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Keele University Physics Laboratory 2
Transient Response of LCR Circuits
1. Introduction and Theory
1.1 Circuit elements
Inductors, capacitors and resistors are the basic elements of all electrical circuits. They are an essential part of all electronic and electrical instrumentation. It would therefore be difficult to overstate their fundamental importance to physics and engineering. One way to achieve the necessary understanding of these circuit elements is through the differential equations that they satisfy i.e.
Here R is a resistance (measured in Ohms, ), L an inductance (measured in Henrys, H) and C is a capacitance (measured in Farads, F). V represents the voltage across each element when I is the electric current flowing through that element. Equations (1.a-b) apply in all cases no matter how complicated the time variation of I and V. These equations can therefore be used to predict the response of any electrical circuit to a time dependent applied e.m.f. of any arbitrary waveform shape by use of Kirchoff's Laws
V (^) R =^ IR^ V (^) L =^ L dIdt^ (1.a)
V (^) c =^ c
C I dt^ I^ =^ C
dV or dt (1.b)
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Kirchoffs 1st Law
The first law of Kirchoff simply expresses the conservation of charge. I (^) 1 I (^) 2 I 3 (2)
i.e. The algebraic sum of the currents at each junction is zero (counting incoming currents as positive and outgoing as negative). Kirchoffs 2nd Law
In any mesh having a total e.m.f. of E E V V V Vi i
1 2 3 (^) (3)
The e.m.f., E , is by definition the work done in transporting unit positive charge (i.e. 1C) around the complete circuit of the mesh in question. The voltages, Vi , across each circuit element are, again by definition, the work done in transporting unit charge between the terminals of each element. Thus this second law of Kirchoff simply expresses the conservation of energy. Combining equations (1.a) and (1.b) with (2) and (3) gives a differential equation for each mesh. eg.
E = IR + L dIdt + (^) C^1 I dt (4) or
dE dt = L d I dt
+ R dI dt
+ I
C
2 2^ (5)
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1.2 The Oscilloscope The cathode ray oscilloscope (C.R.O.) or, more briefly, the oscilloscope is an indispensable instrument in all electrical and electronic observations. Since almost all physical measurements nowadays involve some electrical or electronic features the use of the C.R.O. is widespread in physics and increasingly common in other allied subjects. An introduction to the C.R.O. is therefore one important purpose of this experimental assignment. In performing the experimental measurements that follow you should acquire an understanding of the basic features of this instrument. These include: (1) the timebase controls and the measurement of time periods and frequencies. (2) the dual beam input attenuation controls and the measurement of waveform amplitudes. (3) the trigger controls (automatic and voltage level modes). (4) X and Y shift controls, brightness and focus controls etc. For those who have little or no previous experience of the C.R.O. it would be best to start by asking a demonstrator to give an explanation and demonstration of this instrument. This initial instruction should be supplemented by further consultations as the experiments are worked through and as individual needs dictate.
Keele University Physics Laboratory 6
- RC Circuit
Fig. 2 An RC circuit
For this circuit Kirchoff's laws give
or
For the square waveform used here
and therefore
R dIdt (^) CI = 0 for 0 < t < T 2 (11)
Equation (11) has a solution of the form
the constant of integration, A , is fixed by the initial conditions at t = 0. If the generator is set so that T / 2 >> RC these initial conditions will be
E = V + V = IR + 1
C
R c Idt^ (8)
dE dt =^ R
dI dt +^
I
C^ (9)
dE dt = 0 0 < t < T 2 for (10)
I = A exp( t / RC ) (0 < t < T 2 ) (12)
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Fig. 4 Simple potential divider circuit
Clearly for the circuit of fig. 4
Unfortunately the reduced voltage, Vo , is often required to be delivered to some distant point through some cable connected across R 2 in fig. 4 (e.g. to an oscilloscope perhaps). This has the effect of adding some unintended "stray capacitance" across R 2 which would in turn alter the waveform shape of Vo so that Vo would not then be a true representation of the waveform E. In such cases a "compensating capacitor" is inserted, across R 1 , to ensure that equation(14) remains valid.
Fig. 5. A capacitance compensated potential divider
Applying Kirchoffs laws to the three meshes of fig. 5 gives dE dt
I I
C
I I
C
I I
C R
dI dt I I C R^
dI dt
^
^
^
1 1
2 2 1 1 1
1
2 2 2
2
(15.a) (15. b) (15. c)
Prove that an "undistorted" Vo satisfying equation (14) is obtained if
V (^) o =^ IR 2 =^1
E R
R + R
Keele University Physics Laboratory 9
R C 1 1 R C 2 2 (16)
Procedure Use the circuit board and a decade resistance box for R 2 to set up the circuit of Fig. 5. Display Vo and E on a dual beam oscilloscope and observe that these waveforms have the same shape only for a particular value of R 2. Hence verify equations (14) and (16) using your previously measured value of R 1.
- RL circuit
Fig. 6 An RL circuit
For the RL circuit of Fig. 6 Kirchoff's Laws give
E = (^) E = IR + L dI dt (v < t < T 2
I = ER + A tR L
o o (^) exp( / )
(17.a) (17. b)
The constant of integration, A , is fixed by the requirement that I=0 at t=0( if t 2 >> (^) RL)
Therefore:
I = ER (1 - tR L ) V = E tR L
o
L
exp( / ) exp( / )
(18.a) (18. b)
Procedure Use the circuit board to set up the RL circuit of Fig. 6. Display VL and E on the oscilloscope as before. Measure the time, t 1 , for VL to decay to half its value and so obtain an experimental value for the time constant as done earlier in Section 2. Does the experimentally determined time constant agree with the theoretical value L/R predicted by equation(18)? You will probably find that it does not. Observe the waveform of Eo. You should observe that this is not now perfectly square. Both of these effects have a
