Electrical Engineering Cheatsheet

Cheatsheet Content

Network Theory Thevenin / Norton Equivalence Thevenin Voltage ($V_{Th}$): Open-circuit voltage across the terminals. Thevenin Resistance ($R_{Th}$): Resistance looking into the terminals with all independent sources turned off (voltage sources shorted, current sources opened). Norton Current ($I_N$): Short-circuit current between the terminals. $I_N = V_{Th} / R_{Th}$. Norton Resistance ($R_N$): Same as $R_{Th}$. Load Current/Voltage: Use equivalent circuit with load resistor $R_L$. $I_L = V_{Th} / (R_{Th} + R_L)$ $V_L = I_L \cdot R_L$ Superposition Theorem Used for circuits with multiple independent sources. Turn off all but one independent source (voltage sources short, current sources open). Calculate the response (voltage/current) due to that single active source. Repeat for each independent source. Sum individual responses to get the total response. Dependent sources remain active. Source Transformations Voltage source $V_S$ in series with $R_S$ $\leftrightarrow$ Current source $I_S$ in parallel with $R_S$. $I_S = V_S / R_S$ $V_S = I_S \cdot R_S$ Mesh / Nodal Analysis Nodal Analysis: Apply KCL at each non-reference node. Solve for node voltages. Sum of currents leaving a node = 0. Current through resistor: $I = (V_A - V_B) / R$. Mesh Analysis: Apply KVL around each independent loop (mesh). Solve for mesh currents. Sum of voltages around a loop = 0. Voltage across resistor: $V = I \cdot R$. Transients (RC/RL Circuits) Time Constant ($\tau$): RC Circuit: $\tau = RC$ RL Circuit: $\tau = L/R$ Initial Conditions ($t=0^+$): Capacitor: Acts as short circuit if uncharged, acts as voltage source if charged ($V_C(0^+) = V_C(0^-)$). Inductor: Acts as open circuit if no initial current, acts as current source if current flowing ($I_L(0^+) = I_L(0^-)$). Final Conditions ($t \to \infty$): Capacitor: Acts as open circuit (fully charged). Inductor: Acts as short circuit (steady current). General Solution (First-order): $f(t) = f(\infty) + [f(0^+) - f(\infty)]e^{-t/\tau}$ Sinusoidal Steady State Analysis Phasors: $v(t) = V_m \cos(\omega t + \phi) \leftrightarrow \mathbf{V} = V_m \angle \phi$. Impedance ($Z$): Resistor: $Z_R = R$ Inductor: $Z_L = j\omega L$ Capacitor: $Z_C = 1/(j\omega C) = -j/( \omega C)$ Admittance ($Y$): $Y = 1/Z$. Resistor: $Y_R = 1/R$ Inductor: $Y_L = 1/(j\omega L) = -j/(\omega L)$ Capacitor: $Y_C = j\omega C$ Ohm's Law (Phasor): $\mathbf{V} = \mathbf{I}Z$ or $\mathbf{I} = \mathbf{V}Y$. Power: Apparent Power ($S$): $S = \mathbf{V}_{rms} \mathbf{I}_{rms}^* = P + jQ$ (VA). Real Power ($P$): $P = |\mathbf{V}_{rms}| |\mathbf{I}_{rms}| \cos(\theta_v - \theta_i)$ (Watts). $P = \text{Re}(S)$. Reactive Power ($Q$): $Q = |\mathbf{V}_{rms}| |\mathbf{I}_{rms}| \sin(\theta_v - \theta_i)$ (VAR). $Q = \text{Im}(S)$. Power Factor (PF): $\cos(\theta_v - \theta_i) = P/|S|$. Two-Port Networks Z-parameters: $V_1 = z_{11}I_1 + z_{12}I_2$, $V_2 = z_{21}I_1 + z_{22}I_2$. Y-parameters: $I_1 = y_{11}V_1 + y_{12}V_2$, $I_2 = y_{21}V_1 + y_{22}V_2$. h-parameters (hybrid): $V_1 = h_{11}I_1 + h_{12}V_2$, $I_2 = h_{21}I_1 + h_{22}V_2$. ABCD-parameters (transmission): $V_1 = AV_2 - BI_2$, $I_1 = CV_2 - DI_2$. Signals & Systems Convolution Continuous-time: $y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau$. Discrete-time: $y[n] = x[n] * h[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k]$. Properties: Commutative ($x*h = h*x$), Associative ($(x*h)*g = x*(h*g)$), Distributive ($x*(h_1+h_2) = x*h_1 + x*h_2$). Fourier Transform (FT) Continuous-time FT: $X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt$. Inverse CTFT: $x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) e^{j\omega t} d\omega$. Common Pairs: $\delta(t) \leftrightarrow 1$ $e^{-at}u(t) \leftrightarrow \frac{1}{a+j\omega}$ $rect(t/\tau) \leftrightarrow \tau \text{sinc}(\omega \tau / 2)$ Properties: Linearity: $ax(t) + by(t) \leftrightarrow aX(j\omega) + bY(j\omega)$. Time Shift: $x(t-t_0) \leftrightarrow e^{-j\omega t_0} X(j\omega)$. Frequency Shift (Modulation): $e^{j\omega_0 t} x(t) \leftrightarrow X(j(\omega - \omega_0))$. Time Scaling: $x(at) \leftrightarrow \frac{1}{|a|} X(j\omega/a)$. Differentiation: $\frac{d}{dt}x(t) \leftrightarrow j\omega X(j\omega)$. Integration: $\int_{-\infty}^t x(\tau) d\tau \leftrightarrow \frac{1}{j\omega} X(j\omega) + \pi X(0)\delta(\omega)$. Convolution: $x(t) * h(t) \leftrightarrow X(j\omega)H(j\omega)$. LTI System Properties Causality: Impulse response $h(t)=0$ for $t Stability (BIBO): System is stable if $\int_{-\infty}^{\infty} |h(t)| dt Linearity: $T[ax_1(t)+bx_2(t)] = aT[x_1(t)]+bT[x_2(t)]$. Time Invariance: If $y(t) = T[x(t)]$, then $y(t-t_0) = T[x(t-t_0)]$. Sampling Nyquist Rate: $f_s \ge 2f_{max}$, where $f_{max}$ is the highest frequency component in the signal. Aliasing: Occurs if $f_s Reconstruction: Low-pass filter used to recover original signal from sampled version. Laplace Transform Unilateral Laplace Transform: $X(s) = \int_{0^-}^{\infty} x(t) e^{-st} dt$. Inverse Laplace Transform: Use partial fraction expansion and lookup tables. Region of Convergence (ROC): Range of $s$ for which $X(s)$ converges. Important for causality and stability. System Function: $H(s) = Y(s)/X(s)$. Poles and zeros of $H(s)$ determine system behavior. Stability: For causal systems, all poles must be in the left-half of the s-plane ($\text{Re}(s) Control Systems Routh-Hurwitz Stability Criterion Used to determine the stability of a linear system from its characteristic equation $P(s)=a_n s^n + \dots + a_1 s + a_0 = 0$. Procedure: Form the Routh table from coefficients. If any coefficient $a_i$ is zero or negative while others are positive, there are RHP roots (unstable). Count sign changes in the first column of the Routh table to find the number of RHP roots. Bode Plots Logarithmic plots of magnitude (dB) and phase (degrees) versus log frequency. Magnitude (dB): $20 \log_{10} |H(j\omega)|$. Phase: $\angle H(j\omega)$. Gain Margin (GM): Amount of gain change required to make the system unstable. Measured at phase crossover ($ \angle H(j\omega) = -180^\circ$). Phase Margin (PM): Amount of phase change required to make the system unstable. Measured at gain crossover ($ |H(j\omega)| = 1$ or $0$ dB). Crossover Frequencies: $\omega_{gc}$ (gain crossover), $\omega_{pc}$ (phase crossover). Root Locus Graphical method to show pole locations of the closed-loop system as a parameter (usually gain $K$) varies from $0$ to $\infty$. Rules: Number of branches = number of open-loop poles. Symmetric about the real axis. Start at open-loop poles ($K=0$), end at open-loop zeros ($K=\infty$) or infinity. Real axis segments to the left of an odd number of poles and zeros. Angle of asymptotes: $\frac{(2k+1)180^\circ}{n-m}$, where $n$ is poles, $m$ is zeros. Centroid of asymptotes: $\frac{\sum \text{poles} - \sum \text{zeros}}{n-m}$. Breakaway/break-in points: Solve $dK/ds = 0$. Angle of departure/arrival: Use angle condition sum of angles from zeros - sum of angles from poles = $(2k+1)180^\circ$. Time Domain Specifications (for 2nd Order Systems) Characteristic Equation: $s^2 + 2\zeta\omega_n s + \omega_n^2 = 0$. Undamped Natural Frequency ($\omega_n$): Rad/s. Damping Ratio ($\zeta$): $\zeta $\zeta = 1$: Critically damped (fastest without overshoot). $\zeta > 1$: Overdamped (slow, no overshoot). Rise Time ($T_r$): Time for response to go from 10% to 90% of final value. (For underdamped: $T_r \approx \frac{\pi - \arccos(\zeta)}{\omega_n \sqrt{1-\zeta^2}}$). Peak Time ($T_p$): Time to reach the first peak. $T_p = \frac{\pi}{\omega_n \sqrt{1-\zeta^2}}$. Overshoot (%OS): $\%OS = e^{-\zeta\pi/\sqrt{1-\zeta^2}} \times 100$. Settling Time ($T_s$): Time to settle within $\pm2\%$ or $\pm5\%$ of final value. $2\%$ criterion: $T_s \approx \frac{4}{\zeta\omega_n}$. $5\%$ criterion: $T_s \approx \frac{3}{\zeta\omega_n}$. Steady-State Error ($e_{ss}$) For a unity feedback system with open-loop transfer function $G(s)$. Position Error Constant ($K_p$): $K_p = \lim_{s \to 0} G(s)$. Step input ($R(s)=1/s$): $e_{ss} = \frac{1}{1+K_p}$. Velocity Error Constant ($K_v$): $K_v = \lim_{s \to 0} sG(s)$. Ramp input ($R(s)=1/s^2$): $e_{ss} = \frac{1}{K_v}$. Acceleration Error Constant ($K_a$): $K_a = \lim_{s \to 0} s^2G(s)$. Parabolic input ($R(s)=1/s^3$): $e_{ss} = \frac{1}{K_a}$. Block Diagram Reduction Series: $G_1(s)G_2(s)$. Parallel: $G_1(s)+G_2(s)$. Feedback (Negative): $\frac{G(s)}{1+G(s)H(s)}$. Feedback (Positive): $\frac{G(s)}{1-G(s)H(s)}$. Electrical Machines DC Machines Induced EMF ($E_a$): $E_a = K_a \Phi \omega_m = K_a \Phi (2\pi N/60)$, where $K_a$ is machine constant, $\Phi$ is flux per pole, $\omega_m$ is rotor speed (rad/s), $N$ is speed (RPM). Torque ($T$): $T = K_a \Phi I_a$, where $I_a$ is armature current. Voltage Equation (Motor): $V_t = E_a + I_a R_a$. Voltage Equation (Generator): $E_a = V_t + I_a R_a$. Transformers Turns Ratio ($a$): $a = N_1/N_2 = V_1/V_2 = I_2/I_1$. Equivalent Circuit: Referred to primary or secondary. Includes magnetizing branch ($R_c, X_m$) and leakage impedances ($R_1, X_1, R_2', X_2'$). Open-Circuit (OC) Test: Determines $R_c$ and $X_m$. Short-Circuit (SC) Test: Determines equivalent series resistance $R_{eq}$ and reactance $X_{eq}$. Efficiency ($\eta$): $\eta = \frac{P_{out}}{P_{in}} = \frac{P_{out}}{P_{out} + P_{core} + P_{copper}}$. Voltage Regulation (VR): $VR = \frac{V_{NL} - V_{FL}}{V_{FL}} \times 100\%$. Induction Machines Synchronous Speed ($N_s$): $N_s = \frac{120f}{P}$ (RPM), where $f$ is frequency, $P$ is # of poles. Slip ($s$): $s = \frac{N_s - N_r}{N_s}$, where $N_r$ is rotor speed. Rotor Frequency ($f_r$): $f_r = s f$. Rotor Current Frequency: $f_r = sf$. Torque: $T \propto \frac{s V^2 R_2 / s}{(R_1 + R_2/s)^2 + (X_1 + X_2)^2}$. Maximum Torque: Occurs when $R_2/s = \sqrt{R_1^2 + (X_1+X_2)^2}$. Synchronous Machines Synchronous Speed ($N_s$): $N_s = \frac{120f}{P}$ (RPM). Induced EMF ($E_f$): Depends on field current. Synchronous Reactance ($X_s$): Represents armature reaction and leakage reactance. Phasor Diagram: $E_f = V_t + I_a (R_a + jX_s)$. Power ($P$): $P = \frac{3 V_t E_f}{X_s} \sin\delta$. ($\delta$ is power angle). Voltage Regulation (VR): $VR = \frac{E_f - V_t}{V_t} \times 100\%$ (often for generator at constant field current). Power Systems Per Unit (PU) System Base Quantities: $S_{base}$ (MVA), $V_{base}$ (kV). Base Current: $I_{base} = S_{base} / (\sqrt{3} V_{base})$. Base Impedance: $Z_{base} = V_{base}^2 / S_{base}$. PU Value: Actual Value / Base Value. Changing Bases: $Z_{PU, new} = Z_{PU, old} \left( \frac{V_{base, old}}{V_{base, new}} \right)^2 \left( \frac{S_{base, new}}{S_{base, old}} \right)$. Fault Analysis Symmetrical Faults (LLL, LLLG): Use positive sequence network only. $I_f = V_f / Z_{eq}$. Unsymmetrical Faults (LG, LL, LLG): Requires symmetrical components (positive, negative, zero sequence networks). Sequence Networks: Positive Sequence ($Z_1$): Standard impedance, includes source. Negative Sequence ($Z_2$): Same as positive for rotating machines, usually $Z_1$ for static. No source. Zero Sequence ($Z_0$): Different for transformers (grounding), lines, generators. No source. Fault Currents (simplified): LG: $I_{a1} = I_{a2} = I_{a0} = \frac{E_a}{Z_1+Z_2+Z_0+3Z_f}$. $I_f = 3I_{a0}$. LL: $I_{a1} = \frac{E_a}{Z_1+Z_2+Z_f}$. $I_f = \sqrt{3} |I_{a1}|$. LLG: $I_{a1} = \frac{E_a}{Z_1 + (Z_2(Z_0+3Z_f))/(Z_2+Z_0+3Z_f)}$. Symmetrical Components Used to analyze unsymmetrical faults. $\mathbf{V}_{abc} = \mathbf{A} \mathbf{V}_{012}$ $\mathbf{V}_{012} = \mathbf{A}^{-1} \mathbf{V}_{abc}$ $\mathbf{A} = \begin{pmatrix} 1 & 1 & 1 \\ 1 & a^2 & a \\ 1 & a & a^2 \end{pmatrix}$, where $a = 1 \angle 120^\circ = e^{j2\pi/3}$. $\mathbf{A}^{-1} = \frac{1}{3} \begin{pmatrix} 1 & 1 & 1 \\ 1 & a & a^2 \\ 1 & a^2 & a \end{pmatrix}$. Transmission Lines Short Line: $V_S = V_R + I_R Z$, $I_S = I_R$. ($Z = R+jX$). Medium Line (Nominal $\pi$ or T): Uses ABCD parameters. $V_S = AV_R + BI_R$ $I_S = CV_R + DI_R$ Long Line: Hyperbolic functions. ABCD Parameters (General): $A=D$. $AD-BC=1$. Voltage Regulation (VR): $VR = \frac{|V_{S,NL}| - |V_{R,FL}|}{|V_{R,FL}|} \times 100\%$. Power Angle and Stability Power-Angle Curve: $P = \frac{|V_S| |V_R|}{X} \sin\delta$. ($X$ is reactance between sending and receiving end). Steady-State Stability Limit: Maximum power a system can transmit without losing synchronism. Occurs at $\delta = 90^\circ$. Transient Stability: Ability to remain in synchronism after a large disturbance. Engineering Mathematics Linear Algebra Eigenvalues ($\lambda$) and Eigenvectors ($v$): $Av = \lambda v$. Solve $\det(A-\lambda I) = 0$ for $\lambda$. Rank of a Matrix: Number of linearly independent rows or columns. $rank(A) = \text{dimension of column space} = \text{dimension of row space}$. Solving System of Equations ($Ax=b$): Gaussian Elimination/Row Reduction: Convert $[A|b]$ to row echelon form. Inverse: $x = A^{-1}b$ (if $A$ is invertible). Cramer's Rule: For $n \times n$ systems, $x_i = \det(A_i) / \det(A)$. Calculus Derivatives: Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$. Product Rule: $\frac{d}{dx}(uv) = u'v + uv'$. Quotient Rule: $\frac{d}{dx}(u/v) = (u'v - uv')/v^2$. Chain Rule: $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$. Integrals: Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$). $\int \frac{1}{x} dx = \ln|x| + C$. Integration by Parts: $\int u dv = uv - \int v du$. Maxima/Minima: Find critical points where $f'(x)=0$ or $f'(x)$ is undefined. Use second derivative test ($f''(x)>0$ min, $f''(x) Differential Equations First-order ODEs: Separable: $\frac{dy}{dx} = f(x)g(y)$. Linear: $\frac{dy}{dx} + P(x)y = Q(x)$. Use integrating factor $e^{\int P(x)dx}$. Second-order Linear Homogeneous ODEs (Constant Coefficients): $ay''+by'+cy=0$. Characteristic equation: $ar^2+br+c=0$. Roots $r_1, r_2$: Real & Distinct: $y(x) = C_1e^{r_1x} + C_2e^{r_2x}$. Real & Repeated: $y(x) = (C_1+C_2x)e^{rx}$. Complex Conjugate ($ \alpha \pm j\beta$): $y(x) = e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x))$. Particular Solution (Non-homogeneous): Use method of undetermined coefficients or variation of parameters. Probability Mean (Expected Value): $E[X] = \sum x_i P(X=x_i)$ (discrete), $E[X] = \int x f(x) dx$ (continuous). Variance: $Var(X) = E[X^2] - (E[X])^2$. Standard deviation $\sigma = \sqrt{Var(X)}$. Probability of an Event: $P(A) = \text{number of favorable outcomes} / \text{total outcomes}$. Conditional Probability: $P(A|B) = P(A \cap B) / P(B)$. Bayes' Theorem: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$. Common Distributions: Binomial: $P(k) = \binom{n}{k} p^k (1-p)^{n-k}$. Poisson: $P(k) = \frac{\lambda^k e^{-\lambda}}{k!}$. Normal (Gaussian): $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/(2\sigma^2)}$. Vector Calculus Gradient ($\nabla f$): $\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}$. Points in direction of steepest ascent. Divergence ($\nabla \cdot \mathbf{F}$): $\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$. Measures flux density out of a point. Curl ($\nabla \times \mathbf{F}$): $\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \partial/\partial x & \partial/\partial y & \partial/\partial z \\ F_x & F_y & F_z \end{vmatrix}$. Measures rotation/circulation. Divergence Theorem: $\int_V (\nabla \cdot \mathbf{F}) dV = \oint_S \mathbf{F} \cdot d\mathbf{S}$. Stokes' Theorem: $\int_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}$. Transforms Laplace Transform: (See Signals & Systems section). Fourier Series (Periodic Signals): Exponential: $x(t) = \sum_{k=-\infty}^{\infty} c_k e^{jk\omega_0 t}$, where $c_k = \frac{1}{T} \int_T x(t) e^{-jk\omega_0 t} dt$. Trigonometric: $x(t) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t))$.