Analog Electronics (EX-305) - Unit 2 Short Notes

UNIT 2: ANALOG ELECTRONICS

1. SEMICONDUCTOR DIODES & RECTIFICATION

P-N Junction Diode

• V-I Characteristics:

  • Forward Bias: Exponential increase in current after ~0.7V (Si). $$\displaystyle I = I_S \left( e^{\frac{V}{\eta V_T}} - 1 \right) $$, where $$\displaystyle V_T = kT/q \approx 26\,\mathrm{mV} $$ at room temp, $$\displaystyle \eta = 1-2 $$.

  • Reverse Bias: Small saturation current $$\displaystyle I_S $$ (~nA) until breakdown.

  • Dynamic Resistance: $$\displaystyle r_d = \frac{\eta V_T}{I} $$ (inverse slope).

• Applications:

  • Rectification: AC to DC conversion.

  • Clipping: Limiting signal amplitude (series/shunt).

  • Clamping: Shifting DC level (positive/negative clamper).

  • Switching: Fast transition between states; limited by reverse recovery time $$\displaystyle t_{rr} $$.

• Transient Response:

  • Reverse Recovery: When forward-biased diode is suddenly reverse-biased, stored charge $$\displaystyle Q_s $$ must be removed, causing reverse current $$\displaystyle I_R = Q_s / t_{rr} $$.

  • Instant Reversal Problem: If voltage reverses instantly at $$\displaystyle t=0 $$, $$\displaystyle I_R(0) = \frac{Q_s}{t_{rr}} $$ where $$\displaystyle Q_s \approx I_F \cdot t_f $$ ($$\displaystyle t_f $$ = forward conduction time).

Special Purpose Diodes

Rectifiers & Filters

• Half-Wave Rectifier:

  • $$\displaystyle V_{dc} = \frac{V_m}{\pi} $$, $$\displaystyle V_{rms} = \frac{V_m}{2} $$, $$\displaystyle \eta = 40.6\% $$, $$\displaystyle \gamma = 1.21 $$, $$\displaystyle \text{PIV} = V_m $$.

• Full-Wave Rectifier:

  • Center-Tapped: $$\displaystyle \text{PIV} = 2V_m $$.

  • Bridge: $$\displaystyle \text{PIV} = V_m $$.

  • Efficiency Derivation:

$$P_{dc} = \left( \frac{2V_m}{\pi} \right)^2 \frac{1}{R_L}, \quad P_{ac} = \frac{V_m^2}{2R_L} \quad \Rightarrow \quad \eta_{max} = \frac{P_{dc}}{P_{ac}} = \frac{8}{\pi^2} \approx 81.2\%$$

\boxed{\eta_{max} = 81.2\%}

• $$\displaystyle \gamma = 0.48 $$ (no filter).

• Filters:

  | Filter Type | Ripple Factor ($\gamma$) | Circuit Description | |-----------------|----------------------------------------------|--------------------------------------| | Capacitor ($\pi$) | $$\displaystyle \gamma \approx \frac{1}{4\sqrt{3} f R C} $$ (full-wave) | Shunt C after rectifier | | Choke (L-type) | $$\displaystyle \gamma = \frac{R_L}{2\pi f L} $$ | Series L, then shunt C | | LC ($\pi$) | $$\displaystyle \gamma = \frac{1}{8\pi^2 f^2 L C} $$ | L in series, C shunt, then load |

• Voltage Multipliers: Half-wave doubler, full-wave doubler, bridge doubler – cascade capacitors/diodes to multiply peak voltage.

2. BIPOLAR JUNCTION TRANSISTORS (BJT)

Construction & Types

• NPN: Emitter (n+), Base (p), Collector (n). Easier to fabricate, higher mobility.

• PNP: Emitter (p+), Base (n), Collector (p). Symbol arrow inward.

Transistor Operation & Regions

Transistor Parameters

• Common-Base Current Gain: $$\displaystyle \alpha = I_C / I_E $$ (0.95–0.99).

• Common-Emitter Current Gain: $$\displaystyle \beta = I_C / I_B $$ (20–500).

• Relationship Derivation:

$$I_E = I_B + I_C = \frac{I_C}{\alpha} \quad \Rightarrow \quad I_B = I_C \left( \frac{1}{\alpha} - 1 \right) \quad \Rightarrow \quad \beta = \frac{I_C}{I_B} = \frac{\alpha}{1-\alpha}$$

\boxed{\alpha = \frac{\beta}{\beta+1}, \quad \beta = \frac{\alpha}{1-\alpha}}

Configurations & Characteristics

Biasing Techniques

• Fixed Bias: $$\displaystyle I_B = (V_{CC} - V_{BE})/R_B $$. Disadvantage: Poor stability ($$\displaystyle \Delta I_C / \Delta \beta $$ large).

• Voltage Divider Bias (Self-Bias):

  • $$\displaystyle V_B = V_{CC} \frac{R_2}{R_1+R_2} $$, $$\displaystyle V_E = V_B - V_{BE} $$, $$\displaystyle I_E \approx V_E / R_E $$, $$\displaystyle I_C \approx I_E $$.

  • Stability: $$\displaystyle R_E $$ provides negative feedback. Stability factor $$\displaystyle S \approx 1 + \frac{R_B}{R_E} $$ where $$\displaystyle R_B = R_1 \parallel R_2 $$.

  • Design: Choose $$\displaystyle V_E = V_{CC}/10 $$, $$\displaystyle I_E = I_C/\alpha $$, then $$\displaystyle R_E = V_E / I_E $$, $$\displaystyle R_B \approx 10 R_E $$, $$\displaystyle R_1, R_2 $$ from $$\displaystyle V_B $$ and $$\displaystyle R_B $$.

Load Line Analysis

• DC Load Line: $$\displaystyle V_{CE} = V_{CC} - I_C R_C $$. Intersection with output characteristic gives Q-point.

• AC Load Line: Slope $$\displaystyle = -1/R_C' $$ where $$\displaystyle R_C' = R_C \parallel R_L $$. Used to determine maximum signal swing without distortion.

Small-Signal h-Parameter Model (CE)

• $$\displaystyle h_{ie} = \left. \frac{\Delta V_{BE}}{\Delta I_B} \right|_{V_{CE}=\text{const}} \approx r_{\pi} = \beta \frac{V_T}{I_C} $$

• $$\displaystyle h_{fe} = \left. \frac{\Delta I_C}{\Delta I_B} \right|_{V_{CE}=\text{const}} \approx \beta $$

• $$\displaystyle h_{oe} = \left. \frac{\Delta I_C}{\Delta V_{CE}} \right|_{I_B=\text{const}} \approx 1/r_o $$

• $$\displaystyle h_{re} = \left. \frac{\Delta V_{BE}}{\Delta V_{CE}} \right|_{I_B=\text{const}} \ll 1 $$ (often neglected).

• CE Amplifier Gain (with emitter resistor $$\displaystyle R_E $$ unbypassed):

$$A_v = -\frac{h_{fe} R_C'}{h_{ie} + (1+h_{fe})R_E} \approx -\frac{R_C'}{r_e + R_E} \quad \text{where } r_e = \frac{V_T}{I_E}$$

Special Circuits

• Darlington Pair: Two transistors connected (emitter of Q1 to base of Q2). $$\displaystyle \beta_{total} \approx \beta_1 \beta_2 $$, $$\displaystyle V_{BE(total)} \approx 1.4\,\mathrm{V} $$. High input impedance.

• Transistor as Switch:

  • Cut-off: $$\displaystyle V_{in} < V_{BE(on)} $$, $$\displaystyle I_C \approx 0 $$, $$\displaystyle V_{CE} \approx V_{CC} $$.

  • Saturation: $$\displaystyle V_{in} > V_{BE(on)} $$, $$\displaystyle I_C = \frac{V_{CC} - V_{CE(sat)}}{R_C} $$, $$\displaystyle V_{CE} \approx V_{CE(sat)} \approx 0.2\,\mathrm{V} $$.

  • Design: Ensure $$\displaystyle I_B > I_C / \beta $$ for saturation.

3. FIELD EFFECT TRANSISTORS (FET)

JFET (Junction FET)

• Construction: N-channel (more common) or P-channel. Gate forms reverse-biased p-n junction.

• Working Principle: $$\displaystyle V_{GS} $$ controls depletion region width → channel resistance. Pinch-off at $$\displaystyle V_{GS} = V_P $$ (negative for N-channel).

• Shockley’s Equation:

$$I_D = I_{DSS} \left(1 - \frac{V_{GS}}{V_P}\right)^2 \quad \text{(for } V_{GS} \leq 0 \text{ in N-channel)}$$

• Parameters:

  • $$\displaystyle I_{DSS} $$: Drain current at $$\displaystyle V_{GS}=0 $$ (saturation).

  • $$\displaystyle V_P $$: Pinch-off voltage (negative for N-channel).

  • $$\displaystyle g_m = \left. \frac{\Delta I_D}{\Delta V_{GS}} \right|_{V_{DS}=\text{const}} = \frac{2I_{DSS}}{|V_P|} \left(1 - \frac{V_{GS}}{V_P}\right) $$.

  • $$\displaystyle r_{ds} = 1/g_m $$.

• Biasing:

  • Self-Bias: $$\displaystyle V_{GS} = -I_D R_S $$. Solve Shockley and KVL: $$\displaystyle V_{DS} = V_{DD} - I_D (R_D + R_S) $$.

  • Voltage Divider: $$\displaystyle V_{GS} = V_G - I_D R_S $$ where $$\displaystyle V_G = V_{DD} \frac{R_2}{R_1+R_2} $$.

• Design Problem (Example from Jun 2024):

  Given: $$\displaystyle V_P = -5\,\mathrm{V} $$, $$\displaystyle I_{DSS}=12\,\mathrm{mA} $$, $$\displaystyle V_{DD}=12\,\mathrm{V} $$, $$\displaystyle I_D=5\,\mathrm{mA} $$, $$\displaystyle V_{DS}=6\,\mathrm{V} $$. Find $$\displaystyle R_D $$, $$\displaystyle R_S $$ for self-bias.

  Solution:

  1. From Shockley: $$\displaystyle 5 = 12 \left(1 - \frac{V_{GS}}{-5}\right)^2 \Rightarrow \left(1 + \frac{V_{GS}}{5}\right)^2 = \frac{5}{12} \approx 0.4167 $$.

     Since $$\displaystyle V_{GS} $$ negative, $$\displaystyle 1 + V_{GS}/5 = -\sqrt{0.4167} \approx -0.6455 $$ (positive root gives $$\displaystyle V_{GS}>0 $$ invalid).

     $$\displaystyle \Rightarrow V_{GS} = -5(1 + 0.6455) = -8.2275\,\mathrm{V} $$.

  2. $$\displaystyle V_{GS} = -I_D R_S \Rightarrow R_S = \frac{|V_{GS}|}{I_D} = \frac{8.2275}{0.005} \approx 1645\,\Omega $$.

  3. KVL: $$\displaystyle V_{DS} = V_{DD} - I_D (R_D + R_S) \Rightarrow R_D + R_S = \frac{V_{DD} - V_{DS}}{I_D} = \frac{12-6}{0.005} = 1200\,\Omega $$.

     $$\displaystyle \Rightarrow R_D = 1200 - 1645 = -445\,\Omega $$? Error: Check sign. For N-channel, $$\displaystyle V_{GS} $$ negative, so $$\displaystyle V_{GS} = -I_D R_S $$ implies $$\displaystyle R_S = |V_{GS}|/I_D $$. But from Shockley, $$\displaystyle 1 - V_{GS}/V_P = 1 - (-I_D R_S)/(-5) = 1 - I_D R_S/5 $$. Since $$\displaystyle V_P $$ negative, $$\displaystyle V_{GS}/V_P = (-I_D R_S)/(-5) = I_D R_S/5 $$. So:

$$I_D = I_{DSS} \left(1 - \frac{I_D R_S}{|V_P|}\right)^2$$

 $$\displaystyle \Rightarrow \sqrt{I_D/I_{DSS}} = 1 - \frac{I_D R_S}{|V_P|} $$ (since $$\displaystyle I_D < I_{DSS} $$, term <1).

 $$\displaystyle \Rightarrow \frac{I_D R_S}{|V_P|} = 1 - \sqrt{5/12} \approx 1 - 0.6455 = 0.3545 $$.

 $$\displaystyle \Rightarrow R_S = \frac{0.3545 \times 5}{0.005} = 354.5\,\Omega $$.

 Then $$\displaystyle R_D = 1200 - 354.5 = 845.5\,\Omega $$.

 **Answer**: $$\displaystyle R_S \approx 354.5\,\Omega $$, $$\displaystyle R_D \approx 845.5\,\Omega $$.

MOSFET (Metal-Oxide-Semiconductor FET)

• Types:

  • Enhancement: No channel initially. $$\displaystyle V_{GS} > V_{th} $$ (N-channel) creates inversion layer.

  • Depletion: Channel exists at $$\displaystyle V_{GS}=0 $$. $$\displaystyle V_{GS} $$ can be positive/negative to deplete/enhance.

• Symbols: Arrow direction for body diode; enhancement has broken channel line.

• CMOS: Complementary pair (N-MOS + P-MOS) in pull-down/pull-up. Merits: Near-zero static power, high noise margin.

4. AMPLIFIERS & FEEDBACK

Amplifier Classes

Negative Feedback

• Topologies:

  | Type | Feedback Signal | Input Connection | Output Connection | Effect on $$\displaystyle Z_{in} $$ | Effect on $$\displaystyle Z_{out} $$ | |------------------|-----------------|------------------|-------------------|--------------------|---------------------| | Voltage-Series | Voltage | Series | Voltage | Increases | Increases | | Voltage-Shunt | Voltage | Shunt | Voltage | Decreases | Decreases | | Current-Series | Current | Series | Current | Increases | Decreases | | Current-Shunt | Current | Shunt | Current | Decreases | Increases |

• Effects:

  • Gain: $$\displaystyle A_{vf} = \frac{A}{1 + A\beta} $$ (reduced but stabilized).

  • Bandwidth: $$\displaystyle BW_{fb} = BW_{ol} (1 + A\beta) $$ (Gain-Bandwidth product constant).

  • Distortion & Noise: Reduced by factor $(1+A\beta)$.

  • Stability: Oscillation if $$\displaystyle |A\beta|=1 $$ and $$\displaystyle \angle A\beta = 0^\circ $$ (Barkhausen).

Power Amplifiers (Class B)

• Push-Pull: Two transistors (NPN/PNP or complementary) conduct alternately.

• Maximum Load Power:

$$P_{o,max} = \frac{V_{CC}^2}{2R_L} \quad \text{(ideal, with } V_{CE(sat)}=0\text{)}$$

• DC Input Power: $$\displaystyle P_{dc} = \frac{2V_{CC} I_{C(peak)}}{\pi} = \frac{2V_{CC} V_{CC}}{\pi R_L} = \frac{2V_{CC}^2}{\pi R_L} $$.

• Collector Efficiency:

$$\eta = \frac{P_{o,max}}{P_{dc}} = \frac{\pi}{4} \approx 78.5\%$$

\boxed{\eta_{max} = 78.5%}

5. OSCILLATORS

Barkhausen Criterion: For sustained oscillations, $$\displaystyle |A\beta| = 1 $$ and $$\displaystyle \angle A\beta = 0^\circ $$ (or $$\displaystyle 360^\circ $$).

RC Oscillators

• RC Phase-Shift (3 RC sections):

$$f = \frac{1}{2\pi RC \sqrt{6}} \quad \text{(for equal R, C)}$$

• Wein Bridge (Lead-Lag network):

$$f = \frac{1}{2\pi RC}$$

Design: For given $f$, choose $C$, then $$\displaystyle R = \frac{1}{2\pi f C} $$.

LC Oscillators

• Hartley: Feedback via inductive divider ($$\displaystyle L_1 $$, $$\displaystyle L_2 $$).

$$f = \frac{1}{2\pi \sqrt{(L_1+L_2)C}}$$

• Colpitts: Feedback via capacitive divider ($$\displaystyle C_1 $$, $$\displaystyle C_2 $$).

$$f = \frac{1}{2\pi \sqrt{L \cdot \frac{C_1 C_2}{C_1+C_2}}}$$

• Comparison: Hartley uses inductors, Colpitts uses capacitors; Colpitts better for high freq due to lower inductor losses.

Crystal Oscillator

• Principle: Piezoelectric effect. Crystal acts as series/parallel resonant circuit with very high Q.

• Frequency: Determined by crystal cut; extremely stable.

• Circuit: Crystal in feedback path (series or parallel resonant mode).

6. OPERATIONAL AMPLIFIERS (Op-Amp)

Ideal vs Practical

Basic Circuits (Negative Feedback)

• Inverting Amplifier:

$$A_v = -\frac{R_f}{R_{in}}, \quad Z_{in} = R_{in}$$

Virtual ground at $(-)$ input.

• Non-Inverting Amplifier:

$$A_v = 1 + \frac{R_f}{R_{in}}, \quad Z_{in} \approx \infty$$

• Voltage Follower: $$\displaystyle R_f=0 $$, $$\displaystyle R_{in}=\infty $$, $$\displaystyle A_v=1 $$.

Special Applications

• Summing Amplifier (Inverting):

$$V_{out} = -R_f \left( \frac{V_1}{R_1} + \frac{V_2}{R_2} + \cdots \right)$$

• Differential Amplifier:

$$A_d = \frac{R_2}{R_1} \quad \text{(if } R_1=R_3, R_2=R_4\text{)}$$

CMRR (Common-Mode Rejection Ratio):

$$\text{CMRR} = \left| \frac{A_d}{A_c} \right| \quad \text{or} \quad \text{CMRR}_{\mathrm{dB}} = 20 \log \left| \frac{A_d}{A_c} \right|$$

[!TIP] In Jun 2024: Given $$\displaystyle A_d $$ with $10\,\mathrm{mV}$ diff input gives $1\,\mathrm{V}$ output, $$\displaystyle A_c $$ with $10\,\mathrm{mV}$ CM input gives $5\,\mathrm{mV}$ output → $$\displaystyle \text{CMRR} = 200 $$, $$\displaystyle \text{CMRR}_{\mathrm{dB}} = 20\log 200 \approx 46\,\mathrm{dB} $$.

• Integrator:

$$V_{out} = -\frac{1}{RC} \int V_{in} \, dt$$

Output for sine → cosine; square → triangular (with saturation limits).

• Differentiator:

$$V_{out} = -RC \frac{dV_{in}}{dt}$$

Output for square → spikes at edges; sine → cosine (phase lead). Noise susceptible.

• Logarithmic Amplifier:

  Uses diode in feedback: $$\displaystyle V_{out} = -V_T \ln \left( \frac{V_{in}}{I_s R} \right) $$.

• Zero Crossing Detector: Inverting config without feedback → comparator. Output switches when $$\displaystyle V_{in} $$ crosses $0\,\mathrm{V}$.

Active Filters

• Second-Order Low-Pass (Sallen-Key):

  Given $R$, $$\displaystyle R_f $$, $$\displaystyle R_1 $$, $$\displaystyle R_2 $$, $$\displaystyle C_1 $$, $$\displaystyle C_2 $$:

$$f_c = \frac{1}{2\pi \sqrt{R_1 R_2 C_1 C_2}}$$

Passband gain $$\displaystyle A_0 = 1 + \frac{R_f}{R_g} $$ (if non-inverting Sallen-Key).

7. WAVEFORM GENERATION & SHAPING

555 Timer IC

• Internal Block: Voltage divider (three $5\,\mathrm{k}\Omega$), two comparators, SR flip-flop, discharge transistor.

• Astable Multivibrator:

  $$\displaystyle T_{on} = 0.693 (R_A + R_B) C $$, $$\displaystyle T_{off} = 0.693 R_B C $$

$$f = \frac{1.44}{(R_A + 2R_B)C}$$

Duty cycle $$\displaystyle D = \frac{R_A + R_B}{R_A + 2R_B} $$.

Op-Amp Multivibrators

• Astable: With RC feedback, output switches between $$\displaystyle \pm V_{sat} $$. Frequency $$\displaystyle f = \frac{1}{2RC \ln \left( \frac{1+\beta}{1-\beta} \right)} $$ where $$\displaystyle \beta = R_1/(R_1+R_2) $$.

• Schmitt Trigger (Positive Feedback):

  • Hysteresis width: $$\displaystyle V_{UT} - V_{LT} = 2 \frac{R_1}{R_1+R_2} V_{sat} $$.

  • Transfer characteristic: Output switches at different input thresholds.

  • Applications: Noise immunity, waveform conversion (sine to square).

• Bistable Multivibrator: Two stable states; triggered by external pulse.

Clippers & Clampers

• Clippers: Remove part of signal.

  • Series: Diode in series with load.

  • Shunt: Diode parallel to load.

  • Double-Ended: Two diodes opposite directions; clips at $$\displaystyle \pm V_{ref} $$ (equal/unequal).

• Clampers (DC Restorers):

  • Positive Clamper: Shifts signal upward so negative peak $\approx 0\,\mathrm{V}$.

  • Negative Clamper: Shifts downward.

  • Biased Clamper: Adds DC bias $$\displaystyle V_{bias} $$.

  • Clamping Theorem: Output waveform is input shifted by DC level such that peak $$\displaystyle V_{out} = V_{peak(in)} \pm V_{diode} $$ (or $$\displaystyle \pm V_{bias} $$).

8. SPECIAL TOPICS (Short Notes)

MOSFET (Enhancement vs Depletion)

• Enhancement: No channel at $$\displaystyle V_{GS}=0 $$. Requires $$\displaystyle V_{GS} > V_{th} $$ (N) to conduct. Used in digital CMOS.

• Depletion: Channel exists at $$\displaystyle V_{GS}=0 $$. Can be depleted by $$\displaystyle V_{GS} < V_{th} $$ (N) or enhanced by $$\displaystyle V_{GS} > 0 $$. Used in analog switches.

• CMOS Inverter: P-MOS pull-up, N-MOS pull-down. Low static power, high noise margin.

Schottky Diode

• Metal-semiconductor junction (e.g., Al-Si). No minority carrier storage → fast switching ($$\displaystyle t_{rr} \approx 0.1\,\mathrm{ns} $$), low forward voltage ($$\displaystyle V_f \approx 0.3\,\mathrm{V} $$). Used in high-frequency rectifiers, clamping.

Hartley vs Colpitts Oscillator

Schmitt Trigger

• Positive feedback creates hysteresis. Two threshold voltages: $$\displaystyle V_{UT} $$ and $$\displaystyle V_{LT} $$.

• Transfer characteristic: Output switches when $$\displaystyle V_{in} $$ exceeds $$\displaystyle V_{UT} $$ (high) or falls below $$\displaystyle V_{LT} $$ (low).

• Applications: Noise immunity, sine-to-square conversion, debouncing.

Varactor Diode

• Reverse-biased p-n junction with variable depletion region capacitance.

• $$\displaystyle C \propto 1/V^m $$ where $m$ depends on doping profile (abrupt: $$\displaystyle m=0.5 $$, linear: $$\displaystyle m=2 $$).

• Used in voltage-controlled oscillators (VCOs), frequency modulators.

Crystal Oscillator

• Piezoelectric Effect: Mechanical stress ↔ electrical charge.

• Resonance: Series resonance ($$\displaystyle f_s $$) and parallel resonance ($$\displaystyle f_p $$) close together.

• Advantage: Extremely high Q ($$\displaystyle 10^4 $$–$$\displaystyle 10^6 $$) → excellent frequency stability.

• Circuit: Crystal in feedback path; operates at $$\displaystyle f_s $$ or $$\displaystyle f_p $$.

Bistable Multivibrator

• Two stable states (Q1 ON/Q2 OFF or vice versa).

• Triggered by external pulse to switch states.

• Applications: Memory cells, flip-flops, frequency dividers.

Op-Amps Overview

• Key Specs: $$\displaystyle A_{OL} $$, GBW, slew rate, input offset, CMRR.

• Ideal Rules: Virtual short ($$\displaystyle V_+ = V_- $$) in negative feedback, input currents zero.

• Common Configurations: Inverting, non-inverting, differential, integrator, differentiator, filters.

[!TIP] Exam Focus: Past papers frequently ask for derivations (full-wave efficiency, oscillator frequency), design problems (JFET bias, Wein bridge), and comparisons (BJT configs, rectifiers, feedback topologies). Always box final formulas and include clear diagrams in explanations.