Wien bridge oscillator

Wien bridge oscillator is a type of phase shift oscillator which generates sinusoidal waves at the output. The Wien bridge oscillator was named after Max Karl Wien was based on the bridge circuit developed in the 1890s. It is one of the simplest oscillators known for its audio applications.

Construction

The Wien bridge oscillator includes two capacitors (charge storing element) and four resistors. The other component of the circuit is an inverting amplifier like operational amplifiers or transistors. Here, the circuit uses two transistors. The two combinations in the circuit are one series combination, and one parallel combination of RC.

The circuit of the Wien bridge oscillator is shown below:

It has two paths for feedback, positive feedback and negative feedback.

Frequency of Oscillations

The frequency of oscillations is given by:

The frequency of oscillations when both the resistors and capacitors have equal value is given by:

R1 = R2 = R

C1 = C2 = C

A bandpass filter is formed by resistors and capacitors R1, R2, C1, and C2.

Derivation

The derivation will provide us with the condition for sustained oscillations. It is one of the methods to overcome the drawback of the Wien bridge oscillator. The equivalent circuit diagram of the Wien bridge oscillator can be represented as:

Z1 = R

Z2 = R_F

Z3 = (R₁ + 1/sC₁)

Z4 = (R₂ || 1/sC₂)

Where,

Z is the impedance

R_F is the feedback resistance

(R₁ + 1/sC₁) is the series combination of R₁ and C₁.

(R₂ || 1/sC₂) or [(R₂/(1 + R₂sC₂)) is the parallel combination of R₂ and C₂

The positive feedback voltage (V+) is connected at the non-inverting terminal of the amplifier. It is given by:

V+ = V_O [Z4 (Z3 + Z4)]

V+ = V_O [(R₂/1 + R₁sC₂) / ((R₂/1 + R₂sC₂ + R₁ + 1/sC₁))]

B = V+ / V_O = 1/ (1 + R₁sC₂ + R₁/ R₂ + 1/ R₂sC₁ + C₂/ C₁)

Put, s = jω_o (ω_o is the oscillation frequency)

Equating the values of ω₁ = 1/ R₁C₂ and ω₂ = 1/ R₂C₁, we get:

B = V+ / V_O = 1/ (1 + R₁/ R₂ + C₂/ C₁ + j (ω_o/ ω₁ - ω₂/ ω_o))

The capacitor and the resistor can produce the phase shift of total 180 degrees provided by the oscillating frequencies. As discussed, a single RC network provides a maximum phase shift of 90 degrees. Two RC networks can thus contribute a maximum of 180 degrees phase shift.

To find the overall feedback, let's assume,

R1 = R2 = R C1 = C2 = C, and ω₁ = ω₂= ω_o

The feedback is given by:

B = 1/ (1 + R/ R + C/ C + j (ω_o/ ω_o - ω_o / ω_o))

B = 1/ (1 + R/ R + C/ C +0)

B = 1/ (1 + 1 + 1)

B = 1/ 3

If AB = 1

B = 1/3

The gain will be (A) = 3

The feedback from the circuit can be represented as:

B = 1 + R_F/R

3 = 1 + R_F/R

R_F/R = 2

R_F = 2R

It is the condition for the sustained oscillations of the Wien bridge oscillator.

It states that the product of the magnitude and feedback is one. The total feedback of the oscillator is calculated to ensure positive feedback, i.e., the phase shift of 0 or 360 degrees. It further maintains the periodic oscillations of the system. The sustained oscillations are based on the Barkhausen criteria.

Phase shift

In the case of RC phase shift oscillators, the total phase shift for positive feedback (0 or 360 degrees) was utilized from the phase network at the feedback path and the amplifier. But, in the case of Wien bridge oscillator, the two transistors provide a phase shift of 180 degrees each.

Thus, the total phase shift to ensure positive feedback is:

Phase shift = 180 (transistor) + 180 (transistor)

Phase shift = 360 degrees

Phase shift = 360 degrees = 0 degrees

The output and the phase shift graph of the Wien bridge oscillator are shown below:

The above graph depicts that the phase shift is positive at low frequencies, while at high frequencies, it is negative. The negative phase shift produces phase delays. At the middle of the graph, the resonant frequency is determined, which is given by:

As discussed in the above derivation, the output voltage equals to 1/3^rd of the input voltage to allow oscillations to occur.

History

In 1891, the German physicist Max Karl Wien developed a bridge circuit to measure impedances. The oscillators further developed were based Karl Wien bridge circuit.
In 1937, filament lamp and audio oscillators were developed based on the bridge circuit.
In 1937 - 1938, various discussions on the negative feedback of the oscillators between Terman, Bill Hewlett (co-founder of HP), and R. R. Buss. The oscillator was then demonstrated first in Poland.
Bill Hewlett's first product HP 200A manufactured in California, was a precise Wien Bridge oscillator.
Hewlett's oscillator could produce the output in the form of sinusoidal wave with less distortion and stable amplitude.

Advantages

The advantages of the Wien bridge oscillator are as follows:

It is a simple circuit.
It provides constant output.
It produces high gain due to the presence of two transistors.
A potentiometer can be used with the oscillator to change the frequency of oscillations.
Good frequency stability.
Low distortion.
It provides ease of tuning.

Disadvantages

The disadvantages of the Wien bridge oscillator are as follows:

It can cause high output distortion.
It cannot generate high frequencies.
It has high number of components due to the two transistors.

Numerical Examples

Let's discuss two examples based on the Wien bridge oscillators.

Example: 1 In the Wien Bridge oscillator, R1 = 10k Ohms, R2 = 5k Ohms, X_C1 = 10k Ohms, and X_C2 = 5k Ohms. Find the two impedances of the circuit.

Solution: Given: R1 = 10k Ohms = 10⁴ Ohms

R2 = 5k Ohms = 5 x 10³ Ohms

XC₁ = 10k Ohms = 10⁴ Ohms

XC₂ = 5k Ohms = 5 x 10³ Ohms

X is the reactance and is defined as the combined effect of resistance and resonance opposed by the current. There are two impedances in the Wien bridge oscillator, series and parallel. Let the two impedances be Z1 and Z2.

Z1 = R1 + X_C1 = 20k Ohms

1/Z2 = 1/R2 + 1/ X_C2

1/Z2 = 1/5000 + 1/5000

1/Z2 = 2/5000

Z2 = 2500 Ohms = 2.5k Ohms

Thus, the two impedances of the circuit are 20k Ohms ad 2.5k Ohms.

Example: 2 In the Wien Bridge oscillator, R1 = R2 = 100k Ohms and C1 = C2 = 200 p Farads. Find the frequency at which the circuit will oscillate.

Solution: Given: R1 = R2 = 100k Ohms = 100 x 10³ Ohms = 10⁵Ohms

C1 = C2 = 200 pF = 200 x 10^-12 Farads = 2 x 10^-10 Farads

The frequency of oscillations can be calculated using the formula given by:

F = 1/ (2 x 3.1415 x 10⁵ x 2 x 10^-10)

F = 10⁵/ 12.566