# Series RLC Circuit

The input impedance of a series RLC circuit, shown in Figure 1 is given by,

(1)

Figure 1. Series R-L-C circuit

At resonant frequency, , . Therefore, resonant frequency is given by,

(2)

Quality factor(Q) of series R-L-C circuit is,

(3)

where is the peak current in the circuit
The characteristic impedance of the circuit is
From Eq-(3), the Q-factor of series R-L-C circuit at resonance is

(4)

Substituting the relations Eq.(2) and Eq.(3) in Eq.(1), the input impedance of the series resonant circuit as a function of and is given by,

(5)

The input impedance of Series RLC circuit is shown in Fig.2 as a function of ().

At ‘‘, the impedance seen by the source is equal to which is the minimum and real

For small deviation in frequencies from center frequency,, the input impedance is

(6)

For , is approximated as,

(7)

The magnitude transfer function of series rlc circuit is,

(8)

Figure 3 illustrates the for different Q-factors of the circuit. At resonant frequency |H(j\omega_o)=1| irrespective of resistance (or Q-factor).

At resonance, voltage across resistance is maximum and therefore power dissipation() is also maximum in resistor. When voltage drop across resistor drops by , the power dissipation drops by . The frequencies at which power drop by half from resonant frequency are called half-power frequencies.
Since the power at these frequencies is 3dB less or half that of resonant frequency, these frequencies are called 3dB frequencies or half-power frequencies. At 3dB frequencies,

(9)

The two roots of the equation are

(10)

Bandwidth is the given by

(11)

and resonant frequency is the geometric mean of 3dB frequencies

(12)

With decreasing R, the Q-factor increases whereas the bandwidth decreases

. High Q means higher selectivity or sensitivity. High Q circuits has low BW at a given center frequency, a desired feature in many filtering applications.