Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
98-Phys-A5 — Semiconductor Devices & Circuits — National Exams, May 2014
3 hours duration. Closed book exam (useful constants and equations annexed to the paper). Any FIVE (5) of the SEVEN (7) questions constitute a complete exam paper; all seven are answered here as a complete study resource.
Reference texts: A. S. Sedra & K. C. Smith, Microelectronic Circuits, 8th ed. (diode I–V characteristics and junction physics Ch. 3–4, op-amp circuits and active filters Ch. 2–12, BJT biasing and small-signal amplifiers Ch. 5–6, ADC architectures Ch. 17); C. Kittel, Introduction to Solid State Physics, 8th ed. (crystal structure and packing fraction Ch. 1, free-electron/semiconductor carrier statistics Ch. 8).
Figure P7a: non-inverting Schmitt trigger — $V_i$ through $R_1$ into the "+" input, $R_2$ feedback from $V_o$ to the same "+" input, "−" input grounded. Figure P7c: the same front end, but the output first passes through a series resistor $R$ before $V_o$, with two back-to-back Zener diodes clamping $V_o$ to ground; the feedback $R_2$ is taken from the clamped node $V_o$, not the raw op-amp output.
Find. (a) hysteresis transfer sketch; (b) $R_1$, $R_2$; (c) purpose of the Zener pair and resistor $R$.
Figure P7a — non-inverting Schmitt trigger: $R_1$ carries $V_i$ and $R_2$ carries the output feedback, both into the "+" input; "−" is grounded.
Approach. (a)/(b) superpose $V_i$ and $V_o$ at the "+" input, set the threshold condition $V_+=0$ for each rail value of $V_o$, then size the divider at the boundary of the allowed resistor range; (c) compare where the feedback signal is tapped in Figure P7a vs Figure P7c.
Part (a) — derive the threshold condition. With no current into the "+" input, KCL at that node gives $V_+=\dfrac{R_2V_i+R_1V_o}{R_1+R_2}$. The comparator switches when $V_+=V_-=0$, i.e. $V_i=-\dfrac{R_1}{R_2}V_o$. Since $V_o$ can only be at its two rail values:
$$V_{TH}=-\frac{R_1}{R_2}(-5\,\text{V})=\frac{5R_1}{R_2}\ (\text{switch LOW}\to\text{HIGH, rising }V_i)$$
$$V_{TL}=-\frac{R_1}{R_2}(+5\,\text{V})=-\frac{5R_1}{R_2}\ (\text{switch HIGH}\to\text{LOW, falling }V_i)$$
The transfer characteristic is a rectangular hysteresis loop, sketched below: $V_o=-5\,\text{V}$ for $V_i<V_{TL}$ (and staying there until $V_i$ climbs back above $V_{TH}$); $V_o=+5\,\text{V}$ for $V_i>V_{TH}$ (staying there until $V_i$ falls back below $V_{TL}$), symmetric about the origin.
Transfer characteristic $V_o(V_i)$: a rectangular hysteresis loop, $\pm5\,\text{V}$ output levels, switching at $V_{TL}$ (falling) and $V_{TH}$ (rising).
Part (b) — size $R_1$, $R_2$. Setting $V_{TH}=0.25\,\text{V}$ in the boxed relation above: $\dfrac{R_2}{R_1}=\dfrac{5}{0.25}=20$. To make the resistors "as large as possible" within the stated 1–200 k$\Omega$ range, push the larger one ($R_2$) to the top of the range:
$$R_2=200\,\text{k}\Omega\ \Rightarrow\ R_1=\frac{R_2}{20}=\boxed{10\ \text{k}\Omega},\qquad R_2=\boxed{200\ \text{k}\Omega}$$
Check: $V_{TH}=5(10)/200=0.25\,\text{V}$ ✓, and both values sit inside $[1,200]\,\text{k}\Omega$.
Part (c) — role of the Zener pair and R in Figure P7c. In Figure P7c the feedback resistor $R_2$ is tapped from $V_o$ after the series resistor $R$ and the Zener clamp, not from the op-amp's own output. The two back-to-back Zeners clamp $V_o$ to a fixed, well-defined level: for $V_o>0$ one diode conducts forward ($V_D=0.5\,\text{V}$) while the other breaks down in reverse ($|V_Z|=4\,\text{V}$), pinning
$$|V_o|_{\text{clamped}}=V_Z+V_D=4+0.5=\boxed{4.5\ \text{V}}$$
This is the improvement: an op-amp's own saturation voltage (here nominally $\pm5\,\text{V}$) is not a tightly-specified parameter — it drifts with supply rails, loading and temperature — so basing the hysteresis thresholds on it (as in Figure P7a) gives imprecise, unrepeatable $V_{TH}/V_{TL}$. Clamping $V_o$ with a matched Zener pair fixes the feedback signal's amplitude precisely, so $V_{TH}=5R_1/R_2$ becomes $V_{TH}=4.5\,R_1/R_2$: precise and reproducible regardless of op-amp saturation variation. The series resistor $R$ limits the current the op-amp's output stage must source/sink once the Zeners clamp $V_o$ (without it, the op-amp would try to drive its own unclamped saturation voltage directly into the low dynamic impedance of the breakdown diode, risking excessive current).
