In this lecture the following are introduced for A.C. series resonant circuits.
• The resonance condition.
• The natural frequency for resonance in A.C. series circuits.
• The variation of current with frequency.
• The variation of phase with frequency.
• The quality factor and the bandwidth.
• The magnification of potential differences at resonance.
• Band pass characteristics

Resonance

When a pulse sets a system vibrating, it will do so at the system's natural frequency. This frequency depends on the system's construction.
When a periodic vibration sets a system vibrating then the system is forced to vibrate at that frequency. When the frequency of the forcing vibration is the same as the natural frequency there is a big increase in the amplitude of the vibrations. This big increase in amplitude is called resonance.

A.C. Series Resonance

In A.C. a series circuit resonates when there is a big increase in output potential or current at a particular frequency, as shown in the graphical example below.

As will be demonstrated, this resonance happens when the complex impedance is real, i.e. the circuit becomes purely resistive, the current is in phase with the applied e.m.f., and the power factor is 1.

Starting with the current. The current in the series circuit shown is given on the right. Note the method of rationalising the denominator by multiplying top and bottom with the complex conjugate.

This equation will give a graph similar to the potential difference example above.

Natural Frequency

When the circuit becomes purely resistive, the imaginary part of the complex impedance equals zero. This identifies the circuit's natural angular frequency, ω0 rad.s-1, which is calculated as shown on the right.
When the applied (or forcing) frequency approaches this value then the circuit impedance approaches its minimum (of R) and there is a maximum in the magnitude of the current.

Current versus Frequency

A graph of current versus applied frequency for various load resistances, shows peaks at the natural frequency

When the circuit resistance is small, the graph has large resonant current and a narrow peak. When the circuit resistance is large then the resonant current is small and the peak is broad. With the impedance, the factor makes the inductive part large at high frequencies and the capacitive part large at low frequencies.

Phase versus Frequency

The phase angle between the current and the applied e.m.f. is nominally negative and given by: The phase angle is zero at resonance but changes rapidly near resonance.

At low frequencies the capacitor puts the phase of the current ahead of the applied potential.
At high frequencies the inductor puts the phase of the applied potential ahead of the current.

The Quality factor and the Bandwidth

The Quality factor, Q, is a measure of the sharpness (or selectivity) of the resonance peak. It is the ratio of the resonant frequency to the "bandwidth".
The bandwidth, Δω, is the difference between the two frequencies where the power dissipated by the circuit falls to one half of the peak power.
The bandwidth gives the range for which the power dissipated is at least half of the peak power.	The half-power current is 71% of the peak current.
Now To find the two half power frequencies (higher and lower than resonance), substitute the half power current into E/Z = I, as shown on the right.
Since R equals both the plus and minus expressions, each expression equals the other as shown on the right, and one pair of ω will be the upper half-power frequency and the other will be the lower. The resonant frequency is the geometric mean of the half-power frequencies.
Adding the two equations for R gives:

It can be seen that the bigger the resistance the wider the bandwidth. Also, the bandwidth depends only on the ratio of resistance to inductance. Bandwidth does not depend on the capacitance.

The Q for the circuit is now found to be: 

Potential Magnification at Resonance

Consider the AC series circuit shown below supplied at the resonant frequency ω0.

The Q factor is also called the magnification. This is because there is a simple multiplicative relationship between the potentials at resonance.

At resonance,
• the resistor has all the supply potential difference across it, and
• the inductor and capacitor have Q times the supply potential difference across them.

Since Q may be 1000 or more, the inductor and capacitor may have 1000x the supply potential across them!

Example

	For the circuit shown, with an emf that can vary, find (a) the resonant frequency (Hz), (b) the rms current at resonance, (c) the rms p.d. across each component, (d) the Q factor, and (e) the band width (Hz)
(a) At resonance, the reactance -> 0, i.e.	(b) and (c) With zero reactance at resonance, the impedance is purely resistive.
(d) and (e) The Q factor and half power frequency difference.
The current as a function of angular frequency.	The graph of this resonance is shown below.

Example

For the circuit shown, find the (a) inductance for a 500 Hz resonant frequency, (b) resistance for a Q of 15.

To find the inductance.

To find the resistance.

Band pass characteristics

A resonance peak can be used to let only a range of frequencies pass on to another circuit.

The circuit shown below forms a band pass filter with cut-off frequencies at half power. The output potential is shown below.

The circuit current is:

Summary:

A circuit is in resonance when the complex impedance is real.
The natural frequency for resonance is:

The phase angle between the current and the applied e.m.f. is:

The Q factor is:

The bandwidth relationships are:

At resonance the potentials are:

Using the resistor as an output for a band-pass filter:

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