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.
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.
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,
$$Z_{R-L} = R_l + j X_L$$
$$ \begin{split}
Y_{R-L} &= { 1 \over R_l + j X_L}\\
&={ R_l \over R_l^2 + X^2_L} - j { X_L \over R_l^2 + X^2_L}\\
&={ 1 \over { R_l^2 + X^2_L \over R_l}} + { 1 \over j { R_l^2 + X^2_L \over X_L } }\\
&={ 1 \over R_p} + { 1 \over jX_{Lp}}
\end{split}$$
$$ \bbox[10px,border:1px solid grey]{R_p = { R_l^2 + X^2_L \over R_l}}$$
and
$$\bbox[10px,border:1px solid grey]{X_{Lp} = { R^2_l + X^2_L \over X_L } }$$
as shown in
Fig. 3.
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.
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.
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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