Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
98-Comp-A1, Electronics — National Exams, December 2015. Open-book, 3 hours; seven 20-mark questions, five to be marked (all seven answered below as a complete study resource, per the standing "answer all M" rule). Unless stated otherwise, diode drops $V_D=0.7\text{V}$.
Reference texts: Sedra & Smith, Microelectronic Circuits (diode limiters/rectifiers/clampers, MOSFET/BJT small-signal amplifiers, active loads and current mirrors, active-RC filters, differential pairs with current-mirror loads, op-amp astable multivibrators, CMOS inverter design and delay, clocked cross-coupled latches, current-steering DACs) — the single reference text covering every question on this paper.
Given. $V_{sat}=\pm12\text{V}$; $R_1=10\text{k}\Omega$ (ground to $+$), $R_2=100\text{k}\Omega$ (output to $+$, positive feedback); $R=100\text{k}\Omega$ (output to $-$, charges $C$); $C=0.1\,\mu\text{F}$ ($-$ input to ground).
Given data
Quantity
Value
$V_{sat}$
$\pm12\text{ V}$
$R_1$
$10\text{ k}\Omega$
$R_2$
$100\text{ k}\Omega$
$R$
$100\text{ k}\Omega$
$C$
$0.1\,\mu\text{F}$
Find. Qualitative operation; $V_c(t),V_o(t)$ waveforms; expression for $V_c(t)$; oscillation frequency.
Approach. The $R_1$-$R_2$ divider off the output sets a hysteretic switching threshold $\pm\beta V_{sat}$ at the $+$ input (a Schmitt trigger); the $R$-$C$ network integrates the (constant, saturated) output voltage toward that same output level, so $V_c$ ramps exponentially until it crosses the current threshold and flips the comparator, which flips the threshold and reverses the ramp — a self-sustaining relaxation oscillator.
Part (a) — operation. Whenever $V_o=+V_{sat}$, the $+$ input sits at $+\beta V_{sat}$ (with $\beta=R_1/(R_1+R_2)$) and $C$ charges through $R$ toward $+V_{sat}$. Once $V_c$ (at the $-$ input) rises past $+\beta V_{sat}$, the op-amp's output flips to $-V_{sat}$, the $+$-input threshold flips to $-\beta V_{sat}$, and $C$ begins discharging/recharging toward $-V_{sat}$ through $R$ — repeating indefinitely and producing a square wave at $V_o$ and a triangular-ish (exponential-segment) wave at $V_c$.
Part (c) — threshold and $V_c(t)$.
$$\beta=\frac{R_1}{R_1+R_2}=\frac{10}{110}=0.0909,\qquad V_{th}=\beta V_{sat}=\boxed{\pm1.091\text{ V}}$$
Each half-cycle is a first-order $RC$ charge toward the current output rail starting from the opposite threshold; e.g. starting at $V_c=-V_{th}$ just after $V_o$ flips to $+V_{sat}$:
$$\boxed{V_c(t)=V_{sat}+(-V_{th}-V_{sat})e^{-t/RC}}\quad\text{(this half-cycle, }RC=100\text{k}\times0.1\mu\text{F}=10\text{ms)}$$
and symmetrically $V_c(t)=-V_{sat}+(V_{th}+V_{sat})e^{-t'/RC}$ on the other half, each valid until $V_c$ reaches $\pm V_{th}$ and the cycle flips.
Part (d) — frequency. Each half period is the time for the exponential to travel from $-V_{th}$ to $+V_{th}$ (or vice versa) toward a rail $\pm V_{sat}$:
$$T=2RC\ln\!\left(\frac{1+\beta}{1-\beta}\right)=2(0.01\text{s})\ln\!\left(\frac{1.0909}{0.9091}\right)=2(0.01)(0.1823)=3.646\text{ ms}$$
$$f=\frac{1}{T}=\boxed{274.2\text{ Hz}}$$
Fig. Q5(b) — astable multivibrator waveforms. $V_c$ exponentially relaxes between the $\pm1.09\text{V}$ thresholds ($\tau=RC=10\text{ms}$) while $V_o$ squares between $\pm12\text{V}$, period $T=3.65\text{ms}$.
