Electronics I

This is the fifth course in University Electrical Engineering: electronics I 模拟电子学 — the first course on devices. Wave 1 and EE04 analysed circuits made of ideal components; here the components themselves are the subject, and the core skill is modelling 建模: choosing the simplest device model that answers the question at hand.

• Unit 5.1 introduces semiconductors and the diode's ladder of models.
• Unit 5.2 applies diodes: rectifiers, supplies and zener regulators.
• Units 5.3–5.4 bias the two transistor families (BJT, MOSFET).
• Unit 5.5 linearises about the bias point: small-signal amplifiers.
• Units 5.6–5.7 add feedback and the frequency-domain view.

The checking habit: every bias point is verified against the device's operating regions (the assumption that made the equation legal), every small-signal answer at a limiting case. Each unit ends with the common mistakes and a three-line summary.

Pure silicon conducts poorly; doping 掺杂 adds carriers — donors make n-type (electrons), acceptors make p-type (holes). A pn junction PN 结 conducts one way: forward bias (p side positive) lets current flow exponentially; reverse bias widens the depletion region and blocks.

The diode has a ladder of models — pick by the question:

• Ideal 理想模型: 0 V when conducting, open when off.
• Constant-drop 恒压降模型: 0.7 V when conducting (silicon).
• Piecewise-linear 折线模型: 0.7 V + slope resistance $r_{d} = nV_{T}/I_{D}$.
• The exponential 指数模型 Shockley law $i = I_{S}(e^{v/nV_{T}} - 1)$ when the curve shape matters, with $V_{T} = kT/q \approx 26$ mV at room temperature.

Why the ladder works: differentiate the Shockley law at the operating point and the diode's dynamic resistance is $r_{d} = nV_{T}/I_{D}$ — a few ohms at milliamps. And because $i \propto e^{v/nV_{T}}$, ten times the current needs only $\ln 10 \times nV_{T} = 2.303 \times 26 = 59.9$ mV more voltage (n = 1): the curve is a near-vertical wall, so the resistor in the loop, not the diode, sets the current — which is exactly when the constant-drop model is accurate.

Worked example 1 (model selection, three ways). A 5 V source feeds a diode in series with 1 kΩ. Ideal: 5 mA. Constant-drop: $(5 - 0.7)/1000 = 4.3$ mA. Shockley truth ($I_{S} = 10^{-12}$ A, $nV_{T} = 26$ mV, bisection on $I_{S}(e^{v/26\mathrm{m}} - 1) = (5 - v)/1000$): $v_{D} = 0.5775$ V, $i = 4.423$ mA. The constant-drop answer is within 2.8% — because the resistor dominates the loop. Check why: the constant-drop model is off by $0.7 - 0.5775 = 0.12$ V in the diode voltage, and the resistor turns that into only $0.12/1000 = 0.12$ mA of current error — 2.8% of 4.42 mA.

Worked example 2 (the dynamic resistance, escalated). At that operating point $r_{d} = nV_{T}/I_{D} = 26\,\mathrm{mV}/4.42\,\mathrm{mA} = 5.9\ \Omega$. Now let the source wiggle by $\pm 0.1$ V (small signal): the wiggle current is $0.1/(1000 + 5.9) = 99.4\ \mu\mathrm{A}$ — within 0.6% of the resistor-only 100 µA. The 5 V worth of DC sits across the pair almost unchanged; the 0.1 V of signal sees an ordinary $1\ \mathrm{k}\Omega + 5.9\ \Omega$ loop. This is the whole small-signal idea of unit 5.5, met first at the diode: split the problem into a bias (where the nonlinear law sets the point) and a wiggle (where the linearised resistance takes over).

Common mistakes. Using the exponential model when the resistor dominates (needless bisection; the constant-drop model is the right tool there). Quoting "$r_d$ = 26 mV" — that is $V_T$; the resistance is $V_T/I_D$, current in the denominator. Forgetting n = 2 at high current (59.9 mV/decade becomes 120 mV/decade).

Summary: a diode is a one-way valve with a near-vertical forward curve; the loop resistor sets the current; $r_d = nV_T/I_D$ is the linearised view; pick the model the question needs and check the answer against the next model up.

The half-wave rectifier 半波整流器 conducts one half-cycle; the full-wave bridge 桥式整流 conducts both (four diodes, two conducting at a time, so the diode drops cost 1.4 V and each diode must survive a peak inverse voltage 反向峰值电压 of one secondary peak). A filter capacitor 滤波电容 holds the output between peaks: the ripple 纹波 $\Delta V \approx I_{load}/(f\cdot C)$ (half-wave) or $I_{load}/(2f\cdot C)$ (full-wave) — the capacitor must supply the load for the whole gap between recharge peaks.

Worked example 1 (the bridge supply, checked). A bridge rectifies a 12 V RMS, 50 Hz secondary; the load draws 100 mA and the capacitor is 2200 µF. Peak output: $12\sqrt{2} - 1.4 = 17.0 - 1.4 = 15.6$ V. Ripple (full wave, recharged every 10 ms): $\Delta V = 0.1/(100 \times 2200\mu) = 0.455$ V. So the output lives between 15.6 and 15.1 V. Checks: ripple ≪ peak (2.9% ✓ — the formula's premise); half-wave would double the ripple to 0.91 V for the same capacitor — the bridge buys a 2× for four diodes; PIV per diode ≈ 17.0 V (one secondary peak) — rate the diodes above that.

Worked example 2 (a zener regulator designed at both extremes, escalated). Regulate 12 V down to 5.1 V with a 5.1 V zener 齐纳二极管; the load varies 0–10 mA and the zener needs at least 2 mA to stay in breakdown. Worst case for the zener current is full load: $R \le (12 - 5.1)/(10\mathrm{m} + 2\mathrm{m}) = 575\ \Omega$ → pick 560 Ω. Now check the other extreme — no load: $I_{Z} = (12 - 5.1)/560 = 12.3$ mA, so $P_{Z} = 5.1 \times 12.3\mathrm{m} = 63$ mW and the resistor dissipates $(12-5.1)^{2}/560 = 85$ mW worst case. Full load leaves $I_{Z} = 12.3 - 10 = 2.3$ mA ≥ 2 mA ✓. Both extremes verified — that is a complete regulator design: one inequality from each end of the load range.

Common mistakes. Using f instead of 2f for the bridge ripple (the capacitor recharges every half-cycle). Forgetting the 1.4 V of two diode drops in the peak output. Sizing the zener resistor at the typical load only — the zener current must be checked at BOTH extremes, and its power at the no-load extreme.

Summary: bridge = both half-cycles, 1.4 V cost, PIV one peak; ripple $I/(2fC)$ set by the gap between recharges; the zener holds the voltage while the series resistor absorbs the surplus — designed at the two load extremes.

The bipolar junction transistor 双极型晶体管 has three regions — cut-off, active 放大区 (the amplifier region: $I_C = \beta I_B$, $V_{BE} \approx 0.7$ V), and saturation ($V_{CE} < 0.2$ V, both junctions forward). Biasing means placing the quiescent point (the bias point 工作点, Q-point) firmly in the active region and making it independent of β — the parameter that varies 2:1 between devices of the same part number.

Worked example 1 (voltage-divider bias, exact). $V_{CC} = 12$ V; $R_{1} = 68$ kΩ (to $V_{CC}$), $R_{2} = 12$ kΩ (to ground), $R_{C} = 2.2$ kΩ, $R_{E} = 1$ kΩ, $\beta = 100$. Thévenin the divider: $V_{B} = 12 \times 12/80 = 1.8$ V, $R_{B} = 68\mathrm{k} \parallel 12\mathrm{k} = 10.2$ kΩ. Quick estimate (ignore base current): $V_{E} = 1.8 - 0.7 = 1.1$ V → $I_{E} \approx 1.1$ mA. Exact (base loop with $I_E = (\beta+1)I_B$): $I_B = (1.8 - 0.7)/(10.2\mathrm{k} + 101 \times 1\mathrm{k}) = 9.9\ \mu$A → $I_C = 0.989$ mA, $V_{CE} = 12 - 0.99\mathrm{m} \times 2.2\mathrm{k} - 0.99 = 8.8$ V — active ✓ ($V_{CE} \gg 0.2$ V). The quick estimate was 11% high; the culprit is base current loading the divider — exactly what the exact loop accounts for.

Worked example 2 (β doubled, the point of the circuit, escalated). Same circuit, $\beta = 200$: $I_B = 1.1/(10.2\mathrm{k} + 201\mathrm{k}) = 5.2\ \mu$A → $I_C = 1.04$ mA, $V_{CE} = 8.7$ V — still active ✓. So β × 2 moved the collector current by 5% (0.99 → 1.04 mA) and the Q-point barely budged. The mechanism: any collector-current rise pushes the emitter voltage up, which reduces $V_{BE}$ and throttles the base current — negative feedback inside the bias, provided by $R_E$. With $R_E = 0$ the same β change would double $I_C$. Both error sources are now quantified: 11% from divider loading (fixable with a stiffer divider), 5% from β spread (absorbed by $R_E$).

Common mistakes. Checking $V_{CE} > 0$ instead of $> 0.2$ V (saturation starts at about 0.2 V). Computing $V_C$ from the estimated 1.1 mA and then "checking" against a Q-point computed from the exact current — carry one chain, then check it. Believing the divider alone makes the bias independent of β — it is $R_E$ (with the divider stiff enough to hold $V_B$) that does the work.

Summary: Thévenin the divider, solve the base loop exactly, check the region; $R_E$ buys β-independence at the price of headroom; estimate-then-verify is fine when the two are labelled as such.

The MOSFET 金属氧化物半导体场效应晶体管 is voltage-controlled: the gate draws no DC current, and in saturation 饱和区 (the amplifier region — the opposite naming to the BJT!) the square-law model 平方律模型 gives

with threshold 阈值 $V_t$. The three regions: cut-off ($v_{GS} < V_t$), triode/ohmic ($v_{DS} < v_{GS} - V_t$ — a resistor, not an amplifier), saturation (the inequality holds — a current source).

Worked example 1 (assume saturation, then check). $V_{DD} = 10$ V, $\tfrac{1}{2}k_n(W/L) = 0.5$ mA/V², $V_t = 1.5$ V, $R_D = 3$ kΩ, gate held at $V_G = 3$ V (divider — no DC gate current). Assume saturation: $I_D = 0.5 \times (3 - 1.5)^{2} = 1.125$ mA; $V_{DS} = 10 - 1.125\mathrm{m} \times 3\mathrm{k} = 6.6$ V. Check: $6.6 \ge 3 - 1.5 = 1.5$ ✓ — saturation stands, and the Q-point is (1.125 mA, 6.6 V).

Worked example 2 (the check that fails, escalated). Raise the gate to $V_G = 5$ V, same circuit. Assume saturation: $I_D = 0.5 \times 3.5^{2} = 6.1$ mA, $V_{DS} = 10 - 6.1\mathrm{m} \times 3\mathrm{k} = -8.4$ V — impossible (negative drain voltage), so the assumption is dead: the resistor cannot drop 18 V from a 10 V rail. Re-solve in triode with $i_D = 0.5(2 \times 3.5\, v_{DS} - v_{DS}^{2})$ and the load line $10 = 3\mathrm{k}\, i_D + v_{DS}$: the quadratic gives $v_{DS} = 1.00$ V, $i_D = 3.00$ mA. Checks: region $1.0 < 3.5$ ✓ triode; the triode equation returns $0.5(2 \times 3.5 \times 1 - 1) = 3.0$ mA ✓ consistent with the load line; and saturation at this gate would have demanded 18.4 V of supply. The lesson generalises: the model is an assumption — the region check is what makes it legal.

Common mistakes. Trusting the saturation current without the region check (the failed example above sails through to a wrong answer if you only check $V_{DS} > 0$). Mixing up the region names with the BJT (MOSFET saturation = amplifier; BJT saturation = ruined). Forgetting the gate current is zero at DC — the divider is unloaded, which is precisely why MOSFET dividers can use megohm resistors.

Summary: square law in saturation, boundary $v_{DS} = v_{GS} - V_t$ is the test; assume, solve, check — and when the check fails, the triode quadratic takes over; gate current zero at DC.

Amplification is linear only about a bias point (the Q-point of unit 5.3). The small-signal 小信号 model replaces each device by its linearised parameters at that point: BJT — $g_m = I_C/V_T$ (differentiate the exponential), $r_\pi = \beta/g_m$, $r_o$; MOSFET — $g_m = k(W/L)(V_{GS} - V_t) = 2I_D/(V_{GS} - V_t)$ (differentiate the square law), $r_o$. DC sources become shorts; coupling capacitors become shorts at signal frequency.

Worked example 1 (the common-emitter stage). At the unit-5.3 bias point $I_C = 0.99$ mA, $\beta = 100$: $g_m = 0.99\mathrm{m}/26\mathrm{m} = 38.1$ mS; $r_\pi = 100/38.1\mathrm{m} = 2.6$ kΩ. Voltage gain with $R_E$ bypassed and $r_o = 50$ kΩ: $A_v = -g_m(R_C \parallel r_o) = -38.1\mathrm{m} \times 2.11\mathrm{k} = -80$. Limiting checks: $r_o \to \infty$ gives $-g_m R_C = -84$ (r_o costs 5%); and the gain must be negative — the CE stage inverts.

Worked example 2 (degeneration and the MOSFET comparison, escalated). Lift the bypass so $R_E = 1$ kΩ participates: the exact gain is $-g_m R_C/(1 + g_m R_E) = -84/(1 + 38.1) = -2.14$, against the $-R_C/R_E = -2.2$ textbook form — within 2.4% because $g_m R_E = 38 \gg 1$ (that is the form's validity condition, now visible). Degeneration traded 84 for 2.2 — linearity and a stable gain bought with the same $R_E$ that stabilised the bias. Same story in MOSFET land: the unit-5.4 Q-point gives $g_m = 2I_D/V_{ov} = 2 \times 1.125\mathrm{m}/1.5 = 1.5$ mS and a common-source gain $-g_m R_D = -4.5$. The BJT at the same 1.125 mA has $43.3$ mS — 29× the MOSFET's — transconductance and current gain versus infinite input resistance: the fundamental trade.

Common mistakes. Computing $g_m$ from the estimated bias current after the exact solve exists — one chain, one set of parameters. Using $-R_C/R_E$ where $g_m R_E \sim 1$ (the form needs $g_m R_E \gg 1$). Believing the small-signal model is a different circuit — it is the same circuit, differentiated at the Q-point.

Summary: differentiate the device law at the Q-point to get $g_m$; $r_\pi = \beta/g_m$; bypassed gain $-g_m R_C$, degenerated $-R_C/R_E$ when $g_m R_E \gg 1$; the BJT wins gm-per-milliamp, the MOSFET wins the gate.

Wave 1's golden rules return with the finite-gain view. Negative feedback closes a loop; what matters is the loop gain 环路增益 $A_{OL}\beta_{fb}$. Derivation (inverting stage): if the op-amp can hold $v_- \approx v_+$, the input resistor carries $(v_{in} - 0)/R_{in}$, the feedback resistor carries the same current, and $v_o = -R_f v_{in}/R_{in}$. Writing the exact node equation with open-loop gain $A_{OL}$ and solving gives

so the golden-rule answer is exact in the limit $A_{OL}\beta_{fb} \to \infty$ and off by the factor $1/(1 + A_{OL}\beta_{fb})$ in real life.

Worked example 1 (the four topologies and GBW). Inverting $-R_f/R_{in}$ (input resistance $R_{in}$); non-inverting $1 + R_f/R_{in}$ (input resistance → ∞); transimpedance 跨阻放大器 $v_o = -I_p R_f$ (photodiode: $2\ \mu\mathrm{A} \times 1\ \mathrm{M}\Omega = 2$ V); summing $-(v_1 R_f/R_1 + v_2 R_f/R_2)$. An op-amp with gain-bandwidth product 增益带宽积 GBW = 1 MHz at closed-loop 40 dB (100×) has bandwidth 10 kHz; at 20 dB (10×), 100 kHz ✓ (the product stays 1 MHz along the hyperbola).

Worked example 2 (how wrong the golden rules are, escalated). The $1/(1 + A_{OL}\beta_{fb})$ error factor, computed. Closed-loop gain 100 ($\beta_{fb} = 0.01$): with $A_{OL} = 10^{5}$, $A_{CL} = 100/(1 + 1/1000) = 99.90$ — a 0.1% error, invisible on a scope. With a tired or high-frequency op-amp where $A_{OL}$ has fallen to $10^{3}$ (remember unit 5.7: the open-loop gain is already rolling off): $A_{CL} = 100/1.1 = 90.9$ — a 9.1% error. So the golden rules are a loop-gain statement: where the loop gain is large they are exact, and near the corner they quietly die. Check: at 10 kHz with GBW 1 MHz the open-loop gain has fallen to $10^{6}/10^{4} = 100$ — equal to the closed-loop gain, loop gain 1, and indeed that is the −3 dB corner found in example 1.

Common mistakes. Using the inverting formula for a non-inverting stage ($1 + R_f/R_{in}$ differs by exactly 1 — at unity gain it is everything). Forgetting the transimpedance stage's feedback resistor sets gain from current, so the photodiode's capacitance still shapes the noise gain.

Summary: loop gain $A_{OL}\beta_{fb}$ is the truth meter for the golden rules; error $= 1/(1 + A_{OL}\beta_{fb})$; GBW is a hyperbola — closed-loop gain times bandwidth is conserved until the corner, where the rules die.

Every amplifier is a band-pass: coupling capacitors 耦合电容 set the low corner $f_L = 1/(2\pi R_{eq}C)$ per stage ($R_{eq}$ = what the capacitor sees, sources zeroed), and at the high end Miller capacitance 米勒电容 plus internal poles take over. The Miller derivation: a capacitor $C$ from output to input of an inverting stage with gain $-A$ carries $i = C\,d(v_{in} - v_o)/dt = C(1 + A)\,dv_{in}/dt$ — the input node behaves as if it saw $C_M = C(1 + A)$ to ground. The dominant pole 主极点 approximation: one pole does most of the rolling, $f_H \approx 1/(2\pi R_{eq}C_{eq})$.

Worked example 1 (the full band-pass, both corners). The unit-5.5 CE stage: the coupling capacitor sees $R_{eq} = R_B \parallel r_\pi = 10.2\mathrm{k} \parallel 2.6\mathrm{k} = 2.08$ kΩ, so a 10 µF coupling gives $f_L = 1/(2\pi \times 2.08\mathrm{k} \times 10\mu) = 7.7$ Hz. High end: $C_{gb} = 5$ pF between base and collector where the stage gain is −80: $C_M = 5\mathrm{p} \times 81 = 405$ pF, plus 20 pF of stray → $f_H = 1/(2\pi \times 2\mathrm{k} \times 425\mathrm{p}) = 187$ kHz. The stage passes 7.7 Hz to 187 kHz at gain −80. Checks: the coupling cap sees the parallel combination (sources zeroed), not the sum — the classic $f_L$ slip; and the −80 inside the Miller factor is the mid-band gain, already falling near $f_H$ — the dominant-pole answer is optimistic by a few percent, the honest reading.

Worked example 2 (the cascode and the cascade, escalated). A cascode holds the collector nearly constant, killing the gain across the 5 pF: $f_H = 1/(2\pi \times 2\mathrm{k} \times 25\mathrm{p}) = 3.18$ MHz — 17× for one transistor: Miller explains the disease and the cure. Cascade instead: two identical 187 kHz stages in series are each 3 dB down there, so the pair is 6 dB down — the overall −3 dB point sits at $187 \times \sqrt{2^{1/2} - 1} = 120$ kHz. A multistage amplifier pays for its gain twice — once in Miller, once in cascade.

Common mistakes. Summing instead of paralleling resistances when finding what the coupling capacitor sees; reporting the per-stage corner as the system corner (the cascade shrinks it by $\sqrt{2^{1/n} - 1}$).

Summary: $f_L$ from coupling caps against zeroed-source resistance; Miller multiplies by $(1 + |A|)$; the cascode removes it; cascades shrink corners by $\sqrt{2^{1/n} - 1}$.