RLC Series Circuits
Resonance Behavior -- Reference = Serway Chapt. 33
There is a deep analogy between mechanical and electrical oscillators. Figures 1
and 2 demonstrate the situation and give the energy conservation equations for the
ideal case with no losses to friction or resistance.
This mechanical oscillator has energy E and angular frequency ω.
2
E = 1 m(d x ) + 1 k x2
dt
2
2
ω0
=
k
m
Figure 1.
This electrical oscillator has energy E and angular frequency ω.
2
E = 1 LI 2 + Q
2
2C
magnetic electrical
ω =
1
LC
Figure 2.
We note the analogies Q × x, L × m, 1/C × k. Extending the analogy, we can
associate a series RLC circuit with a mass-spring system with friction. Newton’s
second law applied to the mass-spring system with viscous friction (i.e. a frictional
force proportional to the velocity of the mass) F = -b dx/dt gives:
d2x/dt2 + (γ/m)dx/dt + (k/m)x = 0;
γ = b/2m
1a
the analogous electrical formula is:
d2Q/dt2 + (R/L)dQ/dt + 1/LC = 0;
ω0 = 1/(LC)
1b
for free oscillations of the resistively damped system.
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RLC Series Circuits
By providing an AC voltage source, we can form an electrical analog of the
driven harmonic oscillator from mechanics:
ε
to O-scope
Figure 3.
i=
ε / (R2 + (XL - XC)2);
XL = ωL, XC =1/ωC
This result can be obtained directly from Kirchoff's Circuit equations, or by
using the phasor calculations as in Serway’s Chapter 33. The expression ωL and 1/ωC
are known as the inductive reactance and capacitive reactance, respectively; using
dimensional analysis, you should be able to show that each has units of Ohms. At the
frequency where the inductive and capacitive reactances are equal, the system is in
resonance. Prove to yourself that the resonance frequency is equal to the free oscillation
frequency of the system (i.e. the oscillation frequency for no resistor and no AC voltage
source, as in Figure 2).
The AC current at all points in a series AC circuit has the same amplitude and
phase, and it may be expressed according to the following expression:
i = Im sin(ωt - φ)
We use Serway’s notation (p. 930) with instantaneous current i and peak current Im. For
any sinusoidally varying quantity we can find an average rms (root mean square) value
equal to the peak value divided by •2; thus the average current is related to the peak
current by Irms = Im/•2. The instantaneous voltage across the resistor is in phase with
this instantaneous current i, and is proportional to the resistance R. The average power
delivered by the AC voltage source is dissipated as heat in the resistor, and this average
power is given by:
Pav = (Irms 2)(R);
Irms = Vrms / •(R2 + (XL - XC)2)
The sharpness of the resonance curve as a function of frequency is given by the
quality factor Q0, the ratio of the resonance frequency to the width between half-power
points in frequency units. You should be able to show that the width between halfpower points is given by ∆ω = R/L, so that the quality factor is given by Q0 = ω0 (L/R).
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RLC Series Circuits
Procedure
Connect the AC power source, inductor, capacitor, and resistor in series as shown in
Figure 3. Initially, use component values C = 10-7 farad , L = 10-2 Henry, and R = 600
Ohms. Set the signal generator at sinusoidal output, and use the oscilloscope to
measure the instantaneous voltage VR across R , which is proportional to I.
Using the oscilloscope:
Use the “Channel “Y” input, and the “x1 probe” values for the voltage
settings.
Adjust the trigger threshold until you see a trace on the screen
Adjust the time scale to show one or more cycles
Adjust the vertical scale to show the entire trace full scale
Set the vertical input to “AC coupled”
Make sure the fine adjustment knobs are locked into position
Use the vertical control knob to center the trace vertically
Use the horizontal control knobs to move the trace horizontally
Using the AC power supply:
Use the sinusoidal setting.
Use the 600 ohm output (“High”) and connect the end of the circuit to one
of the common grounds (“Low”)
Use the frequency adjustment knob to change the frequency
Use the amplitude adjustment knob for initial amplitude setting
Measuring resonance frequency and half-power points:
Sweep across frequencies to find the maximum amplitude -- you may
want to expand the scope vertical scale and/or adjust the vertical offset
for more precise determination of the maximum frequency.
Return to centered settings when sweeping frequencies to find the halfpower points (half-power points are discussed in question b below)
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RLC Series Circuits
Questions
a) Study the forced oscillations of an RLC circuit, using component values C = 10-7
farad L = 10-2 Henry, and R = 600 Ohms. Keeping the signal generator output
constant, measure the magnitude of the maximum amplitude of VR (i.e. VRmax) as a
function of frequency. Plot your result and note the position of the peak. Compare
the experimental resonance frequency with the oscillation frequency calculated from
Equation 1b. Note that this formula gives the angular frequency of the resonance, in
radians/sec.
b) The average power dissipated in the resistor is given by:
P = (1/2)(Imax2)(R) = (Irms2)R
Measure the frequencies corresponding to the two half-power points for the circuit
used in (a) above.
c) Repeat (a) and (b) above, substituting a new resistor value R = 300 Ohms for the
previous R = 600 Ohms. By what factor does the width of the resonance (distance
between half-value points in frequency units) change? [Note -- the formulas in the
text above assume zero resistance associated with the AC voltage source, while your
real power supply will have some internal resistance and impedance.]
d) Using the R = 600 Ohm resistor as in part (a) above, substitute capacitors with C = 106 farad and C = 10-8 farad. Find the resulting peak resonance and half-power
frequencies. Determine the fractional resonance width (distance between halfpower frequencies divided by the peak frequency) for each of the 3 capacitor values.
EXTRA CREDIT:
1. Compare the phase of the voltage across the inductor with driving voltage, by using
two oscilloscope inputs simultaneously. Does the Inductor voltage lag or lead the
driving voltage? What is the phase difference?
2. Compare the phase of the voltage across the capacitor with driving voltage, by using
two oscilloscope inputs simultaneously. Does the Capacitor voltage lag or lead the
driving voltage? What is the phase difference?
3. Compare the phase of the voltage across the resistor with the driving voltage, by
using two oscilloscope inputs simultaneously. Does the resistor voltage lag or lead
the driving voltage?
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RLC Series Circuits