Download Some physical problems: The driven, damped, harmonic oscillator

Notes for the RC and LRC Circuits Experiments
P Persans, Spring 2004
Experiment
You will use a function generator and an oscilloscope to measure the frequency
dependent impedance of an inductor-capacitor-resistor circuit.
Read Chapters 2 (background) and 4 in Napolitano. Chapter 4 goes through a similar
experiment and data analysis in detail. Chapter 6 reviews propagation of uncertainty.
Chapter 9 reviews fitting.
1) Determine the resistance of your resistor using two dc multimeters and a voltage
source. Vary applied voltage over a reasonable range and measure current
through the resistor as a function of the voltage across the resistor. Find
resistance by fitting a straight line (Ohm's Law) to the I-V line. Estimate
uncertainty using the techniques described in Chapter 9.
2) Set up the resistor and capacitor in series as shown and measure the magnitude
and phase shift of the voltage across the capacitor as a function of frequency from
10Hz to 106 Hz. Take data on a 1-2-5-10 scale in frequency. For each frequency,
you should also measure the voltage out of the function generator so you can plot
the gain across the capacitor. From this data determine the capacitance by plotting
the theoretical log gain against log frequency as discussed in Napolitano Ch 4 and
guessing C until it produces a good fit to the data.
You are not required to do a statistical fit to the data; just try to make a good fit to
the eye.
o-scope
ch1
wf
gen
o-scope
ch2
3) Set up the R-L-C circuit in series across the function generator and measure the
magnitude and phase of the voltage across the resistor as above. Make sure the
grounds of the two oscilloscope channels and the wavefunctions generator are at
the same point! You will observe a peak in the output amplitude (gain > 0.2)
across the resistor. Take sufficient data about the frequency of the peak so that
you can compare your observation to a Lorentzian function and thus determine
the inductance.
You are not yet expected to numerically fit the data in parts 2 and 3 of this
experiment. It is sufficient to overplot the data with theoretical curves and to find the
best parameters (C and L) by visual observation.
Your discussion should include plausible estimates of your uncertainty in R, C, and L.
Uncertainty in R can be quantitatively estimated from the quality of your fit to the
data and estimates of the uncertainty in the raw I, V measurements.
Theory for the LRC circuit driven with a cosine voltage at frequency 
R
V
L
C
The differential equation for the charge on the capacitor and flowing through the other
elements is,
q
V
R'
1
q
q  0 cos  t or q   q  02 q  a cos  t where R '  R  RL
L
LC
L
includes all of the resistances in the circuit, not just the single known resistor. This is just
the equation for a driven harmonic oscillator.
The homogeneous solution is:
qh (t )  q0 e t  cos(2t    ; with 2  02   2
The real part of the solution to q   q   0 q  ae will satisfy the equation
above, so we will solve the complex equation and then take the real part. Assuming a
solution qp(t)=Re(z0eit), substitution gives:
i t
2
 2 z0 eit   i z0 eit   02 z0 eit  aeit
or
 
2
  i  
2
0
z
0
 a hence q p (t ) 

aeit
2
0
   i 
2
.
Simplifying and finding the real part, and finding the real part:
Re( q p (t )) 
with
 


tan  
a


2 2
2
0

2
0
2
 
2

1/ 2
2
cos  t   

so that:
q(t)  Aqh (t )  q p (t )  Ae  t cos( 2 t  0 ) 
 
a
2
0


2 2
  2 2

1/ 2
cos  t   
.
Note that the homogeneous solution decays to zero so that the long time solution for a
harmonically driven oscillator is just a harmonic wave.
Since we actually measure the voltage across the resistor, we want the current
so the gain is
g 

R
  
2
0

2 2
 
2

1/ 2
2
dq
dt
.
Here is some data taken by T. Markkannen in 2002 with theory using L=0.5 H, C=1x10-7
F, R=1 k  , and RL=1.5 k  .
0.5
Gain
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
Angular Frequency (rad/s)
2.5
1.5
2
Angular Frequency (rad/s)
2.5
3
4
x 10
100
Phase
50
0
-50
-100
0
0.5
1
3
4
x 10