academic-circuit-analysis-electronics

# Electrical Circuit Analysis and Microelectronics (Sedra & Alexander-Sadiku) This skill establishes the analytical engineering of linear electrical networks in direct current (DC) and alternating current (AC), transient regimes, frequency response, and small- and large-signal modeling in semiconductors and operational amplifiers. --- ## ⚡ 1. Network Theorems and Steady-State AC Analysis ### 1.1 Fundamental Network Theorems - **Kirchhoff's Laws**: $\sum_{k} I_k = 0$ (KCL at nodes) and $\sum_{k} V_k = 0$ (KVL around loops). - **Thévenin and Norton Theorems**: $$V_{th} = V_{oc}, \quad I_n = I_{sc}, \quad \mathbf{Z}_{th} = \frac{\mathbf{V}_{oc}}{\mathbf{I}_{sc}}$$ - **Maximum Power Transfer Theorem in AC**: Maximum active power is delivered to the load when the load impedance is the complex conjugate of the Thévenin equivalent impedance: $$\mathbf{Z}_L = \mathbf{Z}_{th}^* = R_{th} - j X_{th} \implies P_{max} = \frac{|V_{th}|^2}{4 R_{th}}$$ ### 1.2 Phasors and the Power Triangle For phasor voltage $\mathbf{V} = V_{rms} \angle \theta_v$ and current $\mathbf{I} = I_{rms} \angle \theta_i$: - **Complex Power**: $\mathbf{S} = \mathbf{V} \mathbf{I}^* = P + j Q = |\mathbf{S}| \angle \theta$ ($[\text{VA}]$). - **Active (Real) Power**: $P = V_{rms} I_{rms} \cos(\theta_v - \theta_i)$ ($[\text{W}]$). - **Reactive Power**: $Q = V_{rms} I_{rms} \sin(\theta_v - \theta_i)$ ($[\text{var}]$). - **Power Factor**: $PF = \cos(\theta_v - \theta_i) = \frac{P}{|\mathbf{S}|}$. --- ## ⏱️ 2. Transients in First- and Second-Order Circuits (RL, RC, and RLC) ### 2.1 First-Order Circuits (RC and RL) The complete response to a unit step with initial condition $x(0)$ and steady state $x(\infty)$: $$x(t) = x(\infty) + [x(0) - x(\infty)] e^{-t/\tau}, \quad \tau_{RC} = R C, \; \tau_{RL} = \frac{L}{R}$$ ### 2.2 Second-Order RLC Circuits Differential equation: $\frac{d^2 x}{dt^2} + 2\zeta\omega_0 \frac{dx}{dt} + \omega_0^2 x = f(t)$, with $\omega_0 = \frac{1}{\sqrt{LC}}$ and $\alpha = \zeta\omega_0 = \frac{R}{2L}$ (series RLC): 1. **Overdamped ($\zeta > 1 \iff \alpha > \omega_0$)**: $x(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}$. 2. **Critically Damped ($\zeta = 1 \iff \alpha = \omega_0$)**: $x(t) = (A_1 + A_2 t) e^{-\alpha t}$ (fastest return to equilibrium without oscillation). 3. **Underdamped ($\zeta < 1 \iff \alpha < \omega_0$)**: $x(t) = e^{-\alpha t} (A_1 \cos\omega_d t + A_2 \sin\omega_d t)$ with $\omega_d = \sqrt{\omega_0^2 - \alpha^2}$. --- ## 🔬 3. Semiconductor Device Modeling (BJT and MOSFET) ### 3.1 MOSFET Field-Effect Transistor (N-Channel) - **Triode / Linear Region ($V_{DS} < V_{GS} - V_{th}$)**: $$I_D = \mu_n C_{ox} \frac{W}{L} \left[ (V_{GS} - V_{th}) V_{DS} - \frac{1}{2} V_{DS}^2 \right]$$ - **Saturation Region ($V_{DS} \ge V_{GS} - V_{th} = V_{OV}$)**: $$I_D = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2 (1 + \lambda V_{DS})$$ - **Small-Signal Parameters (Hybrid-$\pi$ Model)**: $$g_m = \left. \frac{\partial I_D}{\partial V_{GS}} \right|_{Q} = \frac{2 I_D}{V_{OV}} = \sqrt{2 \mu_n C_{ox} \frac{W}{L} I_D}, \quad r_o = \frac{1}{\lambda I_D} \approx \frac{V_A}{I_D}$$ ### 3.2 Bipolar Junction Transistor (BJT) - **Collector Current in Forward Active**: $I_C = I_S e^{V_{BE}/V_T} (1 + V_{CE}/V_A)$, with thermal voltage $V_T = \frac{k_B T}{q} \approx 25.8\text{ mV}$ at $300\text{ K}$. - **Small Signal**: $g_m = \frac{I_C}{V_T}$, $r_\pi = \frac{\beta}{g_m} = \frac{V_T}{I_B}$, $r_e = \frac{\alpha}{g_m} \approx \frac{V_T}{I_E}$, $r_o = \frac{V_A}{I_C}$. --- ## 🎛️ 4. Operational Amplifiers and Active Filters ```mermaid flowchart LR subgraph OpAmp["Amplificador de Instrumentação (INA - 3 Op-Amps)"] IN1["V1 (+)"] --> OP1["Op-Amp 1 (Buffer/Ganho Diferencial)"] IN2["V2 (-)"] --> OP2["Op-Amp 2 (Buffer/Ganho Diferencial)"] OP1 & OP2 --> RG["Resistor de Ajuste de Ganho RG"] OP1 & OP2 --> OP3["Op-Amp 3 (Estágio Subtrator Diferencial)"] OP3 --> VOUT["Vout = (1 + 2R1/RG) * (R3/R2) * (V1 - V2)"] end ``` ### 4.1 Second-Order Sallen-Key Low-Pass Active Filter Transfer function: $$H(s) = \frac{V_{out}(s)}{V_{in}(s)} = \frac{K \omega_0^2}{s^2 + \frac{\omega_0}{Q} s + \omega_0^2}$$ where $\omega_0 = \frac{1}{\sqrt{R_1 R_2 C_1 C_2}}$ and the quality factor $Q$ determines the response: - **Butterworth ($Q = \frac{1}{\sqrt{2}} \approx 0.707$)**: Maximally flat response in the passband with no ripple. - **Chebyshev ($Q > 0.707$)**: Sharper transition in the stopband at the cost of passband ripple.