# 13.7: The RLC Series Acceptor Circuit

A resistance, inductance and a capacitance in series is called an "acceptor" circuit, presumably because, for some combination of the parameters, the magnitude of the inductance is a minimum, and so current is accepted most readily. We see in figure \(\text{XIII.5}\) an alternating voltage \(V=\hat{V}e^{j\omega t}\) applied across such an \(R\), \(L\) and \(C\).

FIGURE \(\text{XIII.5}\)

The impedance is

\[\label{13.7.1}Z=R+j\left (L\omega -\dfrac{1}{C\omega}\right ).\]

We can see that the voltage leads on the current if the reactance is positive; that is, if the inductive reactance is greater than the capacitive reactance; that is, if \(\omega > 1/\sqrt{LC}\). (Recall that the frequency, \(\nu\), is \(\omega/(2\pi)\)). If \(\omega < 1/\sqrt{LC}\), the voltage lags behind the current. And if \(\omega = 1/\sqrt{LC}\), the circuit is purely resistive, and voltage and current are in phase.

The magnitude of the impedance (which is equal to \(\hat{V}/\hat{I}\)) is

\[\label{13.7.2}|Z|=\sqrt{R^2+\left (L\omega - 1/(C\omega )\right )^2},\]

and this is least (and hence the current is greatest) when \(\omega = 1/\sqrt{LC}\), the resonant frequency, which I shall denote by \(\omega_0\).

It is of interest to draw a graph of how the magnitude of the impedance varies with frequency for various values of the circuit parameters. I can reduce the number of parameters by defining the dimensionless quantities

\[\label{13.7.3}\Omega = \omega / \omega_0\]

\[\label{13.7.4}Q=\dfrac{1}{R}\sqrt{\dfrac{L}{C}}\]

and

\[\label{13.7.5}z=\dfrac{|Z|}{R}.\]

You should verify that \(Q\) is indeed dimensionless. We shall see that the sharpness of the resonance depends on \(Q\), which is known as the *quality factor* (hence the symbol \(Q\)). In terms of the dimensionless parameters, Equation \ref{13.7.2} becomes

\[\label{13.7.6}z=\sqrt{1+Q^2(\Omega -1/\Omega )^2}.\]

This is shown in figure \(\text{XIII.6}\), in which it can be seen that the higher the quality factor, the sharper the resonance.

FIGURE \(\text{XIII.6}\)

In particular, it is easy to show that the frequencies at which the impedance is twice its minimum value are given by the positive solutions of

\[\label{13.7.7}\Omega^4 -\left (2+\dfrac{3}{Q^2}\right )\Omega^2+1=0.\]

If I denote the smaller and larger of these solutions by \(\Omega_- \text{ and }\Omega_+\), then \(\Omega_+ -\Omega_-\) will serve as a useful description of the width of the resonance, and this is shown as a function of quality factor in figure \(\text{XIII.7}\).

FIGURE \(\text{XIII.7}\)