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Keele University Physics Laboratory 11
A.C. Circuits
1. Introduction
In experiment F the properties of LCR circuits are studied from the point of view of the differential equations that these circuits satisfy. For input waveforms of arbitrary shape (e.g. transient steps or square waves) the output waveforms were generally of a different shape, sometimes markedly so. A.C. circuit theory is concerned with the response of LCR circuits to a very special waveform type, namely the sinusoidal form. There are at least two reasons why this waveform shape is special and important.
- The differential equations for LCR circuits are linear (i.e. they involve, I, V, dIdt etc. not I^2 , V^2 ,
dI^2 dt
^
^ etc.). From this it follows mathematically that for an applied sinusoidal voltage the resulting currents and voltages everywhere in a LCR network will also be sinusoidal with exactly the same frequency.
- A mathematical theorem (Fourier's theorem) tell us that even non-sinusoidal waveforms (square, triangular etc.) can be expressed as a sum of sinusoids of different frequencies. Thus, rather surprisingly perhaps, a knowledge of the response of LCR circuits to sinusoidal inputs of all frequencies (the "frequency response of the circuit") characterises these circuits completely and determines their response even to non sinusoidal inputs.
The calculation of such frequency responses is the subject matter of A.C. circuit theory. As you will learn from the lectures (PHYS-120 + PHYS-126) it is a characteristic feature of A.C. theory that sinusoidal waveforms are represented by complex exponential functions rather than simple trigonometric functions. The possibility of such representations rests essentially on the mathematical identity
Similar mathematical methods are used elsewhere in Physics, everywhere in fact where oscillations of some kind occur. This includes such areas of physics as wave theory, physical optics, diffraction and scattering theory, quantum theory and wave mechanics, mechanics of vibrations etc. It should be realised
cos( t) = 12 ( exp( i t ) + exp( i t ) ) = Re( exp( i t ) ) (1)
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therefore that an understanding of A.C. theory is not only important in itself but is also essential to an understanding of the mathematical basis of a very much wider realm of physics.
2. Equipment and Measurements The circuit board provided gives several simple LCR circuits with which to test A.C. theory. The quoted “R” values are ± 2% whereas the quoted “C” values are only ± 10%. Also provided are a signal generator, a double beam cathode ray oscilloscope (C.R.O.) and a digital frequency meter. Set the signal generator to sinusoidal output and select an output level of a few volts. If your generator has a choice of output impedance select the high Z output. The amplitude and relative phase of the voltage waveforms will be measured by use of the oscilloscope. It is important therefore that the attenuator switches for both channels should be in their calibrated positions. For good experimental accuracy it is also important to make use of the full height and width of the C.R.O. screen. Ensure that peak to peak voltages (equal to twice the amplitude) correspond to more than half the screen height. Similarly set the timebase control so that one half
wavelength (corresponding to a phase change of 180° or radians) occupies more than half the screen width. The relative phase between two waveforms is obtained from measurements of the relative displacement of waveform maxima on the screen. In determining the sign of the phase shift between two waveforms remember that a positive phase shift corresponds to a waveform advanced or "leading" in time and a negative phase shift to a retarded or "lagging" waveform. Remember also that time advances from left to right as you view the C.R.O. screen. Adjust the intensity and focus controls to obtain fine lines on the screen and minimise parallax errors by viewing the screen square on. These C.R.O. measurements will be the dominant source of experimental error in these experiments. The error in the frequency measurements will be negligible in comparison.
3. RC circuit For the RC circuit of figure 1 Kirchoff's laws give
and therefore
V (^) i = ZI= i^1 C+R I
^ (2)
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Procedure a) Use R and C 1 on the circuit board to set up the circuit of figure 1. Trigger the timebase of the C.R.O. from Vi and observe Vi and Vo on a dual beam display.
b) Vary the frequency, f , to find the value that makes A = V V
o = 1 / 2 i
. Use equation (4a) to obtain a
value for the unknown C 1.
c) Measure the relative amplitude A and relative phase of the RC circuit at several other frequencies to cover a wide range on either side of this measured frequency. (e.g. 100 Hz to 50 kHz). Choose about six frequencies which will give approximately equal spacing in log( f ) (100 Hz, 500 Hz, 1 kHz, 5 kHz, 10 kHz, 50 kHz perhaps).
d) The Linefit program can be used to plot your experimental values of A and and to compare them
with calculated curves from equations (4a) and (4b). To plot a graph of A with Linefit enter values of log 10 ( f ) as your x-values, values of A as your y-values and your errors in A as your error values. Then select Function A.C. Equations Eqn. 4-A on the menu bar and you will be prompted for an RC 1 value for your circuit. Enter the value and click OK. Then click on Plot (note do not click on Fit) and your data and calculated curve should be plotted. You follow a similar procedure to plot the values (Eqn. 4-Phi). Note the phi values entered should be in degrees not radians.
- RL circuit For the RL circuit of fig. 2 Kirchoff's laws give
and
which lead therefore to
R i L R A i L V
V
i
^1
2 1 /^2 (^0) ˆexp( ) 1 exp tan (7)
Vi = ZI = (R + i L) I (5)
V (^) o = RI =^ i
R
Z = V^
1 + i^^ RL
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where 2 1 /^2 ˆ 1
R
A^ L (7a)
and
R
tan^1 L (7b)
Fig. 2 An RL circuit
Procedure a) Use patch leads to set up the LR circuit of figure 2. Observe Vi and Vo on the C.R.O. as before.
b) Vary the frequency, f , until A = V V
o = 1 / 2. i
Use equation (7) to obtain a value for L.
c) Measure A and for the RL circuit over the same range of frequencies as for the RC circuit.
e) The Linefit program can be used to plot your experimental values of A and and to compare them
with calculated curves from equations (7a) and (7b). To plot a graph of A with Linefit enter values of log 10 ( f ) as your x-values, values of A as your y-values and your errors in A as your error values. Then select Function A.C. Equations Eqn. 7-A on the menu bar and you will be prompted for an L/R value for your circuit. Enter the value and click OK. Then click on Plot (note do not click on Fit) and your data and calculated curve should be plotted. You follow a similar procedure to plot the values (Eqn. 7-Phi). Note the phi values entered should be in degrees not radians.
- LCR circuit The impedance, Z of the parallel LCR circuit of fig. 3 is given by
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the frequency region of maximum response. For this reason high Q circuits of this kind are often found in filters and in tuning circuits where sharp frequency discrimination is required. For high Q circuits (Q > 10 say) then equation (9) can be simplified by the approximation
Z
Q R + i LQ 1 -
Q 1 - +
= Z i
2 2 2 o 2 2 2 o
z
^
^
^
0
2 exp^ (12)
where
Z =
R Q 1 + Q 1 -
+ Q^ 1 -
= (^) tan Q 1 -
2 2 o
2 22 o 2 o
2 22 o
z -1 o
2 o
1 2/
^
^
^
^
^
^
^
When this approximation is valid it can be seen from equations (13) and (14) that | Z | has a maximum
value, | Z |max = RQ^2 , at the same frequency ( = o) as that which the phase z = 0. It can also be seen that for high Q circuits
Q = = f f
o (^) o
where f = frequency width of response shown in fig. 4.
Procedure (a) Use patch leads to set up the circuit shown in fig. 5. Observe Vi and Vo on the C.R.O. as before. Since Vi = (Z + r) I and Vo = r I it follows that Z + r = V^ r V
i o
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Fig. 4 A typical resonant response curve
Fig. 5 Circuit to determine the Z of LCR
You should observe that in the region of resonance |Z| >> r so that it is justified to make the approximation
z = r VV i = Z i
o z
exp (17)
You should consider how well this approximation is justified in the following measurements and make appropriate corrections if required.
