Wien
Bridge Sine Wave Oscillators
This article explains how a Wien Bridge oscillator
works and how it can be used to design a sinewave
oscillator.
Introduction
One of the simplest sinewave oscillators is the Wien
Bridge Oscillator. Any circuit requires 2 conditions
to oscillate. Tracing the path from the input, round
the feedback network, back to the input there must
be an overall phase shift of 0 degrees at one
particular frequency. In other words, any signal
travelling around this loop, must be in phase with
the original signal as it arrives back at the input
and thus add to the input signal.
As the signal travels around the loop, there will be
a loss in the system (heat dissipation in the
components, losses in the amplifier etc). Therefore
there must be some form of gain in the loop, such
that the signal arriving back at the input (having
travelled around the loop) is larger than the original signal. If these 2
conditions are met, the oscillations will be
sustained.
FIG 1 shows a typical Wien bridge oscillator. The
circuit relies on the series RC network (made up of
R1 and C1) and the parallel RC network (made up of
R2 and C2) causing a phase shift of 0 degrees at one
particular frequency at the non inverting input.
Thus from what we have said, taking the signal at
the + input, through the op amp, through R1 and C1
and back to the + input causes a phase shift of 0
degrees at one particular frequency.
FIG 1
The feedback resistors R3 and R4 around the
inverting input set the gain to maintain
oscillation.
The following involves some equations that look a
bit hair raising, but if you compare FIG1 with a
normal non inverting op amp circuit and have a rough
grasp of how capacitors vary their characteristics
with frequency, you should be able to follow them.
An easy way of analysing this circuit is to consider
the gain and phase shift caused by the Wien bridge
network (the series and parallel RC components) and
ignore the other components in the circuit, as shown
in FIG 2
FIG 2
The series RC circuit (R1 and C1) has an impedance:
w
is the frequency and is simply equal to 2∏f (where f
is the frequency in Hertz)
The term
is the impedance of the capacitor. The j term
represents a phase shift. When combined with
the
frequency term, w, implies we have an
impedance that changes over frequency. These are
obviously fundamental to the operation of our
oscillator.
So multiplying top and bottom by jwC gives
Likewise, the parallel RC network (R2 and C2) has an
impedance
The parallel combination of 2 resistors is equal to
the product divided by the sum of the resistors. The
same is true with a parallel combination of a
resistor and a capacitor.
The overall impedance of the parallel network is
simply the product divided by the sum of the
capacitor impedance and the resistance
Multiplying top and bottom by jwC gives:
Now, the transfer function of a resistor divider is
equal to:
So the transfer function then becomes of our Wien
network becomes:
Now, to keep the maths simple, we assume that both
the capacitors are the same value (and put them
equal to C) and both resistors are the same value
(and put them equal to R).
From here we can see that at
,
the transfer function becomes:
In other words, the gain from the output to the non
inverting input is 1/3.
As the j terms cancel out, there is no phase shift –
i.e. there is no j term on the top or the bottom of
the above number. We therefore have zero phase shift
from output to input, so we have achieved half of
our criteria for oscillation
We know that for oscillation to occur, the gain of
the loop has to be 1 so we need to have a gain of 3
to overcome the ‘loss’ of 1/3 going from output to
input.
With a simple op amp circuit, if an input is applied
to the non inverting terminal (+) then the feedback
resistor (R3 in FIG 1) has to be twice R4 to get a
gain of 3 since
The same is true with the oscillator circuit. To get
a gain of 3, R3 has to be exactly 2x R4. This has to
be maintained throughout the entire operation of the
circuit so a certain amount of variability has to be
built into the gain components. We also need a gain
of >3 to start the circuit up, then the gain must
limit to 3 to maintain correct operation. To achieve
this, a transistor is normally placed in the
position of R4 that changes its resistance according
to output voltage. As the output grows its
resistance increases to reduce the overall gain.
Thus we have achieved conditions for oscillation –
gain of 1 with a phase shift around the loop of
zero.
