EE 242 EXP 7: PARALLEL RESONANCE 1
PURPOSE:
To experimentally demonstrate parallel resonance phenomena and to verify that the circuit theory accurately
predicts the observed circuit behavior.
LAB EQUIPMENT:
Agilent 33120A Function Generator (FG)
1 Agilent 34410A Digital Multimeter
1 Agilent 54621A Oscilloscope
1 Impedance Bridge (1 per lab)
1
2
1
1
5
3
Resistance Decade boxes, 100 Ω/step
5 mH (nominal) standard or high-Q inductor
Bag of short leads
Banana-Banana leads
BNC-Banana leads
DISCUSSION
Our EE 212 discussion of bandpass circuits and functions is, for simplicity, limited to pure-element
equivalent circuits. Unfortunately, applying pure-element analysis to circuits of practical elements
leads to poor results if no compensation is made for the use of non-ideal circuit elements.
Compensation is done via modeling a practical capacitor (or an inductor) as a parallel or series
combination of a pure capacitor (or an inductor) with a resistor as shown below.
Practical Capacitor
CP
RP
Parallel model
RS
RP
LP
Parallel model
RS
CS
Series model
1
Practical Inductor
LS
Series model
Original experiment 7A, amended and revised 02/25/09, John Saghri
1
For the series model, the lower the value of Rs is, the higher is the quality of the inductor or the
capacitor. Conversely, a higher quality capacitor (or inductor) results in lower Rs. For a parallel
model the opposite is true, i.e., the higher the quality of a capacitor is, the higher is its Rp.
The quality factor Q for an inductor and a capacitor is defined below for both series and parallel
models.
QS = XL / RS = XC / RS
QP = RP / XL = RP / XC
where XL = jωL and Xc = 1 / jωC
Note that the quality factor is frequency dependent. Note that for a capacitor the quality is typically
specified by the “Dissipation factor” D which is the inverse of the Q, i.e., D = 1 / Q. Since QS = QL
for an inductor (or a capacitor), we can obtain the conversion formulas below to convert from the
parallel model to the series model, or visa versa.
Inductor
Parallel-series model
conversion formulas
RS = ω LS /Q , RP = ω LP Q
2
LP = LS ( 1 + 1 / Q )
2
RP = RS ( 1 + Q )
Capacitor
Parallel-series model
conversion formulas
RS = D / ( ω CS )
,
RP = 1 / ( ω CP D)
2
CP = CS / ( 1 + D )
2
RP = RS ( 1 + 1 / D )
PRELAB
1. A measurement of one of the lab’s 5 μF (nominal) capacitors with the impedance bridge (set on
series-model at 1000 Hz) yielded a capacitance of CS= 4.98 μF with a dissipation factor of D=
0.036. Obtain the parallel-model capacitance CP and the series-model resistance RS and the
parallel-model resistance RP associated with this practical capacitior inductor.
2. Form a parallel RLC circuit with the parallel equivalent models of practical L and C (with no
external resistor, i.e., R = ∞) as shown below. Convert the circuit to an equivalent parallel RLC
2
½
circuit. Determine the resonant frequency ωo= 1 / ( LC ) rad/s, the bandwidth B = 1 / ( RC )
rad/s, and the quality factor of the circuit (a bandpass filter) Qo = ωo / B (B in rad/s).
3. Plot the magnitude of the total impedance of the circuit versus the frequency and identify the
resonant and cutoff (half-power) frequencies and the bandwidth on the graph.
PROCEDURE
1.
Obtain a 5 μF capacitor. Use the impedance bridge to find CS and D (dissipation factor) at
1000 Hz
(Note that ω = 2πf = 2π1000 = 6282 rad/sec)
2. Obtain the series-model resistance RS using the formulas below
QS = XCs / RS = 1 / (ω RS CS)
RS = 1 / (ω CS Q)
but Q = 1 / D, therefore,
RS = D / (ω CS)
3. Convert to parallel model using the formulas:
2
RP = RS ( 1 + 1 / D )
2
CP = CS / ( 1 + D )
4.
Obtain the 5 mH high-Q inductor. Use the bridge to find LP and Q (should be close to 100).
Obtain the parallel-model resistance RP using the formulas below:
Q = RP / XL = RP / (ω LP)
RP = Q ω LP (LP ≈5 mH and RP ≈ 3142 Ω)
3
5. Set up the circuit of Figure 1. Note that the heavy (or broad) connecting lines represent
connections to be made with short leads. The equivalent circuit is the lower diagram in Figure
1.
Figure 1. Parallel RLC circuit and its pure-element equivalent
6. Change R1 to ∞, 500, and 100 ohms and find the following:
ωo = 1 / ( L C )½
, B = 1 / ( R C),
Q o = ωo / B
Note that R in the above equation is the equivalent of R1 in parallel with Req. Use the bridge
values for L and C.
7. For the R1 = 500 and 100 ohms cases, while maintaining ⎪V2⎪ = 0.5 Vrms , change the frequency
in the range of 600-1200 Hz and measure the input transfer impedance Zin. Note that the 200 Ω
current-measuring resistor is not included in Zin .
⎪Zin⎪ = ⎪V1⎪ /⎪I⎪ = ⎪V1⎪ / [(0.5 / 200)]
4
Take more measurements near the resonant frequency of ωo = 1 / ( L C )½. For the R1 = ∞
case, you will likely not be able to produce and maintain ⎪V2⎪ = 0.5 V. For this case, you can
use a smaller value of ⎪V2⎪ (in the mV range) instead of 0.5 V.
8.
For each value of R1, construct a table with columns representing Frequency, ⎪V1⎪, ⎪Zin⎪,
and ∠Zin . Use the Quick measure utility of the scope to read off the phase difference between
V1 and I (or equivalently between V1 and V2). However, since V2 is measured in the opposite
direction via channel 2 of the scope (to ensure a common ground for both channels), you
should subtract 180 degrees from the phase difference that you read off the scope.
9. On the same graph, plot the magnitude of Zin versus the frequency for each value of R1.
10. On the same graph, plot the phase of Zin versus the frequency for each value of R1
Questions:
1.
If you were to plot ⎪V1⎪ versus the frequency for a given value of R1, how would it compare
with the corresponding shape of your plot of ⎪Zin⎪ versus the frequency? Explain.
2.
For each value of R1, obtain the resonance frequency from your graphs of ⎪Zin⎪ versus the
frequency and compare it with its expected value as determined in step 6. Explain the
difference.
3. From your graph of ∠Zin .versus the frequency, determine ∠Zin at resonance for each value of
R1. Compare these values with the theoretical values and explain the difference.
4.
At resonance, Zin is purely resistive and its magnitude should equal the true value of R in the
parallel RLC circuit (with pure components). Your values of ⎪Zin⎪at resonance should be close
to Req⎪⎪R1. For each value of R1, determine the percent difference between your measured
⎪Zin⎪at resonance and calculated Req⎪⎪ R1.
5. For each value of R1, measure the bandwidth B from the corresponding plot of Zin(ω). Compare
the measured values of B with the theoretical values (B = 1 / RC) and explain the differences.
6. How would you describe the characteristic of your parallel RLC circuit as a filter (i.e., low-pass,
high-pass, band-pass, or band-stop)?
5