Analogue Electronics Formulae

For an analog electronics exam, you'll need formulas covering key topics like circuit analysis, semiconductor devices, amplifiers, filters, and feedback systems. Below is a categorized list of essential formulas, their applications, and derivations where useful.

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1. Basic Circuit Analysis

Ohm’s Law:

V=IRV = IRV=IR

• Application: Used for solving simple resistor networks.

Kirchhoff’s Laws:

• KVL (Kirchhoff’s Voltage Law): The sum of voltages in a closed loop is zero.

∑V=0\sum V = 0∑V=0

• KCL (Kirchhoff’s Current Law): The sum of currents entering a node equals the sum of currents leaving.

∑I=0\sum I = 0∑I=0

• Application: Used for analyzing circuits with multiple loops and nodes.

Thevenin’s & Norton’s Theorems:

• Thevenin Equivalent Resistance:

Rth=VocIscR_{\text{th}} = \frac{V_{\text{oc}}}{I_{\text{sc}}}Rth​=Isc​Voc​​

• Norton Equivalent Current:

IN=VocRthI_{\text{N}} = \frac{V_{\text{oc}}}{R_{\text{th}}}IN​=Rth​Voc​​

• Application: Simplifies complex circuits into a single voltage or current source with equivalent resistance.

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2. Diodes and Rectifiers

Diode Current Equation (Shockley Equation):

ID=IS(eVDnVT−1)I_D = I_S \left( e^{\frac{V_D}{nV_T}} -1 \right)ID​=IS​(enVT​VD​​−1)

Where:

• ISI_SIS​ = Reverse saturation current

• VTV_TVT​ = Thermal voltage (≈25mV\approx 25mV≈25mV at 300K)

• nnn = Ideality factor (1 for Si, 2 for Ge)

• Application: Determines the current flowing through a diode for a given voltage.

Rectifier Circuits:

• Half-wave rectifier output voltage (ideal):

Vdc=VmπV_{\text{dc}} = \frac{V_m}{\pi}Vdc​=πVm​​

• Full-wave rectifier output voltage (ideal):

Vdc=2VmπV_{\text{dc}} = \frac{2V_m}{\pi}Vdc​=π2Vm​​

• Ripple voltage for capacitor filter:

Vripple=IfCV_{\text{ripple}} = \frac{I}{fC}Vripple​=fCI​

• Application: Used for AC-to-DC conversion.

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3. Bipolar Junction Transistor (BJT)

DC Analysis of BJT:

• Base current:

IB=VB−VBERBI_B = \frac{V_B - V_{BE}}{R_B}IB​=RB​VB​−VBE​​

• Collector current (active region):

IC=βIBI_C = \beta I_BIC​=βIB​

• Emitter current:

IE=IC+IB=ICαI_E = I_C + I_B = \frac{I_C}{\alpha}IE​=IC​+IB​=αIC​​

Where:

α=ββ+1\alpha = \frac{\beta}{\beta +1}α=β+1β​

• Application: Determines operating point in amplifier design.

AC Analysis (Small-Signal Model):

• Transconductance:

gm=ICVTg_m = \frac{I_C}{V_T}gm​=VT​IC​​

• Input resistance (common emitter amplifier):

rπ=βgmr_\pi = \frac{\beta}{g_m}rπ​=gm​β​

• Voltage gain (CE amplifier):

Av=−βRCrπA_v = -\beta \frac{R_C}{r_\pi}Av​=−βrπ​RC​​

• Application: Used for designing BJT amplifiers.

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4. Field Effect Transistor (FET & MOSFET)

DC Analysis of MOSFET (Saturation Region):

ID=12μnCoxWL(VGS−Vth)2I_D = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2ID​=21​μn​Cox​LW​(VGS​−Vth​)2

• Application: Determines MOSFET operating point.

Small-Signal Model of MOSFET:

• Transconductance:

gm=2IDVGS−Vthg_m = \frac{2I_D}{V_{GS} - V_{th}}gm​=VGS​−Vth​2ID​​

• Voltage gain:

Av=−gmRDA_v = -g_m R_DAv​=−gm​RD​

• Application: Used in MOSFET amplifier design.

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5. Amplifiers

Common-Emitter Voltage Gain:

Av=−βRCrπA_v = -\beta \frac{R_C}{r_\pi}Av​=−βrπ​RC​​

• Derivation: Using small-signal model, Vout=−ICRCV_{\text{out}} = -I_C R_CVout​=−IC​RC​, and IC=βIBI_C = \beta I_BIC​=βIB​, substituting IB=Vin/rπI_B = V_{\text{in}}/r_\piIB​=Vin​/rπ​ gives:

Av=−βRCrπA_v = -\beta \frac{R_C}{r_\pi}Av​=−βrπ​RC​​

Gain-Bandwidth Product (for Op-Amps):

Gain×Bandwidth=constant\text{Gain} \times \text{Bandwidth} = \text{constant}Gain×Bandwidth=constant

• Application: Used to determine op-amp stability and bandwidth.

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6. Feedback and Stability

Closed-Loop Gain (Negative Feedback):

Af=A1+AβA_f = \frac{A}{1 + A \beta}Af​=1+AβA​

Where:

• AAA = Open-loop gain

• β\betaβ = Feedback factor

• Application: Controls amplifier gain and bandwidth.

Bode Stability Criterion:

• Phase margin:

PM=180∘+∠Loop Gain\text{PM} = 180^\circ + \angle \text{Loop Gain}PM=180∘+∠Loop Gain

• Application: Ensures amplifier stability.

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7. Filters and Frequency Response

First-Order Low-Pass Filter:

H(jω)=11+jωωcH(j\omega) = \frac{1}{1 + j\frac{\omega}{\omega_c}}H(jω)=1+jωc​ω​1​

• Cutoff frequency:

ωc=1RC\omega_c = \frac{1}{RC}ωc​=RC1​

First-Order High-Pass Filter:

H(jω)=jωωc1+jωωcH(j\omega) = \frac{j\frac{\omega}{\omega_c}}{1 + j\frac{\omega}{\omega_c}}H(jω)=1+jωc​ω​jωc​ω​​

• Application: Designs active/passive filters.

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8. Oscillators

Barkhausen Criterion:

Aβ=1,∠Aβ=0∘ or 360∘A \beta = 1, \quad \angle A\beta = 0^\circ \text{ or } 360^\circAβ=1,∠Aβ=0∘ or 360∘

• Application: Ensures sustained oscillations in circuits.

Wien Bridge Oscillator Frequency:

f=12πRCf = \frac{1}{2\pi RC}f=2πRC1​

• Application: Used for sine wave generation.

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9. Power Electronics

DC-DC Converter Efficiency:

η=PoutPin×100%\eta = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100\%η=Pin​Pout​​×100%

• Application: Analyzes power conversion efficiency.

Buck Converter Output Voltage:

Vout=DVinV_{\text{out}} = D V_{\text{in}}Vout​=DVin​

• Application: Designs step-down regulators.

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Final Tips for Application:

1. Understand Derivations – Many formulas come from circuit laws and small-signal models.

2. Use SPICE/LTspice Simulations – Verify results for circuit performance.

3. Memorize Key Expressions – Focus on amplifier gains, filter cutoffs, and transistor models.

4. Practice Complex Problems – Solve past exam questions to solidify concepts.