Draw the circuit diagram of a Wien bridge oscillator and briefly explain its operation. Derive the expression for frequency of oscillation.

Q.) Draw the circuit diagram of a Wien bridge oscillator and briefly explain its operation. Derive the expression for frequency of oscillation.
Subject: Electronic Devices and Circuit

Wien Bridge Oscillator Circuit Diagram

A Wien bridge oscillator is a type of electronic oscillator that generates sine waves without the need for any input signal. It is based on a bridge circuit that includes a series RC (resistor-capacitor) network in one arm and a parallel RC network in the other arm. Here is a basic circuit diagram of a Wien bridge oscillator:

          +Vcc
            │
           ┌┴┐
           │ │ R1
           └┬┘
            │
            ├─────────────┐
            │             │
           ┌┴┐           ┌┴┐
           │ │ R2        │ │ R3
           └┬┘           └┬┘
            │             │
            ├─────┬───────┤
            │     │       │
           ┌┴┐   ┌┴┐     ┌┴┐
           │ │   │ │     │ │
           │C1   │C2     │C3
           │ │   │ │     │ │
           └┬┘   └┬┘     └┬┘
            │     │       │
            ├─────┤       │
            │     │       │
            │    ┌┴┐     │
            │    │ │     │
            │    │R4     │
            │    │ │     │
            │    └┬┘     │
            │     │      │
            │     ├──────┤
            │     │      │
            │    ┌▼┐    ┌▼┐
            └────┤ ├────┤ ├─────┐
                 │OpAmp│OpAmp  │
                 └─┬─┘  └─┬─┘  │
                   │      │    │
                  ─┴─    ─┴─   │
                   │      │    │
                  GND    GND   │
                               │
                              ─┴─
                               │
                              GND

Operation of the Wien Bridge Oscillator

The operation of the Wien bridge oscillator can be explained in the following steps:

Feedback Network: The Wien bridge oscillator uses a feedback network consisting of R1, R2, C1, and C2. The series combination of R1 and C1 and the parallel combination of R2 and C2 form a frequency-selective network that has a phase shift of 0 degrees at a specific frequency.
Amplifier: The operational amplifier (OpAmp) provides the necessary gain to sustain oscillations. The non-inverting input of the OpAmp is connected to the junction of R1 and C1, while the inverting input is connected to the junction of R2 and C2.
Automatic Gain Control: R3 and R4, along with the diodes or a lamp, provide automatic gain control to stabilize the amplitude of the oscillations. The resistance of the lamp or the diodes changes with the output amplitude, controlling the gain of the OpAmp.
Start of Oscillation: Initially, noise or disturbances in the circuit provide a small signal that gets amplified by the OpAmp. If the loop gain is greater than or equal to one and the phase shift around the loop is zero or an integer multiple of 360 degrees, the signal will build up with each cycle, leading to sustained oscillations.
Steady-State Oscillation: Once the oscillations reach a certain amplitude, the automatic gain control mechanism adjusts the gain to ensure that the loop gain is exactly one. This results in steady-state oscillations at the output.

Frequency of Oscillation

The frequency of oscillation for the Wien bridge oscillator can be derived from the condition that the phase shift around the loop must be zero degrees (or an integer multiple of 360 degrees) and the magnitude of the loop gain must be one.

The phase shift through the RC networks is given by:

$$ \phi = \phi_{series} + \phi_{parallel} $$

At the frequency of oscillation, the phase shift through each network is zero degrees, so:

$$ \phi_{series} = -\phi_{parallel} $$

The impedance of the series RC network (Z_series) and the parallel RC network (Z_parallel) are:

$$ Z_{series} = R + \frac{1}{j\omega C} $$ $$ Z_{parallel} = \frac{1}{\frac{1}{R} + j\omega C} $$

At the frequency of oscillation, the imaginary parts of Z_series and Z_parallel cancel each other out, so:

$$ \omega_0 R C = 1 $$

Solving for the frequency of oscillation (f_0):

$$ \omega_0 = 2\pi f_0 $$ $$ f_0 = \frac{1}{2\pi R C} $$

Where:

( f_0 ) is the frequency of oscillation
( R ) is the resistance (assuming R1 = R2 = R)
( C ) is the capacitance (assuming C1 = C2 = C)
( \omega_0 ) is the angular frequency of oscillation

Example

Let's calculate the frequency of oscillation for a Wien bridge oscillator with R1 = R2 = 10 kΩ and C1 = C2 = 100 nF.

$$ f_0 = \frac{1}{2\pi \times 10,000 \times 100 \times 10^{-9}} $$ $$ f_0 = \frac{1}{2\pi \times 10^4 \times 10^{-7}} $$ $$ f_0 = \frac{1}{2\pi \times 10^{-3}} $$ $$ f_0 \approx \frac{1}{6.28 \times 10^{-3}} $$ $$ f_0 \approx 159.15 \text{ Hz} $$

Therefore, the Wien bridge oscillator with the given values will oscillate at approximately 159.15 Hz.