Four-Stage RC Phase-Shift Oscillator: 98-Comp-A1, May 2014, Q5

Question 5: Four-Stage RC Phase-Shift 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. Stage 1: inverting amp, input $R_G$, feedback $R_f=100\text{k}\Omega$. Stages 2-4: three IDENTICAL cascaded RC low-pass ($R=10\text{k}\Omega$, $C$) + unity-gain voltage-follower sections (the follower isolates each RC stage so they cascade without loading each other). Output feeds back to stage 1's input.

Given data
Quantity	Value
$R$ (each RC stage)	$10\text{k}\Omega$
$R_f$	$100\text{k}\Omega$
Target $f_0$ (part d)	$1\text{kHz}$

Find. Loop-gain expression $T(j\omega)$, Barkhausen oscillation condition and $\omega_0$, a value of $R_G$ to guarantee startup, and $C$ for $f_0=1\text{kHz}$.

Fig. Q5 — Signal-flow: an inverting stage (180° phase, gain $-R_f/R_G$) cascades with three identical unity-gain-buffered RC low-pass sections (each contributing gain $1/(1+j\omega RC)$ and up to 90° lag), fed back to the input — oscillation requires the LOOP's total phase to reach 360° and its magnitude to reach unity.

Approach. Multiply the inverting stage's gain by the three identical RC-follower stages' gain to get $T(j\omega)$; apply the Barkhausen criterion (phase $=0^\circ\!\!\mod360^\circ$, magnitude $=1$) to find $\omega_0$ and the required $R_f/R_G$ ratio.

Part (a) — loop gain. Each follower isolates its RC section, so three identical sections simply cascade in gain: $$T(j\omega)=\left(-\frac{R_f}{R_G}\right)\left(\frac{1}{1+j\omega RC}\right)^3$$
Part (b) — oscillation condition and frequency. The inverting stage supplies a fixed $180^\circ$; each RC section supplies $-\tan^{-1}(\omega RC)$. Oscillation needs the total to reach $0^\circ$ (mod $360^\circ$): $$180^\circ-3\tan^{-1}(\omega_0RC)=0^\circ\ \Rightarrow\ \tan^{-1}(\omega_0RC)=60^\circ\ \Rightarrow\ \omega_0RC=\tan60^\circ=\sqrt3$$ $$\omega_0=\boxed{\dfrac{\sqrt3}{RC}}$$ At this frequency each RC section's magnitude is $1/\sqrt{1+3}=1/2$, so the magnitude (Barkhausen) condition is: $$\frac{R_f}{R_G}\left(\frac12\right)^3=1\ \Rightarrow\ \frac{R_f}{R_G}=8\ \Rightarrow\ R_G=\frac{R_f}{8}=\boxed{12.5\text{ k}\Omega\text{ (unity-loop-gain boundary)}}$$
Part (c) — choose $R_G$ to INITIATE oscillation. Startup (growth from noise) needs $|T|$ slightly >1, i.e. $R_f/R_G>8$, so $R_G$ must be chosen somewhat BELOW the 12.5k boundary: $$R_G=\boxed{10\text{ k}\Omega\text{ (gives }R_f/R_G=10>8\text{, a practical startup margin)}}$$
Part (d) — choose $C$ for $f_0=1\text{kHz}$. From $\omega_0=\sqrt3/(RC)$: $$C=\frac{\sqrt3}{\omega_0R}=\frac{\sqrt3}{2\pi(1000)(10\text{k}\Omega)}=\boxed{27.6\text{ nF}}$$

Final Results — Question 5
Quantity	Value
$T(j\omega)$	$(-R_f/R_G)/(1+j\omega RC)^3$
$\omega_0$	$\sqrt3/(RC)$ (i.e. $60^\circ$ lag per RC stage)
Boundary $R_G$ ($\|T\|=1$)	$12.5\text{ k}\Omega$
Chosen $R_G$ (startup margin)	$10\text{ k}\Omega$
$C$ for $f_0=1\text{kHz}$	$27.6\text{ nF}$

Topic: RC phase-shift oscillator (Barkhausen criterion).

Key relations: Barkhausen: $|T(j\omega_0)|=1$ and $\angle T(j\omega_0)=0^\circ\ (\text{mod }360^\circ)$; per-stage RC lag $\tan^{-1}(\omega RC)$.

Why this works: The fixed $180^\circ$ from the inverting stage plus $3\times60^\circ=180^\circ$ from three identical RC sections sums to exactly $360^\circ$ at one specific frequency, and only at that frequency can a self-sustaining loop exist.

Common pitfall: Setting $R_f/R_G$ to exactly the boundary value (8) — component tolerances would then let the loop gain fall just under unity and oscillation would never start; practical designs always bias the gain slightly high (then let a nonlinearity/AGC limit the amplitude).

Source: Sedra & Smith, Microelectronic Circuits, Signal Generators and Waveform-Shaping Circuits – Sinusoidal Oscillators.

Question 5 of 7: Four-Stage RC Phase-Shift Oscillator

Question 5: Four-Stage RC Phase-Shift Oscillator (20 marks)