Experiment 26
Experiment 26: AC Circuits - The RC Circuit
Purpose
To study the properties of an AC circuit containing a resistance R and a capacitance C.
Apparatus
(a) an AC power supply
(b) an AC multimeter
(c) a sample containing a capacitor and three resistors
(d) a resistance decade box
Theory
Alternating Current (AC)In an AC circuit the current flows in one direction for a short time, then reverses and flows in the opposite direction for an equally short time, before making another reversal, and so on. The value of the current i[(] changes in time harmonically.
(Fig. 1). /
The instantaneous current i in Fig. 1 is described by:
i = IMAX · sint = IMAX · sin 2π f t (1)
where IMAX = maximum amplitude of the current; f = frequency; = angular frequency = 2πf . Both f and are measured in hertz (= cycles/second (cps) for frequency and radians/sec for angular frequency).
Phase Relationships
When an AC given by (1) flows through a resistor, the instantaneous voltage* between
the terminals of the resistor is:
vR = i R = IMAX R · sint (2)
and is said to be in phase with the current. This means that vR and i are zero at the same instant of time, and they also reach their maximum values at the same instant of time.
When a capacitor is inserted in the path of an alternating current (1), the current still flows to and from the power supply, as the capacitor is alternately charged and discharged. The instantaneous voltage vC across the capacitor is:
vC = - · IMAX · cost = · IMAX · sin(t + ) (3)
It takes some time for the voltage to build up in the capacitor when the current flows, so that the phase of the voltage is different from the phase of the current. The voltage peaks after the current peaks. We say that the “voltage lags the current by 90º (or , in radians)”, or that the current “leads the voltage by 90º” (Fig. 2).\
RMS Values
In AC circuits, harmonically-varying quantities like voltages and currents are characterized by their amplitudes. It is customary to use “effective values” (RMS values) defined by:
RMS value = peak amplitude (4)
e.g.
IRMS = 1 IMAX VRMS = 1 VMAX
Phasor Diagrams (Vector Diagrams)
Harmonically-varying quantities are customarily represented by phasors (often called vectors in this context) in a PHASOR DIAGRAM (=vector diagram), constructed as follows:
(i) All voltages are plotted using their RMS values, on the chosen scale.
(ii) Voltages in phase with the current are plotted as vectors in the positive
x direction (to the right) - see Fig. 3.
(iii) Voltages lagging the current by 90° are plotted as vectors in the negative
y direction (down) - see Fig. 3.
(iv) All other voltages are plotted as vectors in a similar fashion, according to their
phase with respect to the current.
The reason for this is that the resultant RMS voltage across two or more circuit elements (like resistors, capacitors, inductors) connected in series, is the vector sum of individual RMS voltages. For instance, a resistor in series with a capacitor yields a phasor diagram as in Fig. 3 and formulae (5) apply:
IMPORTANT NOTE: In AC circuits, all phases are given in the range between + 90° and - 90° . Negative angles are to be used when applicable!
Reactances and Impedances
The quantity appearing in equation (3), 1/ wc, is called the reactance of the capacitor (capacitive reactance) Xc:
Xc = 1 (6)
and is measured in ohms (when C is in farads and is in hertz). The RMS values of the current and the voltage across the capacitor are related by
VRMS = IRMS XC (7)
If we have a capacitor and a resistor in series then the voltage across the resistor
alone is:
VR = I R (8)
but for the RC circuit, using the equations (5), this can be rewritten as:
VRC = I Ö R2 + Xc2 = I · ZRC tan ØRC = -Xc (9)
R
where the quantity ZRC = Ö R2 + Xc2 is the impedance of the RC combination and is measured in ohms.
Procedure Part I. Constant Frequency
the AC multimeter.
You will be using the 10 volt AC scale (red scale) on the multimeter and you are
required to read it within 0.05 volts accuracy:
Make sure that you can do this correctly! /
Set the frequency f =2,000 hertz and record this value. Set the voltage output
knob to the MINIMUM POSITION. Engage your voltmeter (multimeter)
between the terminals of the power supply, using the 10 volt AC scale.
CHECK YOUR SET-UP WITH YOUR LAB INSTRUCTOR BEFORE PROCEEDING FURTHER.
b) Upon your instructor’s approval, plug in your power supply and turn the power ON.
By slowly turning the output knob, increase the output voltage to between 9.80 and
9.95 volts (or your maximum if you cannot reach 9.8V) and record it with an accuracy
of 0.05 volts as VOUT .
c) Measure and record the voltages VR and VC across the resistor and the capacitor
separately. Measure and record the voltage VRC across both of them together.
Always maintain the 0.05 volt accuracy. This completes Run #1.
d) Change the resistance to R2 2,000 Ω (in your sample) and record its exact value.
Using the same frequency (f = 2,000 hertz) as before, return the voltage output knob
to MINIMUM POSITION. Engage your voltmeter between the terminals of the power supply and set the output voltage between 9.80 and 9.95 volts. It need not be the same as under (b) earlier, but record its value, whatever it is.
Repeat (c) and label carefully all values. This completes Run #2.
e) Change the resistance to R3 1,000 Ω and repeat (d), always starting with the
MINIMUM POSITION of the output knob. This completes Run #3.
Procedure Part II. Different Frequencies
f) Revert to your original resistance R1 as in (a) (about 4,000Ω). Set the frequency to
f1 = 1,500 hertz. Following the procedures of Part I again adjust the output voltage to
be between 9.80 and 9.95 volts. Repeat (c) and record everything (labeling carefully!).
g) Keeping the same resistance , set the frequency to f2 = 600 hertz. Repeat (f) and
record.
h) Repeat (g) with f3 = 400 hertz and record.
Procedure Part III. High Frequency
j) Using R1 4,000 Ω, set the frequency at 40,000 hertz. Adjust the output as usual. Measure and record VR , VC , VRC.
Lab Report
Part I
1) Using your measured values of VR , and VC , draw a phasor diagram as in Fig. 3,
accurately to a scale (quote what your scale is!), separately for each of the three runs. Use graph paper.
2) Prepare a table as shown below. Quote all units. To find the deduced values of VRC
and ØRC use the measured values of VR , and VC , and formulae (5). To find graphical values of ØRC, measure the angles carefully, with a protractor, in your
phasor diagrams.
TABLE ONE: (CONSTANT FREQUENCY) f = ......RUN
# / R
(ohms) / MEASURED VALUES / DEDUCED
VALUES / GRAPHICAL
VALUES / %
DISCREPANCY IN VRC
VR / VC / VRC / VRC / ØRC / ØRC
1
2
3
Fill out this table using your calculator. For the % discrepancies use the measured VRC as the basis.
Part II
3) Draw phasor diagrams as in (1) above.
4) Prepare a table as shown below. Quote all units.
TABLE TWO: (DIFFERENT FREQUENCIES) R = ......f
(hertz) / MEASURED VALUES / DEDUCED VALUES
VR / VC / VRC / I = VR
R / XC = VC
I / ZRC = R+ XC / VRC
ZRC
1,500
600
400
Fill out this table. To find the deduced values, use the measured values of VR , VC , and VRC, using formulae explicitly quoted in Table Two.
5) From the values of f and XC in Table Two,construct Table Three, as shown, and
calculate the capacitance C of your sample.
Quote your results in microfarads. / TABLE THREE
f / XC / C
AVERAGE:
Part III
6) According to your understanding of AC circuits, and the value of C which you obtained, what should be the value of XC at f = 40,000 hertz? How does this conform with your measurements of VR , and VC , as compared to VRC ? Explain, briefly.
133
[(]* In this instruction sheet, small case letters for currents and voltages will be used to denote instantaneous values which are functions of time.