Wien-Bridge and Phase-Shift Oscillators¶

Possible Exam Questions¶

1. Definition and Principle¶

A Wien-bridge oscillator is a low-distortion RC sine-wave oscillator. A Wien lead-lag network returns frequency-selective positive feedback to a non-inverting amplifier, while a separate resistive negative-feedback path sets amplifier gain.

The Wien network has zero phase shift at one frequency. At that frequency its attenuation is \(1/3\) for equal \(R,C\), so steady amplifier gain must be 3.

2. Circuit¶

• Series \(RC\) from output to non-inverting input forms one bridge arm.
• Parallel \(RC\) from non-inverting input to ground forms the other.
• \(R_f,R_1\) set non-inverting gain \(A_v=1+R_f/R_1\).
• A lamp, thermistor, JFET or nonlinear resistor is commonly included in the gain-setting path for amplitude control.

3. Wien Network Derivation¶

For equal \(R\) and \(C\), let \(s=j\omega\). The series and shunt impedances are

The positive-feedback fraction is

With \(s=j\omega\),

Zero bridge phase occurs when the imaginary term is zero:

At \(f_0\),

4. Gain Condition¶

At steady state, Barkhausen magnitude requires

For the non-inverting amplifier,

For reliable startup, small-signal gain is made slightly greater than 3. Amplitude control then brings effective gain toward 3. If gain stays below 3, oscillation decays; if it stays far above 3, output clips.

General (Unequal-Component) Bridge Form¶

Drawn as a full bridge, the series arm \(R_1,C_1\) and the shunt arm \(R_2,C_2\) set frequency, while \(R_3,R_4\) in the negative-feedback path set gain. For arbitrary values,

With equal components \(R_1=R_2=R\) and \(C_1=C_2=C\) these reduce to \(f_0=1/(2\pi RC)\) and \(R_3/R_4=2\). Since the non-inverting gain is \(A_v=1+R_3/R_4\), this is again \(A_v=3\) (here \(R_3=R_f\), \(R_4=R_1\)); a ratio \(R_3/R_4\) slightly above 2 guarantees startup.

5. Working by Frequency¶

• At very low frequency, the series capacitor has large reactance, so feedback magnitude is small and bridge phase is leading.
• At \(f_0\), bridge phase is \(0^\circ\) and attenuation is \(1/3\); the non-inverting amplifier also contributes \(0^\circ\).
• At high frequency, the shunt capacitor increasingly diverts the feedback node to ground; feedback magnitude again falls and phase lags.

Thus only a band around \(f_0\) has sufficient correctly phased loop gain.

6. Amplitude Stabilisation¶

Incandescent Lamp/Thermistor¶

A temperature-dependent resistance in the amplifier's negative-feedback gain network changes slowly with output level. At startup it permits gain above 3; as it warms, gain falls smoothly toward 3. The thermal averaging produces very low distortion.

JFET/Automatic Gain Control¶

An output detector controls a JFET/variable-gain element. It offers electronic stabilisation over a wide tuning range.

Diode Limiting¶

Back-to-back diodes change feedback resistance at larger amplitude. It is compact but abrupt conduction adds harmonics.

7. Advantages, Disadvantages and Applications¶

Advantages¶

• low-distortion sine wave with smooth amplitude control;
• no inductor;
• simple formula and continuous tuning with ganged \(R\) or \(C\);
• useful audio/laboratory frequency range;
• amplitude and frequency networks are conceptually separable.

Disadvantages¶

• requires two matched/ganged resistors or capacitors for tuning;
• gain must be controlled accurately around 3;
• op-amp gain-bandwidth and slew rate limit high-frequency operation;
• RC tolerances and temperature cause frequency error;
• hard limiting increases distortion.

Applications¶

Audio-frequency generators, function generators, distortion-test sources, laboratory sine-wave sources and sensor excitation.

8. Wien vs Phase Shift¶

Rapid Recall¶

• \(Z_s=R+1/(sC)\); \(Z_p=R/(1+sRC)\).
• \(f_0=1/(2\pi RC)\); general \(f_0=1/(2\pi\sqrt{R_1C_1R_2C_2})\).
• \(\beta(f_0)=1/3\).
• Steady \(A_v=3\); non-inverting \(R_f=2R_1\); general balance \(R_3/R_4=R_1/R_2+C_2/C_1\).
• Smooth gain control produces low distortion.

Model Answer — Wien-Bridge Oscillator Derivation [10 marks]¶

Exam-ready answer

A Wien-bridge oscillator uses a non-inverting amplifier and an RC lead-lag positive-feedback network. A separate negative-feedback divider sets gain; an amplitude-control element makes startup gain slightly above its steady value and then reduces it.

For equal \(R,C\),

\[ Z_s=R+\frac1{sC}=\frac{1+sRC}{sC}, \qquad Z_p=\frac{R}{1+sRC}. \]

Therefore

\[ \beta(s)=\frac{Z_p}{Z_s+Z_p} =\frac{sRC}{s^2R^2C^2+3sRC+1}. \]

Putting \(s=j\omega\),

\[ \beta(j\omega)= \frac1{3+j\left(\omega RC-1/(\omega RC)\right)}. \]

Zero bridge phase requires

\[ \omega_0RC=\frac1{\omega_0RC}, \]

so

\[ \boxed{f_0=\frac1{2\pi RC}}. \]

At \(f_0\), \(\beta=1/3\). Barkhausen steady-state magnitude then requires \(A_v=3\). Since the op-amp is non-inverting,

\[ A_v=1+\frac{R_f}{R_1}=3 \quad\Rightarrow\quad \boxed{R_f=2R_1}. \]

For unequal components the bridge generalises to \(f_0=1/(2\pi\sqrt{R_1C_1R_2C_2})\) with balance \(R_3/R_4=R_1/R_2+C_2/C_1\). Small-signal gain is set slightly above 3 for noise-startup. A lamp/thermistor, JFET AGC or nonlinear resistance in the amplifier gain path lowers effective gain toward 3 as amplitude grows. Smooth thermal/AGC control gives less distortion than diode clipping.

Advantages are low distortion, no inductor, simple frequency law and convenient audio-range tuning. Limitations are component matching, accurate amplitude control, op-amp bandwidth/slew limits and RC drift. Uses include audio/function generators and test sources.

Practice target: 18 minutes; draw both feedback paths, derive \(Z_s\), \(Z_p\), \(\beta\), \(f_0\) and gain 3, then explain startup and settling separately.