Download Wein Bridge Oscillators

Wein Bridge Oscillators
Additional Notes
MathCAD Application File
Wienbridge Oscillator - Transfer Function - Frequency Analysis - P. F. Ribeiro
i  1  20
j  1  20
x  450  i 5
w  5000  500 j
R1  10000
C1  0.01 10
R2  10000
C2  0.01 10
Ra  x
Ra
i
i
j
6
6
i
Rb  250
10
 500
wo 
jj  1
1
C1 R1
4
wo  1  10
Considering the transfer function of the circuit:
f
i j

Ra 

 1  i   R2 C1 jj w
j
Rb 

 jj wj2R1R2C1C2   jj wj(R1C1  R2C1  R2C2)  1
Sensitivity of the Loop Gain versus frequency for different Amplifier Gains
f1 j
f 10  j
wj
f 20  j
wj
wj
s  1  jj 999
Given
A  3
s 2R12C12  s 3R1C1  1
R2 C2 s  A
5
ss  Find ( s )
ss  3.222  10
4
 i  10
t  0  0.0001  0.01
y ( t)  A  e
Re( ss)  t
 sin ( Im( ss )  t)
[Try R1=11395, R2=10005]
y ( t)
Frequency of Oscilation
 
t
1
R1 R2 C1 C2
Gain
Ra 

i

Av   1 
i 
Rb 
4
  1  10
Gain
Ra 

i


Av  1 
i 
Rb 
Avi
i
Sensitivity of the Loop Gain versus Amplifier Gain for different frequencies
3-D of the Magnitude of the Transfer Function
fi 1
f i  10
Rai
Rai
f
MathCAD Application File
An Investigation of the Wien-Bridge Oscillator
Troy Cok and P.F. Ribeiro
The Wien-bridge oscillator, shown below in Figure 1, is a circuit that provides a sinusiodal
output voltage using no voltage source. The RC circuit uses the initial charge on one of the
capacitors to provide voltage to the rest of the circuit.
Figure 1: Wien-Bridge Oscillator Circuit
The gain of this circuit can be examined in terms of the individual component values. The
noninverting amplifier gain is determined by the res istors R1 and R2, according to:
G
1
R2
R1
The loop gain (or transfer function) of the Wien-bridge oscillator is determined by the
noninverting gain and the remaining circuit element s.
T j
R C G j
1  2R2C2  3 jRC
T j
R C G j
1  2R2C2  3 jRC
For stability, the phase shift is preferred to be zero. In order to accomplish this, the real part of
the denominator of the transfer function must be zero. The real part of the denominator will be
zero if the operating frequency is at resonance. The resonant frequency is:
o
1
R C
T j
At resonance, the transfer function reduces to
G
3
So, if the noniverting gain is 3, the loop gain will be 1.
To investigate the circuit in more detail, we can use a PSPICE simulation. To begin, we will try
to get a unity gain. The individual component values are determined according to the transfer
function. Using standard resistor values, R1 will be set to 10 k.
G
1
R2
R1
R1  10k
solve  R2  ( G  1)  R1
G  3
R2  ( G  1)  R1
4
R2  2  10 
For a resonance frequency of 1 kHz, the resistor and capacitor values can be:
o  1kHz
R  10k
o
1
R C
C 
1
R  o
C  0.1 F
Varying the frequency of the transfer function can be examine for the calculated component
values. Both the theoretical and computer simulated data are plotted using radians.
T  
j  i
R C G j 
1  2R2C2  3 jRC
B   20 log  T 
 6
spice  unity  2 
  10  20  100000

 7
Tspice  unity
Gain (dB)
Frequency Response
B(  )
Tspice
0
20
40
10
100
1 10
  spice
Frequency (rad)
3
1 10
4
1 10
The peak gain of the frequency analysis occurs at the resonance frequency for each circuit
model. The two traces exhibit nearly identical Bode plots.
5
A better design would cause the circuit to exhibit a cons tant (not decaying) oscillation. We can attempt to
update the circuit using a form of amplitude stabilization. There are a couple of available design methods, but
one of the better schemes involves the introduction diodes into the circuit. Along with the diodes, two
additional resistors are added to form an amplitude control network. The schematic for this circuit is shown
below.
The new resistors are determined according to the following equation. This ensures that the noninverting gain
of the circuit will be slightly more than 3 when the diodes are off and slightly less that 3 when one is active.
R2  R3
R1
2
So, if R1 is now 15 k, R2 and R3 can be 15 k and 16 k respectively.
R1  15k
R2  R3
R1
R2 
R2  15k
R3  18k
 2.2
R3 R4
R3  R4
R1
2
Here, the parallel combination of R3 and R4 must be slightly less than R2.
Since R3 is a bit greater than R2, a mid-range resistor value of R4 will suffice.
R4  33k
R2 
R3 R4
R3  R4
R1
 1.776
R2 
R3 R4
R3  R4
R1
 1.776
The updated circuit can again be examined using PSPICE. The resulting transient waveforms are shown
below. With the modification, the circuit appears to operate with a steady oscillation as time passes.
 0
tspice  unity
 1
Uspice  unity
Wien-bridge with Amplitude Stabilization
Steady Oscillation over Time
10
Voltage (V)
Voltage (V)
10
Uspice
0
10
0
0.02
tspice
Time (s)
0.04
Uspice
0
10
0
0.5
1
tspice
Time (s)
1.5
2
Wien-Bridge Oscillator Design
Application Notes 1
Wien-Bridge Oscillator Design
Application Notes 2