An oscillator is a circuit that produces a periodic output without any periodic input. It is built by closing a positive-feedback loop around an amplifier with a frequency-selective network, so that one specific frequency is reinforced on every pass and all other frequencies decay. The op-amp provides the gain, the feedback network sets the frequency, and the loop sustains itself.
A linear oscillator is constructed from two blocks: a forward amplifier of gain $A$ and a feedback network of transfer function $\beta(j\omega)$. The output of the amplifier is sampled by the feedback network, scaled, phase-shifted, and returned to the amplifier input. If the signal that reaches the input is identical in amplitude and phase to the signal that produced it, the loop sustains itself and a steady periodic output emerges.
Solving the closed-loop equation $v_\text{out} = A\,(v_\text{in} + \beta\,v_\text{out})$ for the transfer function gives
The closed-loop gain is unbounded at any frequency where $A\beta(j\omega) = 1$. At that frequency the circuit produces an output for zero input: it oscillates. The role of the amplifier is to supply energy at the rate the feedback network requires; the role of the feedback network is to select the frequency at which the loop closes.
The Wien bridge oscillator is the standard introductory op-amp oscillator because every element of its construction maps directly onto the abstract feedback diagram. The op-amp at the centre provides the forward gain. Two paths leave the output: a negative-feedback path that fixes that gain, and a positive-feedback path through a frequency-selective RC network.
The negative feedback path is the familiar non-inverting amplifier wiring. $R_4$ runs from the output to the inverting input, and $R_3$ runs from the inverting input to ground. Together they fix the forward gain at
The positive-feedback path is the network at the bottom of the schematic. Its job is to sample the output, attenuate every frequency except one, and return that one frequency to the non-inverting input. It is built from two RC arms tuned to the same time constant: a series arm of $R_1$ and $C_1$ between the output and the non-inverting input, and a parallel arm of $R_2$ and $C_2$ between the non-inverting input and ground.
The bottom of the schematic, taken on its own, is a passive voltage divider made from two RC arms. Each arm is a single-pole filter. The series arm $R_1, C_1$ behaves as a high-pass section; the parallel arm $R_2 \,\|\, C_2$ behaves as a low-pass section. Their cascade produces a band-pass response, and the band-pass peak is what the oscillator runs on.
The series arm has impedance
At low frequencies, $1/(\omega C_1)$ is large, so $Z_1$ is dominated by the capacitor and approaches infinity as $\omega \to 0$. The series arm therefore blocks dc and low-frequency content, exactly as a passive high-pass filter does. At high frequencies $C_1$ behaves as a short circuit and $Z_1 \to R_1$, so high frequencies are passed with negligible attenuation.
The parallel arm has impedance
At low frequencies $C_2$ is open and $Z_2 \to R_2$. At high frequencies $C_2$ shunts the parallel arm to ground and $Z_2 \to 0$. Treated as the lower half of a divider, the parallel arm therefore passes low frequencies and shorts out high frequencies, exactly as a passive low-pass filter does.
The voltage that reaches the non-inverting input is the divider output
where the second equality uses $R_1 = R_2 = R$ and $C_1 = C_2 = C$. At low $\omega$ the high-pass arm dominates and $\beta \to 0$. At high $\omega$ the low-pass arm dominates and $\beta \to 0$. Between the two limits there is exactly one frequency at which the rising high-pass response and the falling low-pass response cross. That frequency is
A signal that travels once around the loop is multiplied first by the amplifier gain $A$ and then by the feedback transfer $\beta(j\omega)$. For a steady oscillation, the signal that returns to the input must be identical to the signal that left it. Identical means the same magnitude and the same phase.
Plotting $A\beta(j\omega)$ on the complex plane makes the criterion visual. As $\omega$ sweeps from $0$ to $\infty$ the product traces a closed curve. The criterion is satisfied at any frequency where that curve passes through the point $1 + j\,0$, the round-trip identity.
For the Wien network the curve crosses the real axis at $\omega = \omega_0 = 1/RC$, where $\beta(j\omega_0) = \tfrac{1}{3}$ is purely real. The phase condition is therefore met automatically at that single frequency, and the magnitude condition reduces to
Use the slider to move the amplifier gain $A$. The locus scales with $A$: it crosses the real axis at $A/3$. When that crossing falls short of $1$, oscillation decays; when it overshoots, the amplitude grows until the op-amp clips. Only at $A = 3$ does the curve pass cleanly through the Barkhausen point.
When power is first applied, the output of the op-amp contains only thermal noise. Components of that noise at every frequency are recirculated by the loop, but only the component near $\omega_0$ returns in phase and amplified. On every pass the loop gain $A\beta$ multiplies the surviving sinusoid, so its amplitude grows exponentially.
For loop gain slightly greater than unity the envelope grows as $e^{\sigma t}$ with $\sigma = \tfrac{1}{2}(A\beta - 1)\,\omega_0$. The exponential growth continues until the amplifier output approaches the supply rails or the gain-control element reduces $A$. Steady state is reached when the average loop gain has been pulled down to exactly unity by amplitude-dependent saturation.
Oscillators provide the periodic excitation that many measurement and signal-processing circuits depend on. The role of the oscillator is to supply a known, stable frequency from which the rest of the system derives timing, modulation, or carrier energy.
Impedance, capacitance, and conductance bridges require a sinusoidal excitation at a chosen frequency. A Wien bridge or phase-shift oscillator supplies that excitation with low harmonic content.
The injected current used to measure tissue impedance is a small-amplitude sinusoid, typically in the 10 kHz to 100 kHz range. An op-amp oscillator generates that current and a synchronous detector recovers the impedance.
A lock-in amplifier multiplies a noisy input by a reference sinusoid at the modulation frequency. The reference is generated by an oscillator phase-locked to the stimulus.
Audio-frequency oscillators built around op-amps are the analogue heart of bench function generators and of the stimulus sources used in nerve and muscle experiments.
An RC oscillator provides the clock reference for slow-rate analogue-to-digital converters and for timing in low-power instrumentation where a quartz crystal would be excessive.
Telemetry links and isolated medical front-ends modulate a sinusoidal carrier with the signal of interest. The carrier originates in an oscillator of the type discussed here.