6.3.2.2.1 Circuit Configuration of the Wien Bridge Network

The Wien Bridge network, when used in an oscillator, typically forms part of a feedback loop around an amplifier (often an operational amplifier). The bridge itself consists of four impedances:
* Two purely resistive arms (e.g., $R_3$ and $R_4$ in a typical oscillator configuration, forming a voltage divider for negative feedback).
* Two RC arms that are frequency-dependent:
* One arm (e.g., $R_1$ and $C_1$) connected in series.
* Another arm (e.g., $R_2$ and $C_2$) connected in parallel.

For the simplest and most common Wien Bridge Oscillator design, the components in the frequency-determining arms are chosen to be equal: $R_1 = R_2 = R$ and $C_1 = C_2 = C$.

6.3.2.2.2 Frequency Response and Phase Shift

The key characteristic of the Wien Bridge network that makes it useful for oscillators is its unique phase shift property:

• Low Frequencies: At very low frequencies, the series capacitor ($C_1$) acts almost like an open circuit, and the parallel capacitor ($C_2$) acts almost like an open circuit. This significantly attenuates the signal passing through the RC network, and the phase shift approaches $0^\circ$.
• High Frequencies: At very high frequencies, the series capacitor ($C_1$) acts almost like a short circuit, and the parallel capacitor ($C_2$) acts almost like a short circuit, shunting the output to ground. Again, this attenuates the signal, and the phase shift approaches $0^\circ$.
• Resonant Frequency ($f_r$): There is a specific frequency, called the resonant frequency, at which the phase shift introduced by the RC network is exactly zero degrees ($0^\circ$). At this frequency, the voltage transfer ratio (output voltage/input voltage) of the RC network reaches its maximum and is typically $1/3$ (when $R_1=R_2=R$ and $C_1=C_2=C$).

6.3.2.2.3 Derivation of Resonant Frequency

Assuming $R_1 = R_2 = R$ and $C_1 = C_2 = C$, the resonant frequency ($f_r$) at which the phase shift is zero is given by:

$f_r = \frac{1}{2\pi RC}$

This formula shows that the oscillation frequency is easily tunable by varying the values of R or C. Often, a dual-ganged variable capacitor or potentiometer is used to vary both R and C simultaneously, allowing for a wide range of selectable frequencies.

6.3.2.2.4 Role in Wien Bridge Oscillator

In a Wien Bridge Oscillator, the Wien Bridge network is typically placed in the positive feedback path of a non-inverting amplifier. The conditions for sustained oscillation (Barkhausen Criteria) are:
1. Loop Gain = 1: The magnitude of the loop gain (product of amplifier gain and feedback network transfer ratio) must be unity (or slightly greater than unity to start oscillations).
2. Total Phase Shift = $0^\circ$ (or $360^\circ$): The total phase shift around the feedback loop must be zero.

• Phase Shift Condition: Since the non-inverting amplifier provides $0^\circ$ phase shift, the feedback network must also provide $0^\circ$ phase shift. This is precisely what the Wien Bridge network does at its resonant frequency, $f_r$. Therefore, the circuit oscillates at this frequency.
• Gain Condition: At $f_r$, the Wien Bridge network attenuates the signal by a factor of 3 (i.e., its transfer ratio is 1/3). To achieve a loop gain of 1, the amplifier must provide a voltage gain ($A_v$) of at least 3. In practice, the amplifier gain is usually set slightly higher than 3 to ensure oscillations start, and then an amplitude stabilization mechanism (like a thermistor or a lamp, or diodes) is used to bring the effective gain down to exactly 3 for stable, low-distortion output.

6.3.2.2.5 Advantages of the Wien Bridge Network in Oscillators

• Wide Frequency Range: Easily tunable over a broad range of frequencies by varying R or C.
• Low Distortion: Can produce very pure sinusoidal waveforms, especially with proper amplitude stabilization.
• Good Frequency Stability: Relatively stable frequency output.
• Simple Design: Can be built with readily available resistors, capacitors, and an op-amp.
• No Inductors: Avoids the use of bulky and expensive inductors, which can also be a source of electromagnetic interference.

Exercise Solutions

Easy:

1. The primary function of the Wien Bridge network in a Wien Bridge Oscillator is to act as the frequency-selective feedback network that determines the oscillation frequency. Its frequency-determining arms are composed of Resistors (R) and Capacitors (C).
2. The resonant frequency (fr) of the Wien Bridge network is the specific frequency at which the network provides exactly zero degrees (0°) of phase shift between its input and output. This zero phase shift condition is crucial for meeting one of the Barkhausen Criteria for sustained oscillations when combined with a non-inverting amplifier.

Medium:

1. Key characteristics of the Wien Bridge network at its resonant frequency (fr):
   a) Phase Shift: The phase shift introduced by the RC network is exactly zero degrees (0°). This is the critical condition for its use in the positive feedback path of a non-inverting amplifier.
   b) Voltage Transfer Ratio (Attenuation): At this resonant frequency, the network significantly attenuates the signal. Its voltage transfer ratio (output voltage / input voltage) is typically 1/3 (when R1=R2=R and C1=C2=C). This means the output signal from the bridge is one-third of the input signal to the bridge.
2. Given: R1 = R2 = R = 10 kOhm = 10,000 Ohm, C1 = C2 = C = 10 nF = 10 x 10^-9 F.
   a) Calculate resonant frequency (fr):
   fr = 1 / (2 * pi * R * C)
   fr = 1 / (2 * 3.14159 * 10000 Ohm * 10 x 10^-9 F)
   fr = 1 / (2 * 3.14159 * 100 x 10^-6)
   fr = 1 / (0.000628318)
   fr approx = 15915.5 Hz (or approximately 15.92 kHz) b) Minimum amplifier gain:
   At its resonant frequency, the Wien Bridge network attenuates the signal by a factor of 3 (transfer ratio = 1/3). To satisfy the Barkhausen criterion of a loop gain of 1, the amplifier must compensate for this attenuation.
   Loop Gain = Amplifier Gain (Av) * Feedback Network Transfer Ratio
   1 = Av * (1/3)
   Av = 3
   So, the minimum voltage gain the amplifier must provide is 3. In practice, it's set slightly higher to ensure oscillations start.

Hard:

1. The Barkhausen Criteria for sustained sinusoidal oscillations are:
   1. Loop Gain = 1 (or slightly greater than 1 initially): The magnitude of the loop gain, which is the product of the amplifier's gain (Av) and the feedback network's transfer ratio (Beta), must be equal to or slightly greater than unity (Av * Beta >= 1).
   2. Total Phase Shift = 0° (or 360°): The total phase shift around the entire feedback loop (amplifier phase shift + feedback network phase shift) must be zero degrees or a multiple of 360 degrees.
   How the Wien Bridge Network and Amplifier Satisfy these Criteria:
   • Phase Shift Condition:
     • The non-inverting amplifier configuration inherently provides 0° phase shift between its input and output.
     • The Wien Bridge network is specifically designed such that at its unique resonant frequency (fr), the phase shift it introduces is exactly 0°.
     • Therefore, at fr, the total phase shift around the loop (0° from amplifier + 0° from Wien Bridge) is 0°, satisfying the second Barkhausen criterion.
   • Loop Gain Condition:
     • At its resonant frequency (fr), the Wien Bridge network attenuates the signal, providing a transfer ratio (Beta) of 1/3.
     • To achieve a loop gain of 1 (Av * Beta = 1), the non-inverting amplifier must provide a voltage gain (Av) of at least 3.
     • In practice, the amplifier gain is initially set slightly above 3 to ensure oscillations start, and then an amplitude stabilization mechanism reduces the effective gain to precisely 3 for stable, low-distortion output once oscillations build up. This satisfies the first Barkhausen criterion.
2. Three significant advantages of using the Wien Bridge network in an oscillator design:
   1. No Inductors Required: Unlike LC oscillators, the Wien Bridge Oscillator uses only resistors and capacitors. This is a major advantage because inductors can be bulky, expensive, difficult to fabricate in integrated circuits, and susceptible to electromagnetic interference. Using only R and C makes the circuit simpler, smaller, and often more cost-effective.
   2. Wide Frequency Range and Easy Tunability: The oscillation frequency (fr = 1 / (2 * pi * R * C)) is determined by the values of R and C. By using dual-ganged variable resistors or capacitors, the frequency can be easily tuned over a very wide range (e.g., several decades), making Wien Bridge Oscillators ideal for audio frequency generators and function generators.
   3. Low Distortion Sinusoidal Output: With proper design and especially with the inclusion of an amplitude stabilization mechanism (like a thermistor, lamp, or diodes), Wien Bridge Oscillators can produce very pure, low-distortion sinusoidal waveforms. This makes them suitable for applications requiring high-quality sine waves, such as audio testing equipment.
   (Other advantages include: Good frequency stability, simple design with readily available components.)

Quiz Answers

1. b) Resistors and Capacitors (RC)
2. False.
   • At very high frequencies, both capacitors act like short circuits. The series capacitor acts like a short, and the parallel capacitor shunts the output to ground, leading to significant attenuation, but the phase shift approaches 0 degrees.
3. At its resonant frequency (fr), the Wien Bridge network provides a phase shift of exactly 0 degrees.
4. c) 3
5. c) To ensure the output is a pure, low-distortion sinusoidal waveform.
6. c) Approximately 1.59 kHz
   • fr = 1 / (2 * pi * R * C) = 1 / (2 * 3.14159 * 1000 Ohm * 0.0000001 F)
   • fr = 1 / (0.000628318) approx = 1591.55 Hz = 1.59 kHz
7. c) It can be easily tuned over a wide frequency range.