Download introduction - University of Toronto Physics

CURRENTS THROUGH INDUCTANCES, CAPACITANCES AND
RESISTANCES
INTRODUCTION:
This experiment illustrates voltage relations in circuits involving combinations of
inductance, capacitance, resistance. The experiment is divided into two parts. Part 'A'
deals with decay of voltages or currents in circuits momentarily disturbed but then left
with constant (or zero) applied potentials. Part 'A' deals with the response-of these circuit
elements to steady A.C. voltages. Further investigation of resonant circuits is provided in
a later experiment, "The Q of Oscillators".
PART 'A' - CURRENT DECAY FOLLOWING A TRANSIENT VOLTAGE
DISTURBANCE
INTRODUCTION:
Here we study three types of series circuits:
We are considering the characteristics of each of these for the situation in which V is
constant (or zero), following some transient disturbance in V.
For the type L-R circuits (fig. 1) the solutions have the form:
di
  V

i  A exp  
  , and VR = Ri, VL 
dt
 L/R  R
Graphically, this looks like:
Far the type C-R circuit, (fig. 2)-the solutions have the form:
i = A exp(-t/RC), and VR = Ri, VC 
I
 i dt  const .
C
Graphically, this looks like:
The L-C-R circuit, (fig. 3) has three possible solutions, depending on the value of the
resistance, R, in the circuit, relative to L / C .
For:
R  2 L / C , i  A sin( t  ) exp( at ),  (I / LC)  (R 2 / 4L2 )
For:
(so   I / LC ), a  R / 2L
R  2 L / C , i  (A  Bt ) exp( at ), a  R / 2L
For:
R  2 L / C , i  [A exp( bt )  B exp( bt )] exp( at ), a  R / 2L, b  (R 2 / 4L2 )  (I / LC)
Graphically, these solutions look like:
Note that for the first two circuits, the current i, and thus also the voltages VL, VC, VR,
have the form of an exponential decay in time A.exp(-t/T), starting at some value A away
from the fina1 value, and asymptotically approaching a final value at such a rate that after
time T, the function has approached its final value within l/e = 0.37 of its initial value A.
The value T is called the time constant. Note that T for an L-R circuit is L/R, and for a CR circuit is RC, where these are in seconds if L is in Henrys, R is in ohms, and C is in
Farads.
For the L-C-R circuit, i, and also VR, VC and VL which are simply related to i by the
given equations, vary in a more complicated manner. If R were zero, then a = 0, and i
would oscillate at an angular frequency o = I LC per second, its resonant frequency,
with constant amplitude. However, with R finite but small enough so that R < 2 L / C ,
so Q = o(L/R) > (1/2), these oscillations of frequency-a bit lower than the resonant
frequency are damped, and they decay exponentially with a time constant (i/a).
If R is increased sufficiently, the oscillations are damped rapidly, and when the damping
time <r/a) is shorter than the time for undamped oscillations (i/o), the current in the LC-R circuit no longer oscillates, but decays exponentially. The case, R = 2 L / C (Q =
1/2) is the limiting case in which oscillations are no longer present, but damping is
minimum. This case is called critica1 damping. Any further damping due to increased
resistance is called overdamping.
TECHNIQUE FOR OBSERVING TRANSIENT DECAY OF CURRENTS AND
VOLTAGES:
If the time L J are sufficiently long to be followed slowly with the eye (one second or
longer) as is possible in the case of the C-R circuit, then the application of voltage V can
be done manually as in the following circuit:
Each time the switch is changed, the voltage V is changed, but then stays constant.
Behaviour of the current i through the circuit could be observed with a meter, but is more
easily observed on an oscilloscope, with the oscilloscope vertical terminals connected
across the resistance R, whose voltage VR = iR is just proportional to the current i.
Voltage VC may be observed with the oscilloscope connected across the capacitor C.
(The triggering of the sweep of the oscilloscope should be internal.)
If the time constants are short, then it is necessary to do. the switching described above
more rapidly. This is done by means of a pulse generator or a square wave signal
generator which produces a voltage which switches rapidly positive, remains there for a
period of time, then rapidly becomes negative and remains constant for the same time,
and then repeats positive again, etc., to produce a waveform as illustrated: -
If the time between changes is long compared to the time constant of the L and/or C
and/or R circuit under study- (i.e. -the signal frequency is sufficiently lower than l/T) then
this signal can be used instead of the switch suggested previously, to introduce the
transient change of voltage V required to observe the decays of current in L-C-R circuits.
The square wave generator should be connected as illustrated, in order to observe the
voltage VR or current i (note VR = Ri). (see fig. 5)
The above circuit is for the L-C-R circuit. It should be modified appropriately for the L-R
and C-R circuits. (Triggering of the CRO should be done internally off the Y, signal.)
To observe VC or VL, the capacitance or inductance should be interchanged with the
resistance so that 1n all observations the ground terminal of the square wave generator is
connected to the ground terminal of the CRO.
NOTE: It is important that the signal generator should produce a square pulse
independent of the current drawn from it. To improve the constancy of the output of your
generator, two diodes should be connected in para11el in opposite directions across the
generator output, and the output level should be turned to maximum.
EXPERIMENT:
1. Connect the R-C circuit to the dry cell through the switch, with the values of R = 470
k and C = 1.0 f as in fig 4. With sweep speed of 100 ms/cm, first display V and
then display VR. If you aim to, "click" the switch just when the CRO spot flies back
to start, you will see most of the transient decay. Adjust the vertical sensitivity to a
suitable setting, and adjust the Y position so that when the transient being examined
is finished, the spot is moving along a convenient horizontal graticule line. Estimate
the time constant. Compare this to the calculated value, RC. What effect does the
input resistance of the CRO have on this time constant? NOTE: THIS SECTION OF
THE EXPER1MENT SHOULD BE DONE QUICKLY AND YOU SHOULD NOT
TAKE ANY MORE THAN 15 MINUTES ON IT!
2. Connect the R-C circuit to the square wave generator, using the circuit of fig. 5
appropriately modified, and observe V, VR, VC for any value of R between 100 and
100 k, and for C = 0.022 f. Photograph the traces which appear on the CRO with a
polaroid camera. Mark the time scale on the photograph. What are the observed time
constants? How do these compare with the value of RC ?
3. Do the same as in 2., observing V and VL for the L-R circuit, for a value of R
between 100 and 1.0 k, and using the coil provided. (L for this coil is between 30
mH and 300 mH.) From the observed time constant, estimate the inductance of the
coil. (Note that in part 3., the coil is not a pure inductance, but acts as if there were a
perfect inductance in series with a resistance. This effective series resistance is called
the internal series resistance of the coil. What effect does this resistance have on the
observed results in this part of the experiment?)
4. Do the same as in 2., observing V and VL for the L-C circuit, using the same L and C
as in parts 2. and 3.. From the frequency of the oscillations, again calculate the value
of the inductance (assuming C is as indicated on the capacitor.) From the time
constant for decay of the oscillations, calculate the internal series resistance of the
coil. If you have already studied about Q value, calculate the Q of the coil.
NOTE ON PART A: You will have to get used to adjusting the CRO sweep speed,
and the square wave generator repetition frequency in order to have an adequately full
display of the voltages being observed.
PART 'B' - CURRENT - VOLTAGE RELATIONS FOR SINUSIODAL A.C.
INTRODUCTION:
For sinusoidal A.C., theory shows that, for a current I passing through a resistor, a
capacitor, or an inductor, the voltage across each of these is:
VR  R  I,
VC  xC  I ,
VL  x L  I
where
xC = I/C
xL = L
and where VC and VL differ in phase from I and VR is in phase with I. For a series C-R
circuit as in fig. 2, the value of capacitive reactance can be found since XC =
 VC / VR  R . These relations can be checked by plotting the magnitude of VC/VR vs
frequency on log-log graph paper. Also, the phase difference between VC and I can be
determined by observing the phase relation between V and VR as a function of frequency.
Similar measurements for an L-R circuit (fig. 1) can be used to check the above relation
for an inductance.
For the L-C-R circuit (fig. 3), resonance is observed, and the total impedance of the
circuit is given by: Z  (L  I / C)2  R 2 ,
( where V  Z  I ) Z can be found by
measuring the voltage V across the total circuit, compared to the voltage VR = RI across
the resistor, so Z = (V/VR) R
METHODS OF MEASUREMENT:
The circuit in fig. 6 is useful for comparing VC with VR in magnitude and phase, in the CR circuit.
Note that this circuit compares (+VC) with (-VR) because both voltages are measured
relative to the CRO ground. Thus it is important to be careful in interpreting observed
phase differences.
For comparing V with VR in the L-C-R circuit, the circuit in f1g. 7 is useful.
EXPERIMENT:
1. For C = 0.022 f, and R between 100 and 100 k, measure VC/VR in magnitude
and relative phase in the C-R circuit for several frequencies between 10 Hz and 1.0
MHz. (Observe whether VC leads or lags VR in phase.)
**2.
Repeat 1 for the L-R circuit, using the coil of part A of this experiment.
*3.
Measure V/VR in both magnitude and relative phase for the L-C-R circuit for a
selection of frequencies that shows the resonance curve Choose the values of C
and L as in parts 1 and 2, and use R = 100.
NOTES IN WRITING-UP
1. Plot The Z vs frequency curves on 109-109 graph paper, and interpret the shapes and
intercepts of the curves.
*
*Optional experiment
2. Plot the phase vs frequency curves on semi-log graph paper, and interpret the shapes
of the curves.
3. Comment on and Interpret any deviations you may observe of these curves from
those predicted by theory. (How many resonances did you observe in the L-C-R
circuit?)
4. If you have already studied about Q-value, calculate the Q of the L-C-R circuit using
the nominal values of R, L, C, and from your plot of Z vs frequency. Do the two
values agree within experimental errors? Compare Q values with those of Part A.
What is your value for Z at resonance? Is it equal to R? Account for any difference.
(If you wish, you may borrow the GR impedance bridge from the wicket to obtain the
values of your R, L, and C to a precision of  0.1 % along with their loss factors.)
MODEL SOLVING:
THIS IS AN ADDITIONAL EXERCISE TO BE ATTEMPTED IF TIME PERMITS.
You will probably find that your data displays features which cannot be explained on the
assumption that R, L, and C are all pure elements. Some extraneous effects are:
resistance in the inductor windings and capacitance between them, the resistance of the
dielectric of the capacitor. What, then, are the simplest 'equivalent circuits' of these
elements which might explain your data? Study the plots carefully for clues! Once you
have decided upon an equivalent circuit and have estimated the values of the extraneous
resistances and capacitances, test the model on the Computer Assisted Instruction (CAI)
program "AC".
GENERAL NOTE
The following will prove helpful in the execution of the experiment: Draw a diagram of
each circuit you will be using (in the form of fig. 5, for example) for all sections of the
experiment before coming to class. When you write-up the experiment, include with each
set of graphs or CRO photos a small diagram of the basic circuit used (of the form of figs.
1,2, 3). This makes identification of the quantity being measured easier In your report,
compare the shapes of the curves obtained, and the time constants measured, to the
results predicted by the theory. Comment on the correspondence or lack of
correspondence to the theory.
For a deeper investigation of L-C-R circuits, see the experiment "The Q of Oscillators"