over R_l}} + { 1 \over j { R_l^2 + X^2_L \over X_L } }\\ &={ 1 \over R_p} + { 1 \over jX_{Lp}} \end{split}$$
$$ \bbox[10px,border:1px solid grey]{R_p = { R_l^2 + X^2_L \over R_l}}$$
$$\bbox[10px,border:1px solid grey]{X_{Lp} = { R^2_l + X^2_L \over X_L } }$$
as shown in Fig. 3.
Equivalent parallel network for a series R-L combination.
Fig. 3: Equivalent parallel network for a series R-L combination.
Redrawing the network of Fig. 2 with the equivalent of Fig. 3 and a practical current source having an internal resistance $Rs$ will result in the network of Fig. 4.
Equivalent parallel network for a series R-L combination.
Fig. 4: Substituting the equivalent parallel network for the series R-L combination of Fig. 2.
If we define the parallel combination of Rs and Rp by the notation
$$ \bbox[10px,border:1px solid grey]{R = R_s || R_p }$$
the network of Fig. 5 will result. It has the same format as the ideal configuration of Fig. 1.
Equivalent parallel network for a series R-L combination.
Fig. 5: Substituting $R = R_s || R_p$ for the network of Fig. 4.
We are now at a point where we can define the resonance conditions for the practical parallel resonant configuration. Recall that for series resonance, the resonant frequency was the frequency at which the impedance was a minimum, the current a maximum, and the input impedance purely resistive, and the network had a unity power factor.
For parallel networks, since the resistance $Rp$ in our equivalent model is frequency dependent, the frequency at which maximum $VC$ is obtained is not the same as required for the unity-power-factor characteristic. Since both conditions are often used to define the resonant state, the frequency at which each occurs will be designated by different subscripts.

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