Peter's Physics Pages
Peter's Index Physics Home Lecture 14 Course Index Lecture 16
A Semester of First Year Physics with Peter Eyland
Lecture 15 (AC series resonance)
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. 

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. 
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. 

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 halfpower 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 halfpower frequency and the other will be the lower.


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 ω_{0}. 
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) 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 
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 cutoff 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 bandpass filter:
email Write me a note if you found this useful
Copyright Peter & BJ Eyland. 2007  2015 All Rights Reserved. Website designed and maintained by Eyland.com.au ABN79179540930. Last updated 18 January 2015 