Question 5 of 7: Three-Stage RC Phase-Shift Oscillator
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
17-Comp-A1, Electronics — National Exams, December 2019. Open-book, 3 hours; seven 20-mark questions, five to be marked (all seven answered below as a complete study resource). Where a diode is used without a stated drop, $V_D=0.7\text{V}$.
Reference texts: Sedra & Smith, Microelectronic Circuits (diode limiters, MOSFET/BJT small-signal amplifiers, active loads, op-amp imperfections and oscillators, CMOS logic, DACs) — the single reference text covering every question on this paper.
A three-stage $R$-series/$C$-shunt low-pass ladder can supply at most $-180^\circ$ of phase shift (never a full $-360^\circ$/$0^\circ$), so for the loop to satisfy the Barkhausen criterion the op-amp itself must contribute the other $-180^\circ$, i.e. it must be wired as an inverting amplifier around $R_1$ (input) and $R_2$ (feedback) — this is the standard, well-known "RC phase-shift oscillator" (Sedra & Smith). The source's own note flags exactly this ambiguity (whether the feedback resistor is "$R_f$" or "$R_2$"); solved here as the standard inverting-amplifier version, using $R_2$ for the feedback resistor per part (c)'s naming.
Given. Three identical, cascaded (loading) low-pass sections: $R$ from the previous node to the next, $C$ from that node to ground, repeated three times; final section feeds the inverting-amplifier's virtual-ground summing junction through the amplifier's own input resistor $R_1$, with feedback resistor $R_2$.
Given data
Quantity
Value
$R$ (each ladder resistor)
$10\text{k}\Omega$
$C$ (each ladder capacitor)
$0.1\mu\text{F}$
Find. Oscillation (Barkhausen) condition; oscillation frequency and a qualitative statement of amplitude; a choice of $R_1,R_2$ that sustains oscillation.
Approach. Analyze the loaded 3-stage RC ladder's transfer function $\beta(j\omega)$ (accounting for inter-stage loading, since this is a real cascade, not buffered stages); find where its phase is exactly $-180^\circ$ and what its attenuation is there; the inverting amplifier must supply exactly the reciprocal gain to close the loop at unity magnitude.
Part (a) — oscillation condition. For the standard loaded 3-section RC ladder, the feedback factor's phase reaches exactly $-180^\circ$ at
$$\omega_0=\frac{1}{RC\sqrt6}$$
and at that frequency the network attenuates by exactly $1/29$ (a classical result of solving the loaded 3-pole ladder's transfer function). Combined with the amplifier's own $-180^\circ$ (inverting), total loop phase is $-360^\circ\equiv0^\circ$. Barkhausen's magnitude condition then requires
$$|A_v|\cdot\frac1{29}=1\quad\Rightarrow\quad \boxed{|A_v|=\frac{R_2}{R_1}\ge29}$$
Part (b) — frequency and amplitude.
$$f_0=\frac{1}{2\pi RC\sqrt6}=\frac{1}{2\pi(10\text{k}\Omega)(0.1\mu\text{F})\sqrt6}=\boxed{65.0\text{ Hz}}$$
Amplitude is not set by the linear (small-signal) analysis at all — with $|A_v|$ set exactly to 29 the loop gain is marginally 1 and oscillation neither grows nor decays in theory; in practice $R_2/R_1$ is set slightly above 29 so the oscillation amplitude grows until the op-amp's own output saturates (clips near its supply rails), after which the fundamental at $f_0$ is extracted from the resulting near-square wave. A stable, undistorted sinusoidal amplitude requires adding a nonlinear gain-limiting element (e.g. back-to-back diodes or a JFET in the feedback path) — not shown in this figure.
Part (c) — choosing $R_1,R_2$. Pick a convenient $R_1$ and set $R_2$ using the $\ge29$ condition with headroom for reliable start-up (component tolerances shift the ladder's exact attenuation):
$$R_1=10\text{k}\Omega\quad\Rightarrow\quad R_2=29\times R_1=\boxed{290\text{ k}\Omega}\ \text{(design), choose a standard }300\text{k}\Omega\text{ (or trimmer) for guaranteed start-up}$$
Fig. Q5 — Standard RC phase-shift oscillator: inverting amplifier ($-R_2/R_1$, $-180^\circ$) driven through a 3-stage loaded $R$–$C$ low-pass ladder ($-180^\circ$ more at $f_0=1/(2\pi RC\sqrt6)$), closing the loop at $0^\circ$ total phase.
