Network Theory Cheatsheet

Cheatsheet Content

### Network Fundamentals - **Elements:** - **Resistor (R):** $V=IR$. Power $P=I^2R = V^2/R$. - **Inductor (L):** $V=L\frac{dI}{dt}$, $I=\frac{1}{L}\int V dt$. Stores energy: $E = \frac{1}{2}LI^2$. - **Capacitor (C):** $I=C\frac{dV}{dt}$, $V=\frac{1}{C}\int I dt$. Stores energy: $E = \frac{1}{2}CV^2$. - **Sources:** - **Independent:** Voltage ($V_S$) or Current ($I_S$) source whose value is independent of other circuit variables. - **Dependent (Controlled):** Voltage or current source whose value depends on a voltage or current elsewhere in the circuit. (e.g., VCVS, VCCS, CCVS, CCCS). - **Laws:** - **Ohm's Law:** $V=IR$. - **Kirchhoff's Current Law (KCL):** Sum of currents entering a node is zero. $\sum I_{in} = \sum I_{out}$. - **Kirchhoff's Voltage Law (KVL):** Sum of voltages around any closed loop is zero. $\sum V_{rise} = \sum V_{drop}$. #### Analysis Methods - **Nodal Analysis:** 1. Choose a reference node (ground). 2. Assign unknown voltages ($V_1, V_2, ...$) to the remaining $N-1$ non-reference nodes. 3. Apply KCL at each non-reference node, expressing currents in terms of node voltages using Ohm's Law. 4. Solve the resulting system of linear equations for the node voltages. - **Mesh Analysis:** 1. Assign unknown mesh currents ($I_1, I_2, ...$) to each of the $M$ independent meshes. 2. Apply KVL around each mesh, expressing voltages in terms of mesh currents using Ohm's Law. 3. Solve the resulting system of linear equations for the mesh currents. - **Supermesh:** Used when a current source is common to two meshes. - **Superposition Theorem:** In a linear circuit with multiple independent sources, the total response (voltage or current) in any element is the algebraic sum of the responses caused by each independent source acting alone (with all other independent sources turned off – voltage sources shorted, current sources opened). - **Thevenin's Theorem:** Any linear two-terminal circuit can be replaced by an equivalent circuit consisting of a single voltage source ($V_{Th}$) in series with a single resistor ($R_{Th}$). - $V_{Th}$: Open-circuit voltage across the terminals. - $R_{Th}$: Equivalent resistance looking into the terminals with all independent sources set to zero (voltage sources shorted, current sources opened). - **Norton's Theorem:** Any linear two-terminal circuit can be replaced by an equivalent circuit consisting of a single current source ($I_N$) in parallel with a single resistor ($R_N$). - $I_N$: Short-circuit current flowing between the terminals. - $R_N = R_{Th}$. - **Maximum Power Transfer Theorem:** For a given source with internal resistance $R_{Th}$ (or output impedance $Z_{Th}$), maximum average power is delivered to a load when the load resistance $R_L$ (or impedance $Z_L$) is equal to the source's internal resistance (or complex conjugate of the source's output impedance). - DC: $R_L = R_{Th}$ - AC: $Z_L = Z_{Th}^* = R_{Th} - jX_{Th}$ - Max Power: $P_{max} = \frac{V_{Th}^2}{4R_{Th}}$ (DC) - **Reciprocity Theorem:** In any linear, passive, bilateral network, if an ideal voltage source $V$ in branch 'A' produces a current $I$ in branch 'B', then placing the same voltage source $V$ in branch 'B' will produce the same current $I$ in branch 'A'. Similarly for current sources and voltages. - **Compensation Theorem:** If the resistance of a branch $k$ carrying a current $I_k$ in a network is changed by $\Delta R_k$, the change in current in any other branch $j$ is the same as if an ideal voltage source of value $-I_k \Delta R_k$ were inserted in branch $k$ with polarity opposing $I_k$. ### AC Circuit Analysis - **Phasors:** Representation of sinusoidal signals as complex numbers. - $v(t) = V_m \cos(\omega t + \phi) \iff \mathbf{V} = V_m \angle \phi = V_m e^{j\phi}$ - $i(t) = I_m \cos(\omega t + \theta) \iff \mathbf{I} = I_m \angle \theta = I_m e^{j\theta}$ - **Impedance (Z):** Complex opposition to current flow. - Resistor: $Z_R = R$ - Inductor: $Z_L = j\omega L$ - Capacitor: $Z_C = \frac{1}{j\omega C} = -\frac{j}{\omega C}$ - **Admittance (Y):** Reciprocal of impedance $Y = 1/Z$. - Resistor: $Y_R = 1/R = G$ (conductance) - Inductor: $Y_L = 1/(j\omega L) = -j/\omega L$ - Capacitor: $Y_C = j\omega C$ - **Power in AC Circuits:** - **Instantaneous Power:** $p(t) = v(t)i(t)$. - **Average Power (Real Power, P):** $P = V_{rms}I_{rms}\cos\theta = \text{Re}\{\mathbf{V}\mathbf{I}^*\} = I_{rms}^2 R = V_{rms}^2/R$. Measured in Watts (W). - **Reactive Power (Q):** $Q = V_{rms}I_{rms}\sin\theta = \text{Im}\{\mathbf{V}\mathbf{I}^*\} = I_{rms}^2 X = V_{rms}^2/X$. Measured in Volt-Amperes Reactive (VAR). - **Apparent Power (S):** $S = V_{rms}I_{rms} = |\mathbf{V}\mathbf{I}^*| = \sqrt{P^2 + Q^2}$. Measured in Volt-Amperes (VA). - **Complex Power (S):** $\mathbf{S} = \mathbf{V}\mathbf{I}^* = P + jQ$. - **Power Factor (PF):** $\cos\theta = P/S$. Leading PF (capacitive), Lagging PF (inductive). ### Resonant Circuits - **Resonance:** Condition where the inductive and capacitive reactances cancel each other ($X_L = X_C$), leading to purely resistive impedance. - **Series RLC Circuit:** - Resonant Frequency: $f_0 = \frac{1}{2\pi\sqrt{LC}}$ or $\omega_0 = \frac{1}{\sqrt{LC}}$ rad/s. - At resonance: $Z = R$ (minimum impedance), current is maximum. - Quality Factor ($Q$): $Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R} = \frac{1}{R}\sqrt{\frac{L}{C}}$. High Q means sharper resonance. - Bandwidth (BW): $BW = f_2 - f_1 = \frac{f_0}{Q} = \frac{R}{2\pi L}$ Hz. - Half-power frequencies ($f_1, f_2$): Frequencies where power is half of maximum, or current is $I_{max}/\sqrt{2}$. $f_1 = f_0 - BW/2$, $f_2 = f_0 + BW/2$. - **Parallel RLC Circuit:** - Resonant Frequency: - For ideal parallel RLC: $f_0 = \frac{1}{2\pi\sqrt{LC}}$. - For non-ideal (inductor with series resistance $R_L$): $f_0 = \frac{1}{2\pi}\sqrt{\frac{1}{LC} - \frac{R_L^2}{L^2}}$. - If $R_L \ll \omega L$, then $f_0 \approx \frac{1}{2\pi\sqrt{LC}}$. - At resonance: $Y = G = 1/R$ (minimum admittance), impedance is maximum. - Quality Factor ($Q$): $Q = \frac{R}{\omega_0 L} = \omega_0 C R = R\sqrt{\frac{C}{L}}$ (for ideal parallel). - Bandwidth (BW): $BW = \frac{f_0}{Q}$. ### Coupled Circuits - **Self-Inductance (L):** $L = \frac{N\phi}{I}$. - **Mutual Inductance (M):** Occurs when the magnetic field of one coil links with another coil. - $V_2 = M \frac{dI_1}{dt}$ (voltage induced in coil 2 due to current change in coil 1). - $M = k\sqrt{L_1 L_2}$. - **Coefficient of Coupling (k):** $0 \le k \le 1$. $k=1$ for perfect coupling. - **Dot Convention:** - If current enters a dotted terminal of one coil, the induced voltage in the second coil is positive at its dotted terminal. - If currents enter/leave both dotted terminals, the mutual inductance terms add. If one enters dotted and other leaves dotted, terms subtract. - **Equivalent Inductance of Series-Connected Coils:** - **Aiding (dots aligned):** $L_{eq} = L_1 + L_2 + 2M$. - **Opposing (dots unaligned):** $L_{eq} = L_1 + L_2 - 2M$. - **Transformer:** Device that transfers electrical energy from one circuit to another through mutual inductance. - **Ideal Transformer:** No losses, perfect coupling ($k=1$). - Voltage ratio: $\frac{V_1}{V_2} = \frac{N_1}{N_2} = a$ (turns ratio). - Current ratio: $\frac{I_1}{I_2} = \frac{N_2}{N_1} = \frac{1}{a}$. - Impedance transformation: $Z_{in} = a^2 Z_L$. ### Laplace Transform Fundamentals - **Definition:** $\mathcal{L}\{f(t)\} = F(s) = \int_{0^-}^\infty f(t)e^{-st} dt$. Used for solving linear differential equations and transient analysis. - **Complex Frequency (s):** $s = \sigma + j\omega$. Represents both exponential decay/growth ($\sigma$) and oscillation ($\omega$). - **Key Properties:** - **Linearity:** $\mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s)$. - **Time Differentiation:** $\mathcal{L}\{\frac{df(t)}{dt}\} = sF(s) - f(0^-)$. $\mathcal{L}\{\frac{d^2f(t)}{dt^2}\} = s^2F(s) - sf(0^-) - f'(0^-)$. - **Time Integration:** $\mathcal{L}\{\int_0^t f(\tau)d\tau\} = \frac{F(s)}{s}$. - **Time Shift:** $\mathcal{L}\{f(t-a)u(t-a)\} = e^{-as}F(s)$. - **Frequency Shift:** $\mathcal{L}\{e^{-at}f(t)\} = F(s+a)$. - **Initial Value Theorem:** $\lim_{t\to 0^+} f(t) = \lim_{s\to\infty} sF(s)$ (if limit exists). - **Final Value Theorem:** $\lim_{t\to\infty} f(t) = \lim_{s\to 0} sF(s)$ (if poles of $sF(s)$ are in the left-half plane or at the origin). - **Common Transform Pairs:** - Unit Impulse $\delta(t) \iff 1$ - Unit Step $u(t) \iff 1/s$ - Ramp $t u(t) \iff 1/s^2$ - Exponential $e^{-at}u(t) \iff 1/(s+a)$ - Sine $\sin(\omega t)u(t) \iff \omega/(s^2+\omega^2)$ - Cosine $\cos(\omega t)u(t) \iff s/(s^2+\omega^2)$ - **Circuit Element Equivalents in s-domain:** - **Resistor (R):** $R$ - **Inductor (L):** $sL$ (impedance). - For initial condition $i_L(0^-)$: $V_L(s) = sL I_L(s) - L i_L(0^-)$ (series voltage source). $I_L(s) = \frac{V_L(s)}{sL} + \frac{i_L(0^-)}{s}$ (parallel current source). - **Capacitor (C):** $1/sC$ (impedance). - For initial condition $v_C(0^-)$: $V_C(s) = \frac{1}{sC} I_C(s) + \frac{v_C(0^-)}{s}$ (series voltage source). $I_C(s) = sC V_C(s) - C v_C(0^-)$ (parallel current source). #### Transient & Steady-State Response - **Response Types:** - **Natural Response (Transient):** Due to energy storage elements (L, C) and initial conditions. Decays over time. - **Forced Response (Steady-State):** Due to external sources. Persists as long as sources are active. - **Total Response:** Natural Response + Forced Response. - **Poles and Zeros:** - **Poles:** Values of $s$ where the transfer function $H(s)$ goes to infinity. Determine the natural response. - **Zeros:** Values of $s$ where $H(s)$ goes to zero. - **System Stability:** For a causal system, all poles must be in the left-half of the s-plane ($\sigma 1$):** $s_{1,2} = -\zeta\omega_n \pm \omega_n\sqrt{\zeta^2-1}$ (two distinct real roots). Slow, non-oscillatory decay. ### Graph Theory for Circuit Analysis - **Graph:** A topological representation of a circuit, consisting of nodes (junctions) and branches (elements). - **Connected Graph:** A graph where there's a path between any two nodes. - **Tree:** A subgraph of a connected graph that connects all nodes without forming any closed loops. A tree with $N$ nodes has $N-1$ branches. - **Co-Tree (Links/Chords):** The set of branches in the graph that are not in the chosen tree. A graph with $B$ branches and $N$ nodes has $B-(N-1)$ links. - **Fundamental Cut-Set:** A cut-set that contains exactly one tree branch and possibly several links. A cut-set is a minimal set of branches whose removal divides the graph into two parts. - **Fundamental Loop (Tie-Set):** A loop formed by adding exactly one link to a tree. Each link forms a unique fundamental loop. - **Incidence Matrix (A):** An $N \times B$ matrix that describes the connection between nodes and branches. - $A_{ij} = 1$ if branch $j$ leaves node $i$. - $A_{ij} = -1$ if branch $j$ enters node $i$. - $A_{ij} = 0$ if branch $j$ is not incident to node $i$. - Reduced incidence matrix $\mathbf{A}_r$ (for $N-1$ non-reference nodes). - KCL in terms of branch currents: $\mathbf{A}_r \mathbf{I}_b = 0$. - **Fundamental Cut-Set Matrix (Q_f):** A $(N-1) \times B$ matrix. Each row corresponds to a fundamental cut-set, and entries indicate orientation. - KCL in terms of branch currents: $\mathbf{Q}_f \mathbf{I}_b = 0$. - Branch voltages related to tree branch voltages: $\mathbf{V}_b = \mathbf{Q}_f^T \mathbf{V}_t$. - **Fundamental Loop (Tie-Set) Matrix (B_f):** A $(B-N+1) \times B$ matrix. Each row corresponds to a fundamental loop. - KVL in terms of branch voltages: $\mathbf{B}_f \mathbf{V}_b = 0$. - Branch currents related to link currents: $\mathbf{I}_b = \mathbf{B}_f^T \mathbf{I}_l$. - **Relationship between Matrices:** $\mathbf{B}_f \mathbf{Q}_f^T = 0$ and $\mathbf{Q}_f \mathbf{B}_f^T = 0$. ### Two-Port Networks - **Definition:** A circuit or device with two pairs of terminals (ports), one for input and one for output. - **Port Variables:** $V_1, I_1$ (input port), $V_2, I_2$ (output port). Currents are typically defined as entering the network. - **Parameters (Relating $V_1, I_1, V_2, I_2$):** - **Z-parameters (Open-circuit impedance parameters):** $$V_1 = Z_{11}I_1 + Z_{12}I_2$$ $$V_2 = Z_{21}I_1 + Z_{22}I_2$$ - $Z_{11} = V_1/I_1 |_{I_2=0}$ (input impedance with output open) - $Z_{12} = V_1/I_2 |_{I_1=0}$ (reverse transfer impedance with input open) - $Z_{21} = V_2/I_1 |_{I_2=0}$ (forward transfer impedance with output open) - $Z_{22} = V_2/I_2 |_{I_1=0}$ (output impedance with input open) - For reciprocal networks: $Z_{12} = Z_{21}$. - **Y-parameters (Short-circuit admittance parameters):** $$I_1 = Y_{11}V_1 + Y_{12}V_2$$ $$I_2 = Y_{21}V_1 + Y_{22}V_2$$ - $Y_{11} = I_1/V_1 |_{V_2=0}$ (input admittance with output shorted) - $Y_{12} = I_1/V_2 |_{V_1=0}$ (reverse transfer admittance with input shorted) - $Y_{21} = I_2/V_1 |_{V_2=0}$ (forward transfer admittance with output shorted) - $Y_{22} = I_2/V_2 |_{V_1=0}$ (output admittance with input shorted) - For reciprocal networks: $Y_{12} = Y_{21}$. - **h-parameters (Hybrid parameters):** $$V_1 = h_{11}I_1 + h_{12}V_2$$ $$I_2 = h_{21}I_1 + h_{22}V_2$$ - $h_{11} = V_1/I_1 |_{V_2=0}$ (input impedance with output shorted) - $h_{12} = V_1/V_2 |_{I_1=0}$ (reverse voltage gain with input open) - $h_{21} = I_2/I_1 |_{V_2=0}$ (forward current gain with output shorted) - $h_{22} = I_2/V_2 |_{I_1=0}$ (output admittance with input open) - Used extensively for transistor modeling. - **ABCD-parameters (Transmission parameters):** $$V_1 = AV_2 - BI_2$$ $$I_1 = CV_2 - DI_2$$ - Relate input variables to output variables. Useful for cascaded networks. - For reciprocal networks: $AD - BC = 1$. - **Interconnections of Two-Port Networks:** - **Series Connection:** Overall Z-parameters are the sum of individual Z-parameters. - **Parallel Connection:** Overall Y-parameters are the sum of individual Y-parameters. - **Cascade (Tandem) Connection:** Overall ABCD-parameters are the matrix product of individual ABCD-parameter matrices. - **Driving Point Impedance/Admittance:** The impedance/admittance seen looking into one port, typically with the other port terminated in a specific load (e.g., open, short, or resistive load). - **Transfer Impedance/Admittance/Function:** Ratio of a voltage/current at one port to a current/voltage at another port. Represents how a signal is transferred through the network. ### Filter Circuits - **Purpose:** Selectively pass or reject certain frequency components of a signal. - **Classification by Components:** - **Passive Filters:** Use only R, L, C components. No external power, no gain, can be large/heavy (L). - **Active Filters:** Use R, C, and active devices (e.g., Op-Amps). Can provide gain, avoid inductors (smaller, lighter), easier to tune, require power. - **Classification by Frequency Response:** - **Low-Pass Filter (LPF):** Passes frequencies below a cutoff frequency ($\omega_c$) and attenuates frequencies above it. - **High-Pass Filter (HPF):** Passes frequencies above $\omega_c$ and attenuates frequencies below it. - **Band-Pass Filter (BPF):** Passes frequencies within a specific band ($\omega_1$ to $\omega_2$) and attenuates frequencies outside this band. - **Band-Reject Filter (BRF) / Notch Filter:** Attenuates frequencies within a specific band and passes frequencies outside it. - **All-Pass Filter:** Passes all frequencies with equal gain but introduces a frequency-dependent phase shift. - **Key Characteristics:** - **Cutoff Frequency ($\omega_c$ or $f_c$):** Frequency at which the output power is half of the maximum (or voltage/current is $1/\sqrt{2}$ times max), typically -3 dB point. - **Passband:** Range of frequencies passed with minimal attenuation. - **Stopband:** Range of frequencies attenuated significantly. - **Transition Band:** Region between passband and stopband. - **Roll-off Rate:** Rate at which attenuation increases in the stopband (e.g., -20 dB/decade per order). - **Filter Approximations (Magnitude Response):** - **Butterworth Filter:** - Maximally flat response in the passband (no ripples). - Smooth roll-off. - Pole locations lie on a circle in the s-plane. - Magnitude squared: $|H(j\omega)|^2 = \frac{1}{1+(\omega/\omega_c)^{2n}}$, where $n$ is the order. - **Chebyshev Filter:** - Steeper roll-off than Butterworth for the same order. - Has ripples in either the passband (Type I) or stopband (Type II). - Poles lie on an ellipse in the s-plane. - **Bessel Filter:** - Maximally flat group delay (linear phase response). - Best for preserving pulse shapes (minimal distortion). - Slower roll-off than Butterworth or Chebyshev. - **Active Filter Design (using Op-Amps):** - **Sallen-Key Topology:** Common for 2nd-order LPF/HPF design. Uses a voltage follower or non-inverting amplifier. - **Multiple Feedback (MFB) Topology:** Good for BPF and BRF. - **State Variable Filter:** Highly versatile, can simultaneously provide LPF, HPF, BPF outputs. ### SPICE Simulation - **SPICE (Simulation Program with Integrated Circuit Emphasis):** A general-purpose analog electronic circuit simulator. - **Netlist:** Text-based description of the circuit, including components, connections, and analysis commands. - **Analysis Types:** - **.OP (Operating Point Analysis):** Calculates the DC operating point of the circuit (DC voltages and currents). - **.DC (DC Sweep Analysis):** Varies a DC source over a specified range and calculates the DC response. - **.AC (AC Analysis):** Performs a frequency sweep (phasor analysis) to determine the magnitude and phase response of the circuit at different frequencies. - **.TRAN (Transient Analysis):** Calculates the circuit's behavior over time in response to time-varying inputs (e.g., pulses, sinusoids). - **.PARAM (Parameter Analysis):** Allows defining parameters and using them in component values. - **.STEP (Sweep Analysis):** Sweeps a parameter (e.g., resistance, temperature) over a range and performs a specified analysis for each step. - **.NOISE (Noise Analysis):** Calculates the noise generated by components. - **.FOUR (Fourier Analysis):** Performs a Fourier series analysis on transient data. - **Basic Component Models:** Resistors (R), Capacitors (C), Inductors (L), Independent Voltage Sources (V), Independent Current Sources (I), Dependent Sources (E, F, G, H), Diodes (D), Transistors (Q, M), Op-Amps (X). - **Applications:** - Verifying theoretical calculations and circuit designs. - Optimizing component values. - Analyzing power dissipation, signal integrity. - Studying non-ideal behavior (e.g., parasitic effects). - Educational tool for understanding circuit behavior.