Experiment 27

Experiment 27: ACCircuits II: LRLCRCircuits

Purpose

(1)To study the properties of an ACcircuit containing a resistor and an inductor (coil).

(2)To observe the basic properties of a series LCRcircuit.

(3)To observe resonance in an LCRcircuit.

Apparatus

(a) an AC Power Supply; an AC multimeter

(b) a sample containing an inductor (coil)

(c) two resistors, and a capacitor.

Theory

Faraday’s Law

When an inductor (coil) is inserted in the path of an alternating current (AC), then an induced EMF, εIND , appears across the terminals of the coil, according to Faraday’s Law of Electromagnetic Induction. It is commonly called an induced voltage. The instantaneous value of εIND will be denoted by VL and isexpressed by the formula:

VL = L · ÎMAX · cost = L· ÎMAX · sin(t - )(1)

The inductor voltage opposes the change in voltage of the power supply and peaks before the current peaks. We say that the “induced voltage leads the current by 90˚ (or , in radians)” or,alternatively, “the current lags VL by 90˚”. See Fig. 1 and compare it with Fig. 2 in Experiment 26.

The parameter L is theinductance of the coil and the quantity

XL = · L(2)

is the inductive reactance of the coil. If L is measured in henries (SIunit for inductance) and in hertz, then XL is in ohms.

Employing the RMS values (see Exp. 26), the RMS voltage across the inductor is:

VRMS= IRMS XL(3)

(Compare with equations (6) and (7) in Exp. 26.)

The Phasor Diagram of an LRSeries Circuit

The principles of the phasordiagrams were explained in Exp. 26. The phasor diagram for a coil and a resistor connected in series is shown in Fig. 2, together with relevant formulae (4).

Important Note: As well as its reactance, a coil also has a resistance R´ which creates

an additional voltage VR' = R´i across the coil. This voltage is in phase with the current,

so that it lags VL by 90º. What a voltmeter connected across a coil actually measures

is the phasor sum of the RMSmagnitudes of VL and VR' - that is the quantity

VLR = VL2 + VR'2 (4a)

rather than VL. However, if VR' is much smaller than VL then the voltmeter essentially measures VL .

The LCRSeriesCircuit

When a coil, a capacitor, and a resistor are connected (in that order) in series, then

Fig. 3 applies:

Relevant formulae are

VLCR = VR2 + (VL – VC) 2tanØLCR = VL –VC(5)

VR

It is also customary to define:

XTOTAL = XL – XC = TOTAL REACTANCE

ZTOTAL = R2 + XTOTAL2 = TOTAL IMPEDANCE(6)

and the following formulae then hold:

I = VLCRtan ØLCR = XTOTAL

ZTOTALR(7)

Resonance.

The total reactance

XTOTAL = XL –XC = L – 1

C(8)

depends on frequency f (recall: = 2πf ).

When the frequency happens to be

fRES = 1

2π (9)

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

Procedure Part I. The LRCircuit

a) With the ACpowersupply unplugged and OFF,
set up a circuit as in Fig. 4, using resistance
R1 ~ 4,000Ω from your sample. Record its
exact value.
You should know how to use the ACpower
supply and the ACmultimeter from Exp.26
but, if you do not, check with your instructor.
Set the frequency at f1 = 2,000 hertz and
record. Set up your multimeter to read the
10volt ACscale. /

b) Upon your instructor’s approval, turn ON the power, and adjust the output voltage to

between 9.80 and 9.95 volts (or to YOUR maximum voltage if you can’t reach 9.80 V).

Record this as VOUT to the accuracy of 0.05 V.

c) Measure and record the voltages VL across the coil and and VR across the resistor.

Measure and record the voltage VLR across both of them together to 0.05volt accuracy.

Return the outputvoltage knob to its minimum position.

d) Change the frequency to f2 = 4,000 hertz and record. Re-adjust VOUT to be between

9.80 and 9.95 volts and record the exact value. Repeat (c) above.

e) Repeat (d) above with f3 = 6,000 hertz.

Procedure Part II. The LCRCircuit

f) With the power OFF, assemble a circuit as in Fig. 5, carefully observing the sequence L-C-R. Use the resistance R2 ~ 1,000Ω from your sample and record its exact value.
Set f = 2,000 hertz and record. /

g) Upon your instructor’s approval turn ON the power. Use your voltmeter (using the

same 10 V scale) across the inductor (NOT the outputvoltage, as before!) and

adjust VL to be between 9.80 and 9.95 volts (or your maximum value). Record the exact value of VL.

h) Measure and record VL , VC , VR , and VLCR (as in Fig.3) to 0.05volt accuracy.

Procedure Part III. Resonance

i) With your circuit the same as in (f) above, start with the outputvoltage at the MINIMUM. Set f = 1,000 hertz and use the voltmeter to measure VLC (the total voltage across the LCcombination).

Slowly increase power until VLC is about 8.5 volts. Next, slowly increase the frequency. You will notice that VLC will be decreasing until it reaches a minimum

at some frequency fVLCmin. Record the value of fVLCmin and the minimum value

of VLC . This is the resonance frequency.

j) Use the same set-up as in (i) above (start with MINIMUM VOUT) but now use

the voltmeter to measure VR.

Slowly increase power out until VR is about 5 volts (NO MORE!).

Next, increase f and watch VR increasing, until a maximum is reached at some

frequency fVRmax. Record fVRmax and the maximum value of VR . Also record

VOUT in thissituation. This is also the resonance frequency.

BEFORE YOU LEAVE THE LAB:

Make sure that you have recorded the values of L and C marked on your samples and the values of resistances and frequencies which you used.

Please unplug and turn OFF the ACpowersupply and leave your station in order.

Lab Report

PartI. LRCircuit

1) Using your measured values of VL , and VR , draw phasor diagrams (see Fig. 2) for each or your three runs.

2) Draw a table as shown. Quote all physical units. The graphical values are to be measured by ruler and protractor from your phasordiagrams.

TABLE ONE: LRCIRCUIT
FREQUENCY
f / MEASURED VALUES / GRAPHICAL VALUES / %
DISCREPANCY
IN VLR
VL / VR / VLR / VLR / ØLR

Part II

3) Using your measured
values of VL, VC, and
VR, draw the phasor
diagram (see Fig.3). / TABLE TWO: LCRCIRCUIT
MEASURED: / CALCULATED
VLCR(VOLTS) / %
DISCREPANCY
IN VLCR
VL / VC / VR / VLCR

4) Using your measured values of VL, VC, and VR, calculate the expected value of VLCRaccording to formula (5) and complete Table Two.

5) From the values of L, C, andR2on your data sheet, construct and fill out

Table Three. The graphical value of ØLCR comes from yourphasordiagram. The

calculated value of ØLCR comes from Equation (7), the calculated value of ZTOTAL

comes from Equation (6).

TABLE THREE: LCRCIRCUIT. FREQUENCY: 2,000 Hz
CALCULATED VALUES: / GRAPHICAL
VALUE
OF ØLCR
(DEGREES) / ABSOLUTE
DISCREPANCY
IN ØLCR
(DEGREES)
XL / XC / XTOTAL / ZTOTAL / ØLCR
(DEGREES)

Part III. Resonance.

6) Show the average of the frequencies fVRmax and fVLCminfrom your measurements.

Use the given values of L and C to calculate the expected resonance frequency from

Equation (9). Compare your measured value with the calculated valueby displaying the % discrepancy.

7) Answer the questions:

Question #1: What is the RMSvalue of the current in your LCRcircuit at the resonance frequency?

Question #2: Should VL and VC be equal at resonance frequency (explain why “yes”

or “no”).

Question #3: In your experiment (with the parameters you had), can the

value of R′ be ignored (explain why “yes” or “no”).

1

Before reading this, make sure you have read and understood the Theory Section in Experiment 26.