Series-to-Parallel Conversion 2 comments

In practice, resonant circuits can never be ideal series or parallel resonant circuits. In some situations conversion of series to parallel, or parallel to series circuits makes the design calculations simpler. The following transformations are valid in narrow band of frequencies around resonance.

The input impedance of series R-X circuit shown in Figure 1 is

(1)   \begin{equation*}Z_{s}=R_s + j X_s\end{equation*}

and the input impedance of parallel R-X circuit is,

(2)   \begin{equation*}Z_p=\frac{jR_p X_p}{R_p+jX_p}=\frac{R_p X_p^{2}}{R_p^{2}+X_p^{ 2}}+j\frac{X_pR_p^{2}}{R_p^{ 2}+X_p^{2}} \end{equation*}

One way of defining Q-factor of a circuit, with input impedance(Z_i), is Q = {Im(Z_i)\over Re(Z_i)}.

Series — Parallel Transformation

Figure 1. Series and Parallel R-X circuits

Figure 1. Series and Parallel R-X circuits

Using Eq.(1) and this Q-definition, Q_s = {X_s \over R_s}.
Using Eq.(2) and this Q definition, Q_p= {R_p \over X_p}.
X_{p} and X_{s} are frequency dependent terms. Therefore Q_{p} and Q_{s} are also frequency dependent.

The condition under which series R-X and parallel R-X are equivalent is obtained by comparing Eq.(1) and Eq.(2).

(3)   \begin{equation*}R_s=\frac{R_p}{1+Q_p^{2}} \mbox{ and } X_s=\frac{X_p}{1+\frac{1}{Q_p^{2}}}\end{equation*}

After some arithmetic manipulations from Eq.(3) and using the definitions of Q_s and Q_p, we get \boxed{Q_p=Q_s}.
At frequencies where Q_{p/s}>0, R_p>R_s.
When Q_p \gg 1,

(4)   \begin{equation*}R_p \approx {Q_p}^{2}{R_s} \mbox{ and } X_p \approx X_s\end{equation*}

These impedance transformations are very helpful in designing narrow-band impedance matching networks.

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