Experiment 2P AC Circiuts: (I) The RC-Circuit



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

Theory

|Alternating Current (AC) |[pic] |

|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 · sinωt = IMAX · sin 2π f t (1)

where IMAX = maximum amplitude of the current; f = frequency; ω = angular frequency = 2πf . Both f and ω are measured in hertz (= cycles/second (cps) for frequency f and radians/sec for angular frequency ω).

Phase Relationships

When an AC given by equation (1) flows through a resistor, the instantaneous voltage* between the terminals of the resistor is:

vR = i R = IMAX R · sinωt (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 at in equation (1), the current still flows to and from the power supply, since the capacitor is alternately charged and discharged. The instantaneous voltage vC across the capacitor is:

vC = -[pic] · IMAX · cosωt = [pic] · IMAX · sin(ωt + [pic] ) (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 [pic], in radians)”, or that the “current leads the voltage by 90º” (see Fig. 2).

[pic]

RMS Values

In AC circuits, harmonically-varying quantities like voltages and currents are characterized by their amplitudes. It is customary to use effective values defined by:

Effective Value = RMS value* = peak amplitude (4)

[pic]

e.g. IRMS = 1 IMAX VRMS = 1 VMAX

[pic] [pic]

Phasor Diagrams

Harmonically-varying quantities are customarily represented by phasors (vectors in this context) in a PHASOR DIAGRAM (vector diagram), constructed as follows:

(i) Choose a scale in volts appropriate for your data.

(ii) Plot all voltages using their RMS values, in the chosen scale.

(iii) Voltages in phase with the current are plotted as vectors in the positive

x direction (to the right) - see Fig. 3.

(iv) Voltages lagging the current by 90° are plotted as vectors in the negative

y direction (down) - see Fig. 3.

(v) 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:

[pic]

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/ (c, is called the reactance of the capacitor (capacitive reactance) Xc:

Xc = 1 = 1 (6)

[pic] [pic]

and is measured in ohms when C is in farads and ω is in hertz and f 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 ( R2 + Xc2 = I · ZRC tan ØRC = -Xc (9)

R

where the quantity ZRC = ( R2 + Xc2 is the impedance of the RC combination and is measured in ohms.

Procedure Part I. Constant Frequency

Your instructor will explain to you how to use the power supply and the AC multimeter. Set your voltmeter so that you will be using the 10 volt AC scale (red scale) on the multimeter, Read it to 0.05 volt accuracy. Ask your instructor for help if you are not sure that you can do this correctly.

a) With the AC power supply unplugged and its

power switch off, set up the circuit in Fig. 4,

using resistance R1 [pic] 4,000Ω. Record its exact

value. Set the voltage output knob to its

minimum position.

Set the frequency f =2,000 hertz and record this

value. Attach your voltmeter probes to the

terminals of the power supply so you can measure

the output voltage.

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.

Slowly turn the output knob (amplitude knob) and increase the output voltage to your

maximum voltage (~9 volts). Record it to an accuracy of 0.05 volts as VOUT .

c) Measure and record the voltage VR across the resistor and then VC across the

            capacitor.  Measure and record the voltage VRC across both of them together.

            Always record to 0.05 volt accuracy.

d) Change the resistance to R2 [pic]2,000 Ω and record its exact value.

Using the same frequency (f = 2,000 hertz) return the voltage output knob to

its MINIMUM POSITION. Put your voltmeter probes on the terminals of the

power supply and set the output voltage to your maximum. (It may not be the same as

(b) earlier.) Record its value. Repeat the measurements of (c).

e) Change the resistance to R3 [pic] 1,000 Ω and repeat this process, starting as before with

the MINIMUM POSITION of the output knob.

Procedure Part II. Different Frequencies

f) Following the procedures of Part I set up your circuit using R1[pic] 4,000Ω. Set the

frequency f1 = 1,500 hertz again. Adjust the output voltage to your maximum voltage

Repeat (c) and record everything carefully.

g) Using the same resistance set the frequency to f2 = 600 hertz. Repeat (f) and

record.

h) Using the same resistance set the frequency to f3 = 400 hertz and record.

Procedure Part III. High Frequency

j) Using the same resistance R1[pic] 4,000Ω, set the frequency to 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 separate phasor diagram, as in

Fig. 3, for each of the three runs. Use graph paper. Quote what your scale is.

2) Prepare a table as shown below. To find the deduced values of VRC

and ØRC use the measured values of VR , and VC and formula (5). To find graphical values of ØRC, measure the angles with a protractor, in your

phasor diagrams.

| TABLE ONE: (CONSTANT FREQUENCY) f = _______________ |

|RUN |R |MEASURED VALUES |DEDUCED |GRAPHICAL |% |

|# |(ohms) | |VALUES |VALUES |DISCREPANCYIN VRC |

| | |VR |VC |VRC |VRC |ØRC |ØRC | |

|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 |MEASURED VALUES |DEDUCED VALUES |

|(hertz) | | |

| |VR |VC |VRC |I = VR |XC = VC |ZRC = R[pic]+ XC[pic][pic] |VRC |

| | | | |R |I | |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, |TABLE THREE |

|construct Table Three as shown, and | |

|calculate the capacitance C of your sample. | |

|Quote your results in microfarads. | |

| |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.

( In this lab small case letters for currents and voltages will be used to denote instantaneous values which are functions of time.

* RMS stands for Root Mean Square – ie the square root of the average of the squared value.

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. . . .(quote the units!) . . . .

. . . . . . . . .(quote the units!) . . . . . . . . . . .

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