Peter's Physics Pages

Peter's Index Peter's Index  Physics Home Physics Home  Lecture 15 Lecture 15  Course Index Course Index  Lecture 17 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

circuit diagram

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:
Each ,

and

For example, the common current is calculated as:

equation

Knowing the common current, each of the potential differences can be calculated.



Parallel Circuit Strategy

equation

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:

equation

For the three pure circuit elements:

equation

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:

equation


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



equation
                     equation

Admittances in parallel

For the parallel circuit shown below, the strategy is to add the currents.

equation
equation


With Conductance and Susceptance:

equation



Example

parallel ac circuit

For the circuit shown on the left, find
(a) the admittance of the circuit,
(b) the r.m.s. current drawn from the e.m.f.,
(c) the r.m.s. current in each branch,
(d) the power factor for the circuit, and
(e) the total power supplied by the e.m.f..

total admittance
total current

    branch currents

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.

equation

(e) The total power supplied is:

equation


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.

circuit diagram

The total circuit admittance is:

equation

This will be resonant because the susceptance can become zero. The circuit will have two natural frequencies, ω0 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 capacitor will have a potential difference across it, but no current through it.

• The admittance is also real when:

equation

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 equation.

circuit diagram

Writing the current, first with Cartesian admittance and then Polar form:

equation

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 anti-resonance.

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.

current equation

current graph


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:

equation

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 is also zero when:

equation

The phase angle changes rapidly near resonance.

equation


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.

power graph
equation

The Quality factor and the bandwidth

The quality factor for parallel resonance is defined to be the same as in series circuits:

equation

Writing the natural frequency in terms of Q:

equation

For high Q circuits:

equation

For high Q circuits, substituting the current for twice resonance power (eventually) gives the upper, u, and lower, l, frequencies:

equation

For this particular circuit, the frequencies within the band width Δω around ω0 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.

equation

The admittance at resonance is written in terms of Q:

equation

Hence

equation


The current in the inductive branch at resonance is given as:

equation

For high Q circuits

equation

The current in the capacitive branch at resonance is given as:

equation

For high Q circuits

equation

Collecting the currents:

equation

It may be pictured that a circulating current flows as shown.

equation

The circulating current has magnitude Q times the current supplied; and is 900 ahead of it in phase.



Example

For the circuit shown below find the resonant frequency.

equation
equation

For resonance, the imaginary part is zero, i.e.:

equation
equation

Notice that if then the circuit is resonant at all frequencies.

With this circuit, for a single resonance to occur, equation



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 anti-resonance.
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.


Peter's Index Peter's Index  Physics Home Physics Home  Lecture 15 Lecture 15  top of page top of page  Lecture 17 Lecture 17 

email Write me a note if you found this useful