Wien Bridge Oscillator Simulator — Sinusoidal Oscillation Conditions

A Wien bridge oscillator is built around a non-inverting op-amp amplifier (gain $A = 1 + R_f/R_g$) whose output is fed back to the non-inverting input through a divider of a series RC, $Z_s = R + 1/(j\omega C)$, and a parallel RC, $Z_p = R/(1+j\omega RC)$.

$$\beta(j\omega) = \frac{Z_p}{Z_s + Z_p} = \frac{1}{3 + j\!\left(\omega RC - \dfrac{1}{\omega RC}\right)}$$

The imaginary part vanishes at $\omega_0 = 1/(RC)$, where $\beta$ becomes the real value $\beta_0 = 1/3$ with zero phase shift. This single frequency is the oscillation frequency $f_0 = \omega_0/(2\pi) = 1/(2\pi RC)$. Oscillation requires the Barkhausen condition $A\cdot\beta = 1$ (matched phase and amplitude), so the minimum closed-loop gain is $A_{\min} = 3$.

In linear small-signal terms the envelope grows or decays at a rate $\sigma = \pi f_0 (A\beta - 1)$: $A\beta > 1$ gives exponential growth, $A\beta < 1$ gives exponential decay and $A\beta = 1$ gives a constant-amplitude sine wave. Real circuits ride slightly above $A = 3$ at start-up; a non-linear element (tungsten lamp, JFET, diode shaping or AGC) pulls the average gain back to exactly 3 once the desired amplitude is reached, locking the circuit onto a stable limit cycle.

This simulator draws the linearised model $v(t) = V_0 \exp(\sigma t)\sin(2\pi f_0 t)$ and clips $|v|$ at the supply rails $V_{\mathrm{supply}}$ to approximate the saturated regime for $A > 3$.

Precision low-distortion audio signal sources: ever since the HP200A of 1939, Wien bridge oscillators have been the workhorse of audio-band benchtop signal generators. A tungsten lamp as the amplitude-stabilising element typically delivers THD < 0.01% and frequency stability better than 1e-4. With the defaults R = 10 kohm and C = 100 nF the tool returns f0 = 159.2 Hz and a sustained sine wave at A = 3; switching R between 1 kohm and 100 kohm and C between 1 nF and 1 uF spans 1.6 Hz - 160 kHz and covers the full audio band 20 Hz - 20 kHz. Modern distortion analysers from Audio Precision and HP/Keysight 8903 still use refined versions of this topology in their output stage.

Function generators and reference signal sources: low-frequency pure sine waves are needed for farm-machine vibration tests, automotive ECU stimulation, ECG simulators and calibration of seismometers. LC oscillators are impractical at low frequencies because the inductors become unmanageably large, and crystal oscillators are not continuously tunable, so the Wien bridge fills the mHz - MHz niche. Pushing R to 100 kohm and C to 1 uF in the tool drops f0 to 1.59 Hz, well into the seismic and structural-vibration band.

Hands-on analog electronics education: a standard project in undergraduate analog electronics is to build an op-amp + RC sinusoidal oscillator. The exercise drives home the Barkhausen criterion (loop phase = 0 plus loop gain = 1), the distinct roles of positive and negative feedback, and the need for amplitude stabilisation - core feedback-control concepts in one circuit. Sliding A between 2.9, 3.0 and 3.1 lets students see the three regimes change instantaneously, which makes the tool an excellent companion to breadboard sessions. Texas Instruments and Analog Devices application notes regularly use this topology as a teaching example.

Power electronics and motor-control reference signals: three-phase inverter PWM references, servo-motor positioning tests and UPS output-quality benches all need clean low-frequency sinusoids as internal references. Although direct-digital synthesis is taking over, premium analog signal sources (state-variable oscillators with ultra-low THD < 0.0005%) still build on the Wien bridge. Sweeping V_supply between 3 V and 30 V in the tool shows the output amplitude scaling proportionally, illustrating why the same topology fits both logic-level and power-level rails.

The most common mistake is to think you can simply fix A at 3 and get a clean sine wave. In practice A wanders with temperature, supply voltage and component tolerances, so without amplitude stabilisation the circuit either dies (A < 3) or saturates at the rails (A > 3). Real designs sit A just above 3 (3.01 - 3.10 at start-up) and use a non-linear element - tungsten lamp, JFET, diode network or AGC IC - to pull the average gain back to exactly 3. The simulator faithfully shows the linearised case at A = 3.00 as a sustained sine wave, but bear in mind that this state cannot be reached open-loop on real hardware.

Another myth is that f0 = 1/(2*pi*R*C) is insensitive to component tolerances. Because f0 depends on the R*C product, a +-5% resistor and a +-5% capacitor give roughly +-7% on f0 when their tolerances combine in quadrature. Instrument-grade oscillators use 0.1 - 1% metal-film resistors, low-temperature-coefficient capacitors (polystyrene, PPS, NPO ceramic, 50 ppm/K or better) and temperature compensation to reach f0 stability of 1e-4 or better. The simulator computes with nominal values, so add this tolerance budget when comparing with hardware.

Finally, the assumption that Wien bridge oscillators work at any frequency you like ignores the op-amp limits. Even if 1/(2*pi*R*C) suggests a few MHz, the gain-bandwidth product (GBP) and slew rate of general-purpose op-amps (GBP = 1 - 10 MHz) introduce phase lag that breaks the Barkhausen condition above a few hundred kHz. Beyond about 1 MHz designers move to LC oscillators (Colpitts, Hartley) or crystal oscillators. With C at 1 nF and R at 1 kohm the simulator returns f0 = 159 kHz, which is close to the practical ceiling for a general-purpose op-amp implementation.