⏳
Loading cheatsheet...
Diodes, BJTs, FETs, biasing, amplifiers and frequency response fundamentals for ECE.
| Parameter | Silicon (Si) | Germanium (Ge) | GaAs |
|---|---|---|---|
| Bandgap (eV) | 1.12 | 0.66 | 1.43 |
| Intrinsic Concentration nᵢ (cm⁻³) | 1.5 × 10¹⁰ | 2.4 × 10¹³ | 1.8 × 10⁶ |
| Cut-in Voltage V_γ | ~0.7 V | ~0.3 V | ~1.2 V |
| Reverse Saturation Iₛ | ~nA | ~µA | ~pA |
| Max Operating Temp | ~150 °C | ~85 °C | ~200 °C |
| Thermal Voltage V_T (300K) | 26 mV | 26 mV | 26 mV |
┌─────────────────────────────────────────────────────────────────┐
│ DIODE CURRENT EQUATION │
├─────────────────────────────────────────────────────────────────┤
│ │
│ Shockley Diode Equation: │
│ I = Iₛ (e^(V / ηV_T) - 1) │
│ │
│ Where: │
│ Iₛ = Reverse saturation current (typically nA) │
│ η = Ideality factor (1 for Ge, 2 for Si at low current) │
│ V_T = Thermal voltage = kT/q = 26 mV at 300 K │
│ k = Boltzmann constant = 1.38 × 10⁻²³ J/K │
│ q = Electron charge = 1.6 × 10⁻¹⁹ C │
│ │
│ Forward Bias: V > 0 → I ≈ Iₛ e^(V/ηV_T) (exponential rise) │
│ Reverse Bias: V < 0 → I ≈ -Iₛ (tiny leakage) │
│ Breakdown: V = -V_BR → I rises sharply (avalanche/Zener) │
│ │
│ TEMPERATURE DEPENDENCE: │
│ dV_γ/dT ≈ -2 mV/°C for Si (voltage decreases with heat) │
│ Iₛ doubles for every 10 °C rise │
│ │
│ DYNAMIC RESISTANCE: │
│ r_d = ηV_T / I_D = 26 mV / I_D (at room temp, η=1) │
│ At 1 mA: r_d = 26 Ω | At 10 mA: r_d = 2.6 Ω │
└─────────────────────────────────────────────────────────────────┘| Type | V_dc (avg) | Ripple Factor | # Diodes | PIV | Efficiency |
|---|---|---|---|---|---|
| Half-Wave | V_m / π | 1.21 | 1 | V_m | 40.6% |
| Full-Wave (CT) | 2V_m / π | 0.48 | 2 | 2V_m | 81.2% |
| Bridge | 2V_m / π | 0.48 | 4 | V_m | 81.2% |
| Parameter | Formula / Value |
|---|---|
| Zener Current Range | I_Z(min) < I_Z < I_Z(max) |
| Load Current | I_L = V_Z / R_L |
| Series Resistance | R_S = (V_in - V_Z) / (I_Z + I_L) |
| Line Regulation | ΔV_o / ΔV_in = r_Z / (r_Z + R_S) |
| Load Regulation | ΔV_o / ΔI_L ≈ -r_Z (for small changes) |
| Power Dissipation | P_Z = V_Z × I_Z(max) < P_Z(rated) |
┌─────────────────────────────────────────────────────────────────┐
│ CAPACITOR FILTER FOR FULL-WAVE RECTIFIER │
├─────────────────────────────────────────────────────────────────┤
│ │
│ RIPPLE FACTOR WITH C-FILTER: │
│ γ = 1 / (4√3 f C R_L) (full-wave) │
│ γ = 1 / (2√3 f C R_L) (half-wave) │
│ │
│ DESIGN EXAMPLE: │
│ V_in = 12V rms, f = 50 Hz, C = 1000 µF, R_L = 1 kΩ │
│ V_m = 12√2 = 16.97 V │
│ V_dc ≈ V_m - I_dc/(2fC) = 16.97 - (16.97/1000)/(2×50×1000µF) │
│ γ = 1/(4√3 × 50 × 1000µF × 1000) = 0.0029 (0.29%) │
│ │
│ CLIPPER CIRCUITS: │
│ • Positive Clipper: Diode cuts signal above V_ref │
│ • Negative Clipper: Diode cuts signal below V_ref │
│ • Biased Clipper: Battery adds offset to clipping level │
│ • Combination: Cuts both positive and negative portions │
│ │
│ CLAMPER (DC RESTORER) CIRCUITS: │
│ • Positive Clamper: Shifts signal UP by V_m (diode down) │
│ • Negative Clamper: Shifts signal DOWN by V_m (diode up) │
│ • Biased Clamper: Shifts by V_m + V_battery │
│ Output peak = Input peak + V_battery │
└─────────────────────────────────────────────────────────────────┘r_d = 26/I_D mA is the most frequently used approximation. For Zener regulators, always verify that I_Z(min) is maintained under worst-case load andP_Z = V_Z × I_Z(max)stays within the diode's power rating.| Parameter | Formula | Typical Value |
|---|---|---|
| Collector Current (Active) | I_C = β I_B | β = 50–300 |
| Emitter Current | I_E = I_B + I_C = (1+β) I_B | — |
| Thermal Voltage | V_T = kT/q | 26 mV @ 300K |
| Early Voltage | V_A = V_CE × g_m / I_C | 50–200 V |
| Transconductance | g_m = I_C / V_T | ~40 mA/V at I_C=1mA |
| Base-spreading Resistance | r_bb' | 100–500 Ω |
| Unity-gain BW (f_T) | f_T = g_m / (2π C_π) | 100s of MHz |
┌─────────────────────────────────────────────────────────────────┐
│ BJT OPERATING REGIONS (NPN) │
├─────────────────────────────────────────────────────────────────┤
│ │
│ ┌────────────────┬──────────────┬───────────────────────┐ │
│ │ Region │ B-E Junction │ B-C Junction │ │
│ ├────────────────┼──────────────┼───────────────────────┤ │
│ │ Active │ Forward │ Reverse │ │
│ │ Saturation │ Forward │ Forward │ │
│ │ Cut-off │ Reverse │ Reverse │ │
│ │ Inverted Act. │ Reverse │ Forward │ │
│ └────────────────┴──────────────┴───────────────────────┘ │
│ │
│ SATURATION: V_CE(sat) ≈ 0.2 V (Si), transistor acts as switch │
│ CUT-OFF: I_B = 0, I_C = I_CEO ≈ 0, transistor is open │
│ ACTIVE: I_C = βI_B, used for amplification │
└─────────────────────────────────────────────────────────────────┘| Method | Stability (S) | Notes |
|---|---|---|
| Fixed Bias | S = 1 + β (worst) | Simple, β-dependent, rarely used |
| Collector Feedback | S = (1+β)/(1+β·R_C/R_B) | Moderate, auto-stabilizing |
| Voltage Divider | S ≈ (1+β)(R₁‖R₂)/(R₁‖R₂ + (1+β)R_E) | Best, β-independent if R₁‖R₂ << βR_E |
| Emitter Bias | S = (1+β)/(1 + β·R_E/(R_B+R_E)) | Good, needs dual supply (-V_EE) |
Goal: Set Q-point I_CQ, V_CEQ
Given: V_CC = 12V, I_CQ = 2mA, V_CEQ = 5V, β = 100
Step 1: Choose V_E = V_CC/10 = 1.2V
Step 2: R_E = V_E / I_E ≈ 1.2V / 2mA = 600Ω
Step 3: R_C = (V_CC - V_CEQ - V_E)/I_C
= (12 - 5 - 1.2)/2mA = 2.9kΩ
Step 4: R₂ = β·R_E/10 = 100×600/10 = 6kΩ
Step 5: R₁ = (V_CC - V_B)/V_B × R₂
where V_B = V_E + V_BE = 1.2 + 0.7 = 1.9V
R₁ = (12 - 1.9)/1.9 × 6k = 31.9kΩ
Stability Factor: S ≈ 1 + R₁‖R₂/R_E ≈ 1.02 ✓┌─────────────────────────────────────────────────────────────────┐
│ HYBRID-π MODEL (Small Signal, NPN) │
├─────────────────────────────────────────────────────────────────┤
│ │
│ CIRCUIT MODEL: │
│ B ──[r_π]──┬──[C_π]──┤ │
│ │ │ │
│ [V_be] [g_m·V_be] (current source) │
│ │ │ │
│ GND C ────┤──[r_o]──[C_μ]── B │
│ │ │
│ GND │
│ │
│ PARAMETERS (at I_C = 1 mA): │
│ g_m = I_C / V_T = 1mA / 26mV = 38.46 mA/V │
│ r_π = β / g_m = 100 / 38.46m = 2.6 kΩ │
│ r_o = V_A / I_C = 100V / 1mA = 100 kΩ │
│ f_T = g_m / (2π(C_π + C_μ)) ≈ 300 MHz │
│ C_π ≈ g_m / (2π f_T) = 38.46m / (2π × 300M) ≈ 20.4 pF │
│ C_μ = C_ob (from datasheet, typically 2–5 pF) │
│ │
│ MILLER'S THEOREM: │
│ C_μ reflected to input: C_Miller(in) = C_μ(1 + |A_v|) │
│ C_μ reflected to output: C_Miller(out) = C_μ(1 + 1/|A_v|) │
│ For A_v = -100: C_in(miller) = C_μ × 101 ≈ 404 pF! │
│ │
│ HYBRID h-PARAMETERS: │
│ h_ie = input impedance ≈ r_π │
│ h_fe = forward current gain = β │
│ h_re = reverse voltage ratio ≈ 10⁻⁴ (negligible) │
│ h_oe = output admittance ≈ 1/r_o │
└─────────────────────────────────────────────────────────────────┘| Parameter | Formula (CE with R_E bypassed) |
|---|---|
| Voltage Gain | A_v = -g_m (R_C ‖ R_L) / (1 + g_m R_E) (unbypassed); = -g_m(R_C‖R_L) (bypassed) |
| Input Resistance | R_in = R₁ ‖ R₂ ‖ r_π |
| Output Resistance | R_out = R_C ‖ r_o |
| Current Gain | A_i = -β × (R_C ‖ R_L) / (R_C ‖ R_L + r_π) |
| With Emitter Bypass | C_E chosen: X_C_E << r_π at lowest frequency |
| Mid-band A_v (bypassed) | A_v = -β (R_C ‖ R_L) / r_π = -g_m (R_C ‖ R_L) |
R₁‖R₂ << βR_E ensures stability. In the hybrid-π model, always include r_o = V_A/I_C when computing output resistance or gain with large R_C.| Parameter | N-Channel JFET | P-Channel JFET |
|---|---|---|
| Drain Current (Saturation) | I_D = I_DSS(1 - V_GS/V_P)² | I_D = I_DSS(1 - V_GS/V_P)² |
| Pinch-off Voltage | V_P (negative for N-ch) | V_P (positive for P-ch) |
| Gate-Source Voltage | 0 ≥ V_GS ≥ V_P | 0 ≤ V_GS ≤ V_P |
| Transconductance | g_m = -2I_DSS/V_P × (1-V_GS/V_P) | Same form |
| g_m (at V_GS=0) | g_m0 = -2I_DSS/V_P | g_m0 = -2I_DSS/V_P |
| Output Resistance | r_ds = V_A / I_D | r_ds = V_A / I_D |
| Type | V_th | V_GS for ON | Substrate |
|---|---|---|---|
| N-ch Enhancement | Positive (+) | V_GS > V_th | P-type |
| P-ch Enhancement | Negative (-) | V_GS < V_th | N-type |
| N-ch Depletion | Negative (-) | V_GS > V_P or 0 | P-type (built-in channel) |
| P-ch Depletion | Positive (+) | V_GS < V_P or 0 | N-type (built-in channel) |
┌─────────────────────────────────────────────────────────────────┐
│ MOSFET EQUATIONS (N-channel) │
├─────────────────────────────────────────────────────────────────┤
│ │
│ ENHANCEMENT MOSFET: │
│ Cutoff Region (V_GS < V_th): I_D = 0 │
│ │
│ Linear / Triode Region (V_GS > V_th, V_DS < V_GS - V_th): │
│ I_D = K_n [2(V_GS - V_th)V_DS - V_DS²] │
│ where K_n = µ_n C_ox (W/L) / 2 │
│ │
│ Saturation Region (V_GS > V_th, V_DS ≥ V_GS - V_th): │
│ I_D = K_n (V_GS - V_th)² │
│ (ignoring channel-length modulation) │
│ │
│ With CLM (channel-length modulation): │
│ I_D = K_n (V_GS - V_th)² (1 + λV_DS) │
│ r_o = 1 / (λ I_D) = V_A / I_D │
│ │
│ TRANSFORMATIVE RELATIONSHIPS: │
│ g_m = 2I_D / (V_GS - V_th) = 2K_n(V_GS - V_th) │
│ g_m = √(2K_n I_D) = √(2µ_n C_ox (W/L) I_D) │
│ g_mb = η × g_m (body transconductance, η ≈ 0.1–0.3) │
│ │
│ DEPLETION MOSFET (N-channel, V_th = -V_P): │
│ I_D = I_DSS (1 - V_GS/V_P)² (same as JFET) │
│ ON for V_GS from V_P (pinch-off) to 0 (and beyond for enh.) │
│ │
│ DESIGN RULES: │
│ W/L ratio determines current drive capability │
│ Larger W/L → higher I_D, higher g_m │
│ Minimum L set by process technology (e.g., 7 nm) │
└─────────────────────────────────────────────────────────────────┘| Circuit | Formula | Application |
|---|---|---|
| Fixed Bias (Enh.) | V_GS = V_DD × R₂/(R₁+R₂); I_D = K(V_GS-V_th)² | Digital switching |
| Drain Feedback | V_GS = V_DD - I_D R_D; solve quadratic | Simple analog |
| Self Bias (Depl.) | R_G from gate to ground; R_S provides auto-bias | JFET / Depletion MOSFET |
| Voltage Divider | V_G = V_DD R₂/(R₁+R₂); V_GS = V_G - I_D R_S | Most common analog |
| Current Mirror | I_ref → M₁ sets V_GS; M₂ mirrors I_ref (W ratio) | IC design |
K_n = µ_n C_ox (W/L) / 2. For analog design, keep the transistor in saturation (V_DS ≥ V_GS - V_th). The transconductance g_m = √(2K_n I_D) increases with the square root of bias current — unlike BJT where g_m = I_C/V_T is linear in I_C.| Property | Ideal Value | Typical (741) |
|---|---|---|
| Open-loop Gain A_OL | ∞ | ~10⁵ (100 dB) |
| Input Impedance | ∞ | ~2 MΩ |
| Output Impedance | 0 | ~75 Ω |
| CMRR | ∞ | ~90 dB |
| Slew Rate | ∞ | 0.5 V/µs |
| Bandwidth (GBW) | ∞ | 1 MHz |
| Input Offset Voltage | 0 | ~1 mV |
| Input Bias Current | 0 | ~80 nA |
| Rule | Condition | Implication |
|---|---|---|
| Virtual Short | Negative feedback | V₊ = V₋ (no differential input) |
| No Input Current | Negative feedback | I₊ = I₋ = 0 |
┌─────────────────────────────────────────────────────────────────┐
│ OP-AMP CONFIGURATIONS & FORMULAS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ INVERTING AMPLIFIER: │
│ ┌─[R_in]──┬─[R_f]──┐ │
│ │ ─┤- │ A_v = -R_f / R_in │
│ V_in ─┤+ V_out R_in = R_in │
│ │ ─ │ BW = GBW / |A_v| │
│ └──────── GND ─┘ │
│ │
│ NON-INVERTING AMPLIFIER: │
│ V_in ─────┬──┤+ │ A_v = 1 + R_f / R_in │
│ │ ┤- [R_f]──┤ R_in → ∞ (very high) │
│ └──[R_in]───┤ │
│ │ Buffer (R_f=0): A_v = 1 │
│ GND Voltage follower, R_in = ∞ │
│ │
│ SUMMING AMPLIFIER (Inverting): │
│ V_o = -(R_f/R₁)V₁ - (R_f/R₂)V₂ - (R_f/R₃)V₃ │
│ If R₁ = R₂ = R₃ = R: V_o = -(R_f/R)(V₁ + V₂ + V₃) │
│ Weighted: each input scaled by its respective ratio │
│ │
│ DIFFERENCE AMPLIFIER: │
│ V_o = (R₂/R₁)(V₂ - V₁) │
│ Requires: R₁/R₂ = R₃/R₄ for proper common-mode rejection │
│ │
│ INSTRUMENTATION AMPLIFIER (3 op-amps): │
│ V_o = (1 + 2R₁/R_G)(R₃/R₂)(V₂ - V₁) │
│ Very high R_in, high CMRR, adjustable gain via R_G │
└─────────────────────────────────────────────────────────────────┘┌─────────────────────────────────────────────────────────────────┐
│ INTEGRATOR & DIFFERENTIATOR │
├─────────────────────────────────────────────────────────────────┤
│ │
│ INTEGRATOR (replace R_f with C): │
│ V_o(t) = -(1/RC) ∫ V_in(t) dt │
│ │
│ Frequency response: A(jω) = -1/(jωRC) │
│ |A| = 1/(ωRC) → acts as low-pass filter │
│ Gain ∝ 1/f → infinite gain at DC → practical issue! │
│ │
│ PRACTICAL INTEGRATOR: │
│ ┌─[R]──┬──┬──[R_f]──┐ R_f in parallel with C │
│ │ │ │ │ provides DC feedback (limits DC │
│ V_in ─┤- ├──[C]────┤ gain to R_f/R). f_c = 1/(2π R_f C) │
│ ──┤+ V_o │
│ ── ─┘ │
│ │
│ DIFFERENTIATOR (replace R_in with C): │
│ V_o(t) = -RC × dV_in(t)/dt │
│ │
│ Frequency response: A(jω) = -jωRC │
│ |A| = ωRC → acts as high-pass filter │
│ Gain ∝ f → amplifies high-frequency noise! │
│ │
│ PRACTICAL DIFFERENTIATOR: │
│ ┌─[C]──┬──┬──[R_f]──┐ R_in in series with C │
│ │ │ │ │ limits high-freq gain to R_f/R_in. │
│ V_in ─┤- ├──[R]────┤ f_c = 1/(2π R_in C) │
│ ──┤+ V_o │
│ ── ─┘ │
│ │
│ LOGARITHMIC AMPLIFIER: │
│ V_o = -(V_T) ln(V_in / I_s R) (uses diode/transistor) │
│ │
│ ANTI-LOG AMPLIFIER: │
│ V_o = -I_s R × e^(-V_in / V_T) │
└─────────────────────────────────────────────────────────────────┘| Parameter | Formula | Notes |
|---|---|---|
| Gain-Bandwidth Product | GBW = A_v × f_3dB = constant | For single-pole op-amp |
| Closed-loop BW | f_3dB(CL) = GBW / |A_v(CL)| | Higher gain → lower BW |
| Slew Rate | SR = dV_o/dt (max) | Largest signal rate of change |
| Full-Power BW | f_max = SR / (2π V_peak) | Max freq at full output swing |
| Phase Margin | PM = 180° - |phase at |Aβ|=1| | >45° for stability |
| Bode Plot (dominant pole) | Roll-off: -20 dB/decade per pole | Add capacitor for compensation |
slew rate limits large-signal performance: a 741 with SR = 0.5 V/µs can only output a 10 V peak sine wave up to f = 0.5/(2π×10) ≈ 8 kHz without distortion.┌─────────────────────────────────────────────────────────────────┐
│ BARKHAUSEN CRITERION FOR OSCILLATION │
├─────────────────────────────────────────────────────────────────┤
│ │
│ An oscillator produces sustained oscillations when: │
│ 1. |Aβ| ≥ 1 (Loop gain magnitude ≥ 1) │
│ 2. ∠Aβ = 0° or 360° (Loop gain phase = 0° or 2nπ) │
│ │
│ At startup: |Aβ| > 1 (amplitude grows until nonlinearity) │
│ At steady: |Aβ| = 1 (sustained oscillation) │
│ │
│ GENERALIZED FORM: │
│ Oscillation freq = freq where ∠Aβ = 0° │
│ Required gain: A ≥ 1/β at oscillation frequency │
└─────────────────────────────────────────────────────────────────┘| Type | Frequency Formula | Gain Required | Features |
|---|---|---|---|
| Wien Bridge | f = 1/(2π RC) | A ≥ 3 | Low distortion, sine wave; R₁/R₂ = 2 for gain=3 |
| Phase Shift | f = 1/(2π RC√6) | A ≥ 29 | 3 RC sections (60° each); high distortion |
| Twin-T | f = 1/(2π RC) | A ≥ 1 (at notch) | Notch filter feedback; good stability |
| Quadrature | f = 1/(2π RC) | A = √2 | Generates sine + cosine simultaneously |
| Type | Frequency | Features |
|---|---|---|
| Colpitts | f = 1/(2π√(L C₁C₂/(C₁+C₂))) | Capacitive divider feedback; very popular RF oscillator |
| Hartley | f = 1/(2π√(L_eq C)) where L_eq = L₁+L₂+2M | Tapped inductor feedback |
| Clapp | f ≈ 1/(2π√(L C₃)) where C₃ << C₁,C₂ | Series cap C₃ improves freq stability |
| Armstrong | f = 1/(2π√(LC)) | Transformer coupled; simplest LC oscillator |
| Parameter | Value / Description |
|---|---|
| Resonant Frequency | f = 1/(2π√(L_s C_s)) — series resonance |
| Parallel Resonance | f_p = f_s √(1 + C_s/C_0) — slightly higher than f_s |
| Q Factor | 10⁴ to 10⁶ (extremely high!) |
| Temperature Stability | ±0.001% or better (TCXO, OCXO) |
| Typical Frequencies | 32.768 kHz (watch) to 100s of MHz |
| Equivalent Circuit | Series L_s, C_s, R_s in parallel with C_0 (package cap) |
Colpitts oscillator is the most commonly used RF oscillator because capacitive taps are easier to fabricate than inductor taps.| Class | Conduction Angle | Max Efficiency | Distortion | Bias Point |
|---|---|---|---|---|
| A | 360° (full cycle) | 25% (series-fed), 50% (transformer) | Lowest | Active region center |
| B | 180° (half cycle) | 78.5% (transformer-coupled push-pull) | High (crossover) | At cut-off |
| AB | 180°–360° | 50–78.5% | Moderate (reduces crossover) | Slightly above cut-off |
| C | <180° | ~90% (tuned) | Very high | Below cut-off (needs tank circuit) |
| D (Switching) | Pulse / PWM | >90% | Low (filtered output) | Switching mode |
┌─────────────────────────────────────────────────────────────────┐
│ POWER AMPLIFIER — KEY CALCULATIONS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ CLASS A (Transformer Coupled): │
│ η_max = P_ac / P_dc = 50% │
│ P_ac(max) = V_CC² / (2R_L') where R_L' = (N₁/N₂)² × R_L │
│ P_dc = V_CC × I_CQ │
│ P_dissipated(max) = V_CC² / R_L' (when no signal!) │
│ ⚠️ Class A wastes max power with no input signal! │
│ │
│ CLASS B (Push-Pull): │
│ η_max = 78.5% = π/4 │
│ P_ac(max) = V_CC² / (2R_L) │
│ P_dc = (2/π) V_CC I_peak │
│ P_dissipated(each) = V_CC² / (π²R_L) │
│ P_dissipated(total max) = 2 × V_CC² / (π²R_L) │
│ │
│ CROSSOVER DISTORTION: │
│ Occurs in Class B when V_BE < 0.7V for both transistors │
│ Solution: Class AB bias (small idle current through diodes) │
│ │
│ CLASS C (Tuned): │
│ η_max ≈ 100% (theoretical with ideal tank) │
│ Conduction angle θ < 180° │
│ η = (θ - sin θ) / (4(sin(θ/2) - θcos(θ/2)/2)) │
│ Used only with resonant/tuned circuits (RF applications) │
│ │
│ CLASS D (Switching): │
│ Transistors act as switches (ON/OFF, not linear) │
│ Output filtered by LC low-pass → sinusoid │
│ η > 90% (ideal 100%) — used in audio, motor drives │
│ THD depends on switching frequency and filter quality │
└─────────────────────────────────────────────────────────────────┘| Parameter | Formula |
|---|---|
| Junction Temp | T_J = T_A + P_D (θ_JC + θ_CS + θ_SA) |
| Thermal Resistance (junction-case) | θ_JC (from datasheet, °C/W) |
| Thermal Resistance (case-sink) | θ_CS (with thermal compound, 0.1–1 °C/W) |
| Thermal Resistance (sink-ambient) | θ_SA (depends on heatsink size/airflow) |
| Max Power | P_D(max) = (T_J(max) - T_A) / (θ_JC + θ_CS + θ_SA) |
| Derating | Above 25°C: P_D reduces linearly to zero at T_J(max) |
I_peak = (2/π) × V_CC/R_L, not at maximum output. Always use a heat sink rated for worst-case power dissipation.┌─────────────────────────────────────────────────────────────────┐
│ FEEDBACK AMPLIFIER THEORY │
├─────────────────────────────────────────────────────────────────┤
│ │
│ GENERAL FEEDBACK SYSTEM: │
│ A_f = A / (1 + Aβ) [Negative feedback] │
│ A_f = A / (1 - Aβ) [Positive feedback] │
│ │
│ Where: │
│ A = Open-loop gain (forward gain) │
│ β = Feedback factor (β = V_f / V_o typically ≤ 1) │
│ D = 1 + Aβ = Desensitivity factor │
│ │
│ EFFECTS OF NEGATIVE FEEDBACK: │
│ • Gain decreases by factor D (= 1 + Aβ) │
│ • Gain stability improves by factor D │
│ • Bandwidth increases by factor D │
│ • Input impedance changes: │
│ - Series mixing: R_in increases by D │
│ - Shunt mixing: R_in decreases by D │
│ • Output impedance changes: │
│ - Voltage sampling: R_out decreases by D │
│ - Current sampling: R_out increases by D │
│ • Noise/distortion reduced by factor D │
│ • GBW product remains constant: A × BW = A_f × BW_f │
│ │
│ EFFECTS OF POSITIVE FEEDBACK: │
│ • Gain increases (A_f = A/(1-Aβ)) │
│ • Bandwidth decreases │
│ • If Aβ = 1 → OSCILLATION (Barkhausen criterion) │
│ • Reduces stability │
└─────────────────────────────────────────────────────────────────┘| Topology | Mixing | Sampling | R_in Effect | R_out Effect | A_f Formula |
|---|---|---|---|---|---|
| Voltage-Voltage | Series | Shunt (Voltage) | ×D (↑) | ÷D (↓) | A_vf = A_v/(1+A_v β) |
| Current-Voltage | Shunt | Shunt (Voltage) | ÷D (↓) | ÷D (↓) | A_rf = A_r/(1+A_r β) |
| Voltage-Current | Series | Series (Current) | ×D (↑) | ×D (↑) | A_gf = A_g/(1+A_g β) |
| Current-Current | Shunt | Series (Current) | ÷D (↓) | ×D (↑) | A_if = A_i/(1+A_i β) |
| Check | Method | Result |
|---|---|---|
| Output sampling type | Short output → β → 0? | Yes → Voltage sampling; No → Current sampling |
| Input mixing type | Short input → β → 0? | Yes → Shunt mixing; No → Series mixing |
| Voltage-Voltage hint | Feedback from V_out to input (series) | Op-amp non-inverting amp |
| Current-Voltage hint | Feedback from I_out to input (shunt) | Transresistance amp |
| Voltage-Current hint | Feedback from V_out via series at input | Common-base feedback |
| Current-Current hint | Feedback from I_out via shunt at input | Current mirror feedback |
D = 1 + Aβdetermines all improvements. A 741 with A_OL = 10⁵ and β = 0.01 gives A_f = 100, D = 1001, so bandwidth increases by 1001×. To identify topology: short the output — if feedback vanishes, it's voltage sampling.| Type | Passes | Blocks | Ideal Shape |
|---|---|---|---|
| Low-Pass (LPF) | Low freq | High freq | Flat → sharp cutoff |
| High-Pass (HPF) | High freq | Low freq | Sharp cutoff → flat |
| Band-Pass (BPF) | Band (f_L to f_H) | Outside band | Band of frequencies |
| Band-Stop (BSF) | Outside notch | Notch freq | Notch in frequency response |
| All-Pass | All freq equally | None (phase only) | Flat magnitude, phase shift |
| Approx. | Passband | Stopband | Phase | Order for -60dB/dec |
|---|---|---|---|---|
| Butterworth | Maximally flat | Monotonic | Moderate | 6th |
| Chebyshev (Type I) | Ripple | Monotonic | Poor | Lower than Butterworth |
| Chebyshev (Type II) | Flat | Ripple | Better | — |
| Elliptic (Cauer) | Ripple | Ripple | Non-linear | Lowest order (steepest) |
| Bessel | Gentle roll-off | Gradual | Best (linear) | Higher than Butterworth |
┌─────────────────────────────────────────────────────────────────┐
│ FILTER DESIGN — KEY FORMULAS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ RC FIRST-ORDER LOW-PASS FILTER: │
│ f_c = 1 / (2π RC) │
│ |H(jω)| = 1 / √(1 + (f/f_c)²) │
│ Phase = -arctan(f/f_c) │
│ Roll-off: -20 dB/decade │
│ │
│ RC FIRST-ORDER HIGH-PASS FILTER: │
│ f_c = 1 / (2π RC) │
│ |H(jω)| = (f/f_c) / √(1 + (f/f_c)²) │
│ Roll-off: +20 dB/decade (slope of passband) │
│ │
│ n-th ORDER ROLL-OFF: │
│ Roll-off rate = -20n dB/decade = -6n dB/octave │
│ n=1: -20 dB/dec | n=2: -40 dB/dec | n=3: -60 dB/dec │
│ │
│ BUTTERWORTH POLYNOMIAL (denominator coefficients): │
│ n=1: s + 1 │
│ n=2: s² + 1.414s + 1 │
│ n=3: (s + 1)(s² + s + 1) │
│ n=4: (s² + 0.765s + 1)(s² + 1.848s + 1) │
│ │
│ ACTIVE FILTER (Sallen-Key 2nd-order LPF): │
│ f_c = 1 / (2π√(R₁R₂C₁C₂)) │
│ DC Gain = 1 + R_f/R_G (for non-inverting) │
│ Q = √(R₁R₂C₁C₂) / (R₂C₁ + R₂C₂) │
│ For Butterworth: Q = 1/√2 ≈ 0.707 │
│ │
│ BANDPASS FILTER: │
│ BW = f_H - f_L (bandwidth) │
│ Q = f_center / BW (quality factor) │
│ f_center = √(f_L × f_H) for wideband │
│ Narrowband: f_center = (f_L + f_H) / 2 │
│ │
│ PASSIVE vs ACTIVE: │
│ Passive: R, L, C only — no gain, load affects response │
│ Active: Op-amp + RC — gain available, no loading, tunable │
└─────────────────────────────────────────────────────────────────┘| Step | Action |
|---|---|
| 1. Specs | Determine f_c, passband ripple, stopband attenuation, filter type |
| 2. Approximation | Choose Butterworth/Chebyshev/Elliptic based on requirements |
| 3. Order | Calculate minimum order: n ≥ log₁₀(√(10^(A_s/10)-1)) / log₁₀(f_s/f_c) |
| 4. Normalize | Use normalized Butterworth/Chebyshev tables for prototype values |
| 5. Frequency Scale | Denormalize: R_new = R_old × f_c(desired)/f_c(normalized) |
| 6. Impedance Scale | Scale all R and C to practical values: multiply R, divide C (or vice versa) |
| 7. Topology | Choose Sallen-Key (non-inverting) or MFB (inverting, better Q control) |
| 8. Simulate | Verify with SPICE: check magnitude, phase, and transient response |
Chebyshev (accept some passband ripple). TheSallen-Key topology is the simplest 2nd-order active filter but Q is sensitive to component tolerances. For high-Q designs, the Multiple Feedback (MFB) topology offers better Q control.