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Wien Bridge Oscillator Formula, Theory, and Circuit

We know that dc is produced by batteries. But how we produce ac? One way is using oscillators, which are circuits that convert dc to ac.
An oscillator is a circuit that produces an ac waveform as output when powered by a dc input.

Wien Bridge Oscillator

The only external source an oscillator needs is the dc power supply. Ironically, the dc power supply is usually obtained by converting the ac supplied by electric utility company to dc. Having gone through the trouble of conversion, you may wonder why we need to use the oscillator to convert the dc to ac again.
Wien Bridge Oscillator circuit
The problem is that the ac supplied by the utility company operates at a preset frequency of 60 Hz in some country such as United States and 50 Hz in some other nations, whereas many applications such as electronic circuits, communication systems, and microwave devices require internally generated frequencies that range from 0 to 10 GHZ or higher. Oscillators are used for generating these frequencies.

In order for sine wave oscillators to sustain oscillations, they must meet the Barkhausen criteria :
  1. The overall gain of the oscillator must be unity or greater. Therefore, losses must be compensated for by an amplifying device.
  2. The overall phase shift (from input to output and back to the input) must be zero.
Three common types of sine wave oscillators are phase-shift, twin T, and Wien-bridge oscillators. Here we consider only the Wien-bridge oscillator.

The Wien-bridge oscillator is widely used for generating sinusoids in the frequency range below 1 MHz. It is an RC op amp circuit with only a few components, easily tunable and easy to design. As shown in Figure.(1), the oscillator essentially consists of a noninverting amplifier with two feedback paths : The positive feedback path to the noninverting input creates oscillations, while the negative feedback path to the inverting input controls the gain.
Wien Bridge Oscillator circuit
Figure 1. Wien Bridge Oscillator
If we define the impedances of the RC series and parallel combinations as Zs and Zp, then
Wien Bridge Oscillator circuit
(1)
Wien Bridge Oscillator circuit
(2)
The feedback ratio is
Wien Bridge Oscillator circuit
(3)
Substituting Equations.(1) and (2) to (3) gives
Wien Bridge Oscillator circuit
(4)
To satisfy the second Barkhausen criterion, V2 must be in phase with Vo, which implies that the ratio in Equation.(4) must be purely real. Thus, the imaginary part must be zero. Setting the imaginary part equal to zero gives the oscillation frequency ωo as
Wien Bridge Oscillator circuit
or
Wien Bridge Oscillator circuit
(5)
In most practical applications, R1 = R2 = R and C1 = C2 = C, so that
Wien Bridge Oscillator circuit
(6)
or
Wien Bridge Oscillator circuit
(7)
Substituting Equations.(6) and  R1 = R2 = R, C1 = C2 = C into Equation.(4) results
Wien Bridge Oscillator circuit
(8)
Hence, in order to satisfy the first Barkhausen criterion, the op amp must compensate by providing the gain of 3 or greater so that the overall gain is at least 1 or unity. We recall that for a noninverting amplifier,
Wien Bridge Oscillator circuit
(9)
or
Wien Bridge Oscillator circuit
(10)
Due to the inherent delay caused by the op amp, Wien-bridge oscillators are limited to operating in the frequency range of 1 MHz or less.

Wien Bridge Oscillator Circuit

For better understanding let us review the example below :
Design a Wien-bridge oscillators circuit at 100 kHz
Solution :
Using Equation.(7), we get the time constant of the circuit as
Wien Bridge Oscillator circuit
(1.1)
If we select R = 10 kΩ, then we can select C = 159 pF to satisfy Equation.(1.1). Since the gain must be 3, Rf/Rg = 2. We could select Rf = 20 kΩ while Rg = 10 kΩ.

Untuk Bahasa Indonesia baca Osilator Jembatan Wien.
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