Question 5 of 7: RC-Ladder (Wien-Bridge Type) Sinusoidal Oscillator

Question 5: RC-Ladder (Wien-Bridge Type) Sinusoidal Oscillator (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Feedback network: series $C$/shunt $R$/series $R$/shunt $C$ ladder from $V_o$ back to the non-inverting input, with $R=10\text{k}\Omega$, $C=0.1\,\mu\text{F}$; gain-setting pair $R_1$ (to ground), $R_2$ (feedback) at the inverting input.

Find. Barkhausen (oscillation) condition; frequency and amplitude of $V_o$; a suitable $R_1$, $R_2$ pair.

Approach. The drawn $C$-series/$R$-shunt/$R$-series/$C$-shunt ladder is algebraically identical to the classic Wien-bridge feedback network (nodal analysis of the ladder, using that the ideal op-amp's non-inverting input draws no current, reduces to the same $\beta(s)=RCs/(R^2C^2s^2+3RCs+1)$ as the textbook two-impedance form), so the standard Wien-bridge oscillation result applies directly.

1. Part (a) — oscillation condition. The ladder's feedback factor peaks at $\beta_0=1/3$ with zero phase shift at $\omega_0=1/RC$. Barkhausen's criterion (loop gain $=1$, $0^\circ$ phase) then requires the amplifier's non-inverting gain to exactly cancel the $1/3$ attenuation: $$\left(1+\frac{R_2}{R_1}\right)\cdot\frac{1}{3}=1\;\Longrightarrow\;\boxed{R_2=2R_1}$$ (In practice $R_2$ is set slightly above $2R_1$ so the loop gain is marginally $>1$, guaranteeing start-up; some nonlinearity — e.g. back-to-back diodes across $R_2$, or lamp AGC — then limits the amplitude once oscillation builds, since pure linear analysis with $R_2>2R_1$ predicts unbounded growth.)
2. Part (b) — frequency and amplitude. The ladder's zero-phase frequency: $$f_0=\frac{1}{2\pi RC}=\frac{1}{2\pi(10\text{k})(0.1\,\mu\text{F})}=\boxed{159\text{ Hz}}$$ Amplitude is not set by the linear Barkhausen condition itself (an exactly-unity loop gain is a knife-edge, marginal case); with $R_2$ chosen slightly above $2R_1$ for reliable start-up, the amplitude grows until the op-amp's own output saturation (or an explicit limiting network) caps it — the supply rails for Figure 5 are not stated, so the amplitude is reported qualitatively as "grows to the amplifier's saturation level" rather than a specific volt figure (flagged below).
3. Part (c) — choosing $R_1$, $R_2$. Take $R_1=10\text{k}\Omega$; the marginal condition is $R_2=2\times10\text{k}=20\text{k}\Omega$. Choosing $$R_1=\boxed{10\text{ k}\Omega},\qquad R_2=\boxed{22\text{ k}\Omega}$$ gives a loop gain of $(1+22/10)/3=1.067>1$ (a $\sim$7% margin), enough to guarantee reliable start-up from noise without being so large that the amplifier slams hard into saturation every cycle.