In measuring inductance values, it will be found that real components are not ideal.  Thus, the impedance of the device is represented by either a series or a parallel equivalent as shown in the figure at the left.  Conventionally, real capacitors are represented by a parallel circuit of a perfect capacitor and resistor while real inductors are represented by a series circuit of a perfect inductor and a resistor.  A simple LCR meter just gives an estimate of the total impedance converted into Farads or Henrys for capacitors and inductors, respectively, and does not give you any information about the resistive parts.  The impedance bridge does provide this additional information for the components (at a particular frequency, usually 1 kHz) and these are given in the form of dissipation and quality factors.

For the capacitor, the dissipation factor, D, is related to the parallel resistance, R by:

D = 1 / wCR

where  is w is the angular frequency and C is the measured capacitance.  For the real inductor, the quality factor, Q is related to the series inductance by:

Q = wL / r

where r is the series resitance and L is the measured inductance.

The impedance, and hence the response, of any circuit containing reactive elements depends upon frequency.  Thus any circuit containing reactive elements can be called a frequency selective circuit, since it provides a certain response to certain frequencies.  However, the term is usually used to denote only a circuit specifically designed to separate different frequencies.  This is the function of RLC series and parallel circuits, which are "resonant" at a specific frequency.

A series circuit containing R, L, and C is in resonance when the current in the circuit is in phase with the total voltage across the circuit.  Depending on the particular values of R, L, and C, resonance occurs at one distinct frequency.  Because of its distinct frequency characteristics, the series resonant circuit is one of the most important frequency selective circuits.

An important consideration when designing an RLC circuit is the nonideal nature of the reactive components.  Real capacitors closely approximate perfect capacitors so we may neglect the parallel resistance associated with D.  Real inductors, however, have a small series resistance which is shown in the circuit diagram as r.  This cannot normally be neglected since the Q of real inductors is not infinitely large.

The transfer function for this network, V_out / V_in, which we will call F is, by inspection:

This equation is approximate since r is itself a function of frequency.  However, the approximation is fairly good and measured transfer functions do not differ much from it for reasonable ranges of frequency.  F has both a phase and a magnitude as a function of frequency.  Since the circuit is a series resonant circuit, at resonance, the inductive and capacitive reactances have equal values but opposite signs so the imaginary term in the denominator goes to zero and the value of F just becomes R / r + R.

The graph below shows a series of plots of the transfer function for a series RLC circuit where the inductance had a value of 1.0 mH, the capacitor was 25.3 uF and the series resistor across the output, R, was 1.0 ohm.  The three curves are for inductors with three different Q values:  100, 10 and 1.  As you can see, the highest Q circuit had the least loss and the narrowest passband.  The passband is defined as the difference, in Hz, between the two frequencies at which the F is down 3 dB from its peak value.  The lowest value of Q resulted in the greatest loss and the broadest passband.

There is an approximate linear relation between circuit Q and bandwidth which is:

Q = w_o / dw

where dw is the bandwidth between -3 dB points as previously described.

Note that the overall Q of the circuit must also take into account the value of R, the output series resistance, since it is part of the circuit.  That is, for the circuit as a whole, the total resistance is r+R, not just r.  So:

Circuit Q = wL / (r+R)

At resonance, X_L. = X_C, and the resonant frequency is determined using the values of L and C:

f_r= 1 / 2p(LC)^1/2

Parallel resonance is the condition that exists in an AC circuit containing inductive and capacitive branches when the total current is in phase with the voltage across the circuit.  This, of course, implies that the total impedance, or conversely, the total admittance, is real.

A plot of the transfer function for this network is similar in appearance to the series RLC circuit because the output is taken off the parallel LC part of the circuit.  At resonance, the parallel circuit becomes totally real and has a value of just the parallel circuit becomes totally real and has a value of just the parallel equivalent resistance of the inductor alone.  If the parallel equivalent resistance of the inductor, at the resonant frequency, is R_par, then F at resonance is just equal to R_par / R+R_par.