Download AC Series Circuit: Power and Resonance

Experiment 13: AC Series Circuit: Power and Resonance
Objective: To investigate the frequency response and power consumption in an RLC
series circuit.
Prior to lab: Using equation (2) below, develop a table of 20 frequencies relative to the
resonant frequency fo that will map out the resonant frequency curve shown in
Figure 1. Plot your data to determine the best use of you information. Use the data for
L, C, and r + 10  (remember r is the loss term for the inductor) which you determined
for the LRC circuit board last week. Assume the source voltage is 5.0 V. Hint: You
should use frequency values which are grouped near resonance as well as some which
are more widely spaced when away from the resonance frequency.
Apparatus: The same RLC circuit board used last week, audio frequency sine wave
generator, 2 DMMs, wires
Discussion: The impedance of an RLC series circuit is given by
Z  R2  X L  X C 
2
where XL = L and XC = 1/C. Both the inductive and capacitive reactances are
frequency dependent ( = 2f).
Since I = V/Z, the current in the circuit will maximize (for a constant voltage source)
where the impedance is a minimum. The occurs at resonance when X L = XC. Or
f0 
1
(1)
2 LC
Furthermore the power dissipated by the circuit (Pav = I2Z cos  = I2R) will be a
maximum at resonance, and because Z = R, the phase () between the source voltage
and the current will be zero.
The following algebra leads to an expression of the average power as a function of
frequency.
Start with the expression for average power
Pav  I 2R
Substitute for V and Z
V
I
Z
Pav 
V
Z2
2
R
Substitute for Z in terms of R, XL and XC.
Pav 
V2R
R2   XL  XC 
Write the reactances in terms of L, C and .
2
V2R
Pav 
2
1 

R2   L 


C 
Multiply the capacitive reactance by L/L
V2R
Pav 
2
L 

R2   L 


LC 
Substitute the o2 for 1/LC.
Pav 
V2R

L 2o 

R   L 
 

2
2
Factor out an L.
Pav 
V2R

2 
R   L 1  2o 
 

2
2
2 2
Factor out a 1/2.
V2R
2
1
R2  2L2 4 2  2o

Simplify and then multiply R by 2/2.
V2R
Pav 
2
2 L2
R2 2  2 2  2o


Simplify.
Pav 



Pav 
V2R2

R22  L2 2  2o

(2)

2
A graph of this equation is shown in the following figure.
100
Pav
50
0
0
500
1000
1500
2000
f
Figure 1. A graph of the frequency
response of a high Q circuit.
The quality factor of the circuit is determined from the graph by finding the ratio of the
frequency spread at one half maximum power and dividing it into the resonant
frequency as expressed in equation (3) below.
Q
o
f
 o
 f
(3)
The theoretical equations on the previous pages yield Q in terms of R,L and o.
Q
oL
R
(4)
Procedure: Wire the circuit according to the circuit diagram given below (and shown in
the picture of the apparatus.
DMM
DCV (for f)
ac
source

R
L
DMM
ACV
C
Figure 3. The wiring diagram of the
apparatus pictured in Fig. 2.
Figure 2. The experimental apparatus wired to measure
the voltage across the 10  resistor
Notes:
1.  = 2f
2. Use the same circuit board as you used last week.
3. All voltage measurements are rms.
1. The DMM used to measure frequency should be set at 20 DCV..
2. Connect the other DMM across all the elements of the RLC circuit and set it on the
ACV 20V range. (You will be measuring the source voltage.) Be sure the sine wave
function is chosen on the function generator and set the source output voltage to
about 5 V.
3. Reconnect the leads of the AC DMM (just used in step 2) to measure the voltage
across the 10  resistor.
4. Make a table to record frequency, the potential difference (V10) across the 10 
resistor.
5. Vary the frequency and measure the voltage across the 10  resistor. The voltage
across the resistor should vary from about 1.0 V to a maximum and then back down
to 1.0 V as the frequency is increased. . Be sure to collect enough data (a minimum
of 20 data points between 20 Hz and 600 Hz) to be able to plot an appropriate
resonance curve.
6. Determine the resonant frequency of the circuit by adjusting the frequency output
until the potential difference (V10) across the 10  resistor is a maximum.
7. Record the source voltage determined in 3 and the resonant frequency (fo). Include
this measurement in your data table.
Meter measurement uncertainties are
1. AC voltage: 1.5% of reading + 3 digits.
2. Resistance: 0.5% of reading + 1 digit.
Analysis:
You will need to use the calculated or measured values for the L r and C from
the previous AC series circuit experiment. If you do not have the same circuit
board you will have to repeat the previous week’s measurements and
calculations.
1. From the previous experiment on AC series circuit (using the same circuit board)
calculate the resonance frequency from L and C (Eq. 1), and compare to the measured
value.
2. Plot the graphs of power (I2Rtotal) vs. frequency where I=V10/R and R is the resistance
of the 10  resistor and Rtotal is the total resistance of the circuit.
3. Use the theoretical equation for power (Eq. 2) as function of frequency, and plot its
curve on the same graph with the measured power vs. frequency graph. Caution: You
must use the (measured) source voltage and the total resistance of the circuit for
this calculation.
4. Determine the experimental value of Q. (Eq. 3).
5. Calculate the theoretical value of Q (Eq. 4) for the circuit. You must use the total
resistance of the circuit for this calculation.
Report:
Determine appropriate error, and discuss your results and analysis. There is a lot you
have done here, report on all of it.