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Peter's Index Physics Home Lecture 15 Course Index Lecture 17
A Semester of First Year Physics with Peter Eyland
Lecture 16 (AC parallel resonance)
In this lecture the following are introduced:
• Series circuits strategy
• Parallel circuits strategy
• Admittance, conductance and susceptance
• Parallel resonance in A.C. circuits
• Natural frequency for resonance in A.C. parallel circuits
• Current versus frequency
• Phase versus frequency
• The quality factor and the bandwidth for A.C. parallel circuits
• Currents in parallel branches for high Q circuits at resonance
Series Circuit Strategy
In a series circuit as shown on the left, the elements have the same current through them. The strategy for analysing series circuits is to use the current as the common feature and add the potential differences around the circuit. 

In general: 
For example, the common current is calculated as: Knowing the common current, each of the potential differences can be calculated. 
Parallel Circuit Strategy
In a parallel circuit as shown on the left, the elements have the same potential difference across them. The strategy is to use the applied e.m.f. as the common feature and add up the currents from the branches. 

For this circuit: 
This involves adding the reciprocals of the impedances. 
Admittance, Conductance and Susceptance
As foreshadowed above, in parallel circuits,
it is ofen convenient to use a quantity which is the reciprocal of impedance.
Such a quantity is called the admittance Y, and will have real and imaginary parts as does the impedance.
In Cartesian form: 
For the three pure circuit elements: It should be noted that
unless X=0. 
Admittance has the SI unit called the Siemens. The real part of the admittance, G, is called conductance. The imaginary part of the admittance, S, is susceptance. The negative sign in the admittance makes its phase angle, (θ) the same as the impedance (φ).
Working in Polar form:
Example
Find the admittances of the following:
(a) a 1k0 resistor,
(b) a 6μ0 capacitor at 1000 rad.s^{1},
(c) a pure 20m0 inductor at 500 rad.s^{1},
(d) an inductor with 500R and 1m2 at 1 M rad.s^{1}, and
(e) a 300R resistor in series with a 10μ capacitor at 250 rad.s^{1}
Admittances in parallel
For the parallel circuit shown below, the strategy is to add the currents. 
With Conductance and Susceptance: 
Example
For the circuit shown on the left, find 

Note that you can't add 4.2 A and 5.8 A to get the total current because you have to take the phases into account. 

(d) The power factor is the cosine of the phase difference between the circuit e.m.f. and the total current. 
(e) The total power supplied is: 
Parallel Resonance
There are many combinations of elements that will produce parallel resonance. A useful one is shown on the right, it will attenuate a band of frequencies and pass the other frequencies unchanged. The circuit is in resonance when the complex impedance is real. 

The total circuit admittance is: This will be resonant because the susceptance can become zero. The circuit will have two natural frequencies, ω_{0} which make the admittance real. 
• The admittance is real, for . This is a steady D.C. supply. The resistor will have current through it and potential difference across it.
The inductor will have current but no potential difference, since the current does not vary. The frequency cannot be negative so only the positive solution applies. 
In contrast to series resonance the resonant frequency depends on the resistance. With increasing resistance the resonant frequency is lowered.
Current versus Frequency
Continuing with the circuit shown, the applied emf is taken as
. 
Writing the current, first with Cartesian admittance and then Polar form: Where θ is the phase angle between current and applied emf and goes to zero at resonance as discussed below. 
Current vs angular frequency graphs are shown below for various resistances. Because the susceptance is zero at the resonant frequency making the current small instead of large, the situation is sometimes called an antiresonance.
In contrast to series resonance, the resonant frequency does not occur at the minimum of the current. This is illustrated on the right by the green coloured curve of the resonant frequency for different resistances. 

The angular frequency of the current minimum, ω' may be found by setting the derivative of the current with respect to frequency, equal to zero. After some manipulation it can be shown to occur at:
Phase versus Frequency
The phase difference between current and applied emf is given by: 
• The phase angle is zero for a steady D.C. potential. 
The phase angle changes rapidly near resonance.
The Bandwidth currents
Since the total current supplied is small at resonance, the bandwidth is the frequency range between frequencies where the power is at least twice the power at resonance. 
The Quality factor and the bandwidth
The quality factor for parallel resonance is defined to be the same as in series circuits:
Writing the natural frequency in terms of Q: 
For high Q circuits: 
For high Q circuits, substituting the current for twice resonance power (eventually) gives the upper, u, and lower, l, frequencies:
For this particular circuit, the frequencies within the band width Δω around ω_{0} are suppressed and other frequencies are passed almost without attenuation. Hence this circuit could be used as a notch filter to reject frequencies within the bandwidth.
Currents in parallel branches
Continuing with the circuit (as shown below), the currents in the parallel branches are calculated at resonance. 
The admittance at resonance is written in terms of Q: Hence 
The current in the inductive branch at resonance is given as: 
For high Q circuits 
The current in the capacitive branch at resonance is given as: 
For high Q circuits 
Collecting the currents: 
It may be pictured that a circulating current flows as shown. The circulating current has magnitude Q times the current supplied; and is 90^{0} ahead of it in phase. 
Example
For the circuit shown below find the resonant frequency. 

For resonance, the imaginary part is zero, i.e.: 
Notice that if then the circuit is resonant at all frequencies.
With this circuit, for a single resonance to occur,
Summarising:
Series circuits strategy: use the current as the common feature and add the potential differences around the circuit.
Parallel circuits strategy: use the applied e.m.f. as the common feature and add up the currents from the branches.
Admittance:
The admittance Y is in Siemens, S.
G is the conductance and S is the susceptance, (both in S).
A circuit is in parallel resonance when the complex impedance is real.
The current is small at parallel resonance and is sometimes called antiresonance.
The phase angle changes rapidly near resonance.
The Q factor is defined in the same way as in series resonance.
The bandwidth for parallel circuits is the frequency range between frequencies where the power is twice the power at resonance.
Some parallel circuits can be used as notch filters to supress a certain frequencies.
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