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Van der Pol
Convergence

The damped driven oscillator
has both transient and
steady-state behavior.
q
• Transient dies out
• Converges to steady state
q  2q  02 q  f cos t

2 

f cos t  arctan 2
2 
0   

q  ae t cos 02   2 t   
02   2 2  4 2 2


q
Equivalent Circuit

L
vin
C
• Inductance as mass
• Resistance as damping
• Capacitance as inverse
spring constant
v
R
q
C
dq
vR  Ri  R
dt
vout  vC 
vL   L
Oscillators can be simulated
by RLC circuits.
d 2 v R dv 1
2


v


V0 sin t
2
dt
L dt LC
2
di
d q
 L 2
dt
dt
vin  vL  vC  vR  0

R
2L
02 
1
LC
Negative Resistance

Devices can exhibit negative
resistance.
•
•

Negative slope current vs.
voltage
Examples: tunnel diode,
vacuum tube
These were described by Van
der Pol.
R. V. Jones, Harvard University


d 2v d
3
2
2


v


v


v


V0 sin t
0
2
dt
dt
Steady State

v  12 V (t )e  it  c.c.

• Time varying amplitude V
• Slow time variation

1 2
0   2 V  iV
2
 2V0
1
3 2 
 i  
V V  i
0
2
4
2

V 
2
4
3
Assume an oscillating
solution.

The equation for V follows
from substitution and
approximation.

The steady state is based on
the relative damping terms.
Frequency Locking

V (t )  V ei(t )
V
1
3 2 
V   
V V  0 cos 
2
4
2

2
2

  V0 sin 



0
 

2
2V
  d  l sin 

l
V0
2V

3V0
8
The amplitude term can be
separated.
• Two coupled equations
• Detuning term d
• Locking coefficient l

The detuning is roughly the
frequency difference.

For small driving force the
locking coefficient depends
on the relative damping.
Relaxation Oscillator

The Van der Pol oscillator
shows slow charge build up
followed by a sudden
discharge.

The oscillations are self
sustaining, even without a
driving force.


y   1  y 2 y  y  0
Wolfram Mathworld
Limit Cycle

The phase portraits show
convergence to a steady
state.

This is called a limit cycle.
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