Experiment 2Q AC Circiuts: (II) The RL & RCL-Circuits



Experiment 28: Alternating Currents Circuits

Purpose

1) To observe properties of simple AC circuits.

2) To observe resonance in an LRC series circuit.

Apparatus

(a) an AC generator (function generator), an AC voltmeter

(b) a set if circuit elements that includes an inductor, capacitor, and resistor

Theory

A) 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. In this experiment , the current will change sinusoidally as shown in Fig. 1:

[pic]

The instantaneous current i 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).

For AC circuits, we usually give voltage and current magnitudes by their “effective

values” or Root Mean Square (RMS) values. These are average values for sinusoidal

properties taken from the amplitude (maximum value) by the formulae:

V = 1 VMAX I = 1 · IMAX (2)

[pic] [pic]

NOTE: In this experiment, small case script letters will denote instantaneous values of currents and voltages which are functions of time. Capital letters will indicate RMS values.

B) Capacitors

When a current flows through a resistor, the instantaneous voltage across the resistor,

vR, is:

vR = i · R = IMAX · sin[pic]t · R (3)

We say “the voltage is in phase with the current” because vR and i are zero at the

same time, and they also reach their maxima at the same time.

When a capacitor is inserted in a circuit with an alternating current as in equation (1), the current still flows to and from the power supply as the capacitor is alternately charged and discharged. 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 instantaneous voltage vC across the capacitor is:

vC = -[pic] · IMAX · cosωt = [pic] · IMAX · sin(ωt + [pic] ) (4)

[pic]

The capacitor’s voltage peaks after the current’s voltage peaks. We say that the “voltage lags the current by 90º (or [pic], in radians)”, or that the “current leads the voltage by 90º”. Or we may say that “the current leads the voltage by 90º” (see Fig. 2).

C) Phasor Diagrams

Harmonically varying quantities can be represented by phasors. These represent the

phases of the quantities and are added and subtracted by the same rules as vectors(.

To draw a phasor diagram of voltages for circuits, follow these rules:

1) A voltage is represented by an arrow vector. Its magnitude is the

RMS value. Remember to write down the scale you are using in your drawing

such as 1 volt = 1 cm.

2) Voltages which are in phase with the current (as in a resistor) are plotted

in the positive x direction.

3) All other voltages are plotted according to their phase (phase angle) relative

to the current.

4) Voltages lagging the current by 90º are plotted in the negative y direction.

5) When voltages are to be added, as is the case of series connections, the result is the vector sum of the individual voltages. 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.

D) Reactance and Impedance

The quantity appearing in Equation (4), 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.

See Section F for similar quantities for inductors.

E) Inductances and Their Properties

When an inductor is inserted in the path of the current given by (1) then, by

Faraday’s Law, an induced EMF appears. The instantaneous voltage VL across

the inductor will be:

VL = [pic]L · IMAX · cos[pic]t = [pic]L· IMAX · sin([pic]t - [pic] ) (10)

We say that this induced voltage “leads the current by 90º” (or “the current lags the voltage”). The graph of VL is shown in Fig 4.

[pic]

[Compare this situation with that of a capacitor, as in formula (3) and Fig.2 where

the voltage “lags the current by 90 º”.]

The parameter L is known as inductance, and the quantity XL is the inductive

reactance. If [pic] is in radians/second, L is in Henries then XL is in ohms.

XL = [pic] · L (11)

In a phasor diagram the voltage VL is plotted in the positive y direction. The

phasor diagram for an LR series combination is shown in Fig. 5.

[pic]

F) The LCR Series Circuit.

When an inductor, a capacitor, and a resistor are connected in series then

Fig. 6 applies:

As in Section D for RC circuits, the following definitions apply:

XTOT = XL - XC (TOTAL REACTANCE) (14)

ZTOT = R2 + XTOT2 (TOTAL IMPEDANCE)

and:

I = VLCR tanØLCR = XTOT (15)

ZTOT R

G) Resonance

The total reactance

XTOTAL = XL – XC = [pic]L – 1 (16)

[pic]C

depends on frequency f (recall: [pic] = 2πf ). When the frequency happens to be

f RES = 1

2π [pic] (17)

then XTOTAL = 0 and we say the circuit is in resonance with the applied frequency.

Preliminary Procedure

a) On your data sheet, record the code number of your sample, and the values and

units of your circuit elements R, L, and C.

b) Your instructor will explain the operation of your AC generator, and multimeter.

In this experiment, the multimeter will be used as an AC voltmeter and

should be set to the 10 V AC scale. Try to read this scale within 0.05 volt accuracy.

Ask for help if you are not sure how to use these instruments.

Procedure Part I. RC Series Circuit

|c) Assemble your circuit as shown. |[pic] |

|Set the frequency to 2,000 Hz. | |

|Attach the AC voltmeter wires to the | |

|output terminals of your AC generator | |

|so you can measure VRC=Vout. | |

| | |

|Ask your instructor to check your | |

|circuit before turning on the power. | |

| | |

| d) Copy this table to your data |f |VR |VC |VRC |

|sheet. Adjust the output |(Hertz) | | | |

|voltage to the maximum (but not more | | | | |

|than 9.20 volts). | | | | |

| | | | | |

|Measure and record | | | | |

|voltages VR, VC, and VRC | | | | |

|(see Fig. 3 above). | | | | |

| |2,000 | | | |

| |600 | | | |

| |400 | | | |

e) Repeat (d) for frequencies 600 Hz and 400 Hz.

Procedure Part II. LR Series Circuit

| |[pic] |

|f) Assemble the circuit shown here, with | |

|f = 2,000 Hz. Proceed as in (d) and | |

|(e),but use 4,000 Hz and 6,000 Hz | |

|as your frequencies. Measure | |

|VR, VL, and VLR and record them | |

|in a table. | |

Procedure Part III. LCR Series Circuit

| g) Assemble the circuit shown, |[pic] |

|with f = 2,000 Hz. Check the output voltage| |

|to make sure it is still | |

|less than 9.2 V. | |

|. | |

|Measure and record VL, VC, VLC, VR, and VLCR on| |

|your data sheet. (Note: only one frequency | |

|is used in this part.) | |

Procedure Part IV. Series Resonance

h) Using the same circuit as in (g), attach the voltmeter across both L and C – so that it

will read VLC.

Slowly reduce the frequency from 2,000 Hz and watch the voltmeter needle

carefully. The voltage should decrease until at some frequency fRES it will

start increasing again. Go back and forth a few times so you

can determine fRES as accurately as possible. Record fRES on your data sheet.

DISASSEMBLE ALL CONNECTIONS AND UNPLUG THE GENERATOR WHEN DONE.

Lab Report

Part I. RC Circuit

1) Copy the table below. Transfer the measured values from your data sheet. Then complete the table by using formulae (5). Display the % differences between the measured and calculated values of VRC. Use their average as the basis in the denominator.

|TABLE ONE: RC CIRCUIT. BASIC RESULTS |

|FREQUENCY |MEASURED VALUES |CALCULATED VALUES |

|f | | |

|(HERTZ) | | |

| |VR |VC |VRC |ØRC |VRC |% |

| | | | | | |DISCREPANCY |

| | | | | | |IN VRC |

| | | | | | | |

| | | | | | | |

| | | | | | | |

2) Use a sheet of unruled paper. With help of a ruler and a drafting triangle, draw

separate phasor diagrams for each of the three frequencies. Use the measured data and the scale: 1 cm ↔ 1 volt. Display ØRC as measured by a protractor.

|3) Copy this table as |TABLE TWO: CALCULATIONS OF CAPACITANCE |

|shown. Using R and | |

|the measured values | |

|of VR, VC, and VRC | |

|from your data sheet, | |

|fill out the table. | |

| |f | | |ZRC | | |

| |(HERTZ) |I =VR |XC = VC | |VRC |C = 1 |

| | |R |I | |ZRC |[pic]XC |

| | | | | | | |

| | | | | | | |

| | | | | | | |

| | | | | | | |

| AVERAGE | |

Display the value of and compare it with the value given on your sample.

Part II. LR Circuit

4) Construct a table for your LR circuit similar to Table One for the RC circuit.

Complete this table following a similar procedure as in (1) above.

5) Draw phasor diagrams similar to those for the RC circuit.

Note: Calculations like those in (3) are NOT required.

Part III. LCR Circuit

6) Using your measured values:

(6a) Calculate the % discrepancy between (VL – VC) and VLC.

Use VLC as the basis.

(6b) Calculate the % discrepancy between VR2 + VLC2 and VLCR.

Use VLCR as the basis.

Part IV. Resonance

7) Calculate fRES from formula (17), using L and C from your sample. Calculate

the % discrepancy between fRES and your measured value of fRES. Use your

calculated value as the basis.

( Caution: Phasors also have some additional properties which vectors do not have!

-----------------------

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