Master Formula Reference

ENEE 154 — Master Formula Sheet

Rigorous compilation of every governing equation across all 9 chapters. Includes exact parameter definitions, SI units, operational conditions, and common exam pitfalls.

Chapter 1: Network Elements & Initial Conditions

\[v_L(t) = L \frac{di_L(t)}{dt}, \quad i_C(t) = C \frac{dv_C(t)}{dt}\]
Variables: \(v_L\) = inductor voltage [V], \(i_L\) = inductor current [A], \(L\) = inductance [H], \(i_C\) = capacitor current [A], \(v_C\) = capacitor voltage [V], \(C\) = capacitance [F].
Conditions: Fundamental differential v-i relationships. Current through an inductor and voltage across a capacitor cannot change instantaneously.
\[i_L(0^+) = i_L(0^-), \quad v_C(0^+) = v_C(0^-)\] \[\frac{di_L}{dt}(0^+) = \frac{v_L(0^+)}{L}, \quad \frac{dv_C}{dt}(0^+) = \frac{i_C(0^+)}{C}\]
Switching Invariance Rules: Inductor behaves as an open circuit (or constant current source of \(i_L(0^-)\)) at \(t=0^+\). Capacitor behaves as a short circuit (or constant voltage source of \(v_C(0^-)\)) at \(t=0^+\).
Derivative Rule: Always find \(v_L(0^+)\) and \(i_C(0^+)\) from KVL/KCL at \(t=0^+\) first, then divide by \(L\) and \(C\).

Chapter 2: Classical Transient Analysis

\[\text{RL Circuit: } \tau = \frac{L}{R} \text{ [seconds]}, \quad i(t) = i(\infty) + [i(0^+) - i(\infty)] e^{-t/\tau}\] \[\text{RC Circuit: } \tau = R C \text{ [seconds]}, \quad v(t) = v(\infty) + [v(0^+) - v(\infty)] e^{-t/\tau}\]
Usage: Universal first-order step response formula. Applicable to any linear first-order circuit with constant DC sources.
\[\text{Series RLC ODE: } L \frac{d^2 i}{dt^2} + R \frac{di}{dt} + \frac{1}{C} i = \frac{dv_s}{dt}\] \[s^2 + \frac{R}{L} s + \frac{1}{LC} = 0 \implies s^2 + 2\alpha s + \omega_0^2 = 0\] \[\alpha = \frac{R}{2L}, \quad \omega_0 = \frac{1}{\sqrt{LC}}, \quad s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}\]
Damping Classifications:
1. Overdamped (\(\alpha > \omega_0\)): \(i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}\) (two distinct real negative roots).
2. Critically Damped (\(\alpha = \omega_0\)): \(i(t) = (A_1 + A_2 t) e^{-\alpha t}\) (repeated real root).
3. Underdamped (\(\alpha < \omega_0\)): \(i(t) = e^{-\alpha t} [A_1 \cos(\omega_d t) + A_2 \sin(\omega_d t)]\), where \(\omega_d = \sqrt{\omega_0^2 - \alpha^2}\).
\[\text{Parallel RLC: } \alpha = \frac{1}{2RC}, \quad \omega_0 = \frac{1}{\sqrt{LC}}\]
Important Difference: In parallel RLC, larger \(R\) gives smaller damping \(\alpha\) (closer to oscillation), which is the exact opposite of series RLC!

Chapter 3: Laplace Transform Transient Analysis

Element Time Domain s-Domain Impedance Series Initial Source Model Parallel Initial Source Model
Resistor \(R\) \(v(t) = R i(t)\) \(Z(s) = R\) None None
Inductor \(L\) \(v(t) = L \frac{di}{dt}\) \(Z(s) = sL\) Voltage source \(L i(0^-)\) aiding current Current source \(\frac{i(0^-)}{s}\) across \(sL\)
Capacitor \(C\) \(i(t) = C \frac{dv}{dt}\) \(Z(s) = \frac{1}{sC}\) Voltage source \(\frac{v(0^-)}{s}\) in series with \(\frac{1}{sC}\) Current source \(C v(0^-)\) in parallel with \(\frac{1}{sC}\)
\[\mathcal{L}\{e^{-at}\} = \frac{1}{s+a}, \quad \mathcal{L}\{\sin(\omega t)\} = \frac{\omega}{s^2+\omega^2}, \quad \mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2+\omega^2}\] \[\mathcal{L}\left\{\frac{df}{dt}\right\} = s F(s) - f(0^-)\]
Standard Procedure: Transform all independent sources (\(V/s\)), replace \(L\) and \(C\) with their s-domain equivalent models including initial condition generators, write s-domain KVL/KCL, solve for \(I(s)\) or \(V(s)\), and perform Partial Fraction Expansion (PFE) to obtain inverse transform.

Chapter 4: Transfer Function, Bode Plots & Filters

\[H(s) = \frac{N(s)}{D(s)} = K \frac{(s - z_1)(s - z_2)\dots(s - z_m)}{(s - p_1)(s - p_2)\dots(s - p_n)}\] \[|H(j\omega)|_{\text{dB}} = 20 \log_{10} |H(j\omega)|, \quad \angle H(j\omega) = \sum \angle(\text{zeros}) - \sum \angle(\text{poles})\]
Bode Plot Standard Form: Express all factors as \((1 + s/\omega_c)\). The constant multiplier is the Bode gain \(K_B\).
Factor in \(H(j\omega)\) Magnitude Slope Phase Contribution
Constant \(K_B\) \(0\text{ dB/decade}\), level at \(20\log_{10} |K_B|\) \(0^\circ\) (\(+180^\circ\) if \(K_B < 0\))
Pole at origin \(1/s^n\) \(-20n\text{ dB/decade}\) \(-90^\circ \times n\)
Zero at origin \(s^n\) \(+20n\text{ dB/decade}\) \(+90^\circ \times n\)
Simple zero \((1 + s/\omega_c)\) \(+20\text{ dB/decade}\) for \(\omega > \omega_c\) \(0^\circ\) (\(\omega \ll \omega_c\)), \(+45^\circ\) (\(\omega = \omega_c\)), \(+90^\circ\) (\(\omega \gg \omega_c\))
Simple pole \(\frac{1}{1 + s/\omega_c}\) \(-20\text{ dB/decade}\) for \(\omega > \omega_c\) \(0^\circ\) (\(\omega \ll \omega_c\)), \(-45^\circ\) (\(\omega = \omega_c\)), \(-90^\circ\) (\(\omega \gg \omega_c\))
Quadratic pole \(\frac{1}{1 + 2\zeta(s/\omega_n) + (s/\omega_n)^2}\) \(-40\text{ dB/decade}\) for \(\omega > \omega_n\) \(0^\circ \to -90^\circ \to -180^\circ\)
\[\text{Series Resonance: } \omega_0 = \frac{1}{\sqrt{LC}}, \quad Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 C R} = \frac{1}{R}\sqrt{\frac{L}{C}}\] \[BW = \omega_2 - \omega_1 = \frac{\omega_0}{Q} = \frac{R}{L} \text{ [rad/s]}\]
High-Q Approximation (\(Q \ge 10\)): Half-power frequencies are symmetric about \(\omega_0\): \(\omega_1, \omega_2 \approx \omega_0 \mp \frac{BW}{2}\).

Chapter 5: Two-Port Network Parameters

Parameter Governing Matrix Equations Reciprocity Condition Symmetry Condition
Impedance \([Z]\) \(V_1 = Z_{11}I_1 + Z_{12}I_2\)
\(V_2 = Z_{21}I_1 + Z_{22}I_2\)		 \(Z_{12} = Z_{21}\) \(Z_{11} = Z_{22}\)
Admittance \([Y]\) \(I_1 = Y_{11}V_1 + Y_{12}V_2\)
\(I_2 = Y_{21}V_1 + Y_{22}V_2\)		 \(Y_{12} = Y_{21}\) \(Y_{11} = Y_{22}\)
Transmission \([T]\) or \([ABCD]\) \(V_1 = A V_2 - B I_2\)
\(I_1 = C V_2 - D I_2\)		 \(A D - B C = 1\) \(A = D\)
Hybrid \([h]\) \(V_1 = h_{11}I_1 + h_{12}V_2\)
\(I_2 = h_{21}I_1 + h_{22}V_2\)		 \(h_{12} = -h_{21}\) \(\Delta h = h_{11}h_{22} - h_{12}h_{21} = 1\)
\[\text{Cascade Connection: } \begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} A_1 & B_1 \\ C_1 & D_1 \end{bmatrix} \begin{bmatrix} A_2 & B_2 \\ C_2 & D_2 \end{bmatrix}\] \[h_{11} = \frac{\Delta Z}{Z_{22}}, \quad h_{12} = \frac{Z_{12}}{Z_{22}}, \quad h_{21} = -\frac{Z_{21}}{Z_{22}}, \quad h_{22} = \frac{1}{Z_{22}}\]
Important Convention Trap: In the transmission matrix, current \(I_2\) leaves the port (hence \(-I_2\) appears in the algebraic definition). Always double-check signs!

Chapter 6: Magnetic Circuits & Induction

\[\mathcal{F} = N I = \Phi \mathcal{R}, \quad \mathcal{R} = \frac{l}{\mu_0 \mu_r A} \text{ [AT/Wb]}\] \[B = \frac{\Phi}{A} \text{ [Tesla or Wb/m}^2], \quad H = \frac{B}{\mu_0 \mu_r} \text{ [AT/m]}\]
Permeability constants: \(\mu_0 = 4\pi \times 10^{-7}\text{ H/m}\). For air gap, \(\mu_r = 1\), so \(H_g = B / \mu_0\).
\[\text{Total MMF: } N I = H_{\text{iron}} l_i + H_{\text{air}} l_g = \frac{B}{\mu_0 \mu_r} (l_{\text{mean}} - l_g) + \frac{B}{\mu_0} l_g\] \[P_h = \eta B_{\text{max}}^{1.6} f V \text{ [W]}, \quad P_e = K_e B_{\text{max}}^2 f^2 t^2 V \text{ [W]}\] \[e = B l v \sin\theta \text{ [V]}, \quad F = B I l \sin\theta \text{ [N]}\]
Core Loss Variables: \(\eta\) = Steinmetz coefficient, \(f\) = frequency [Hz], \(V\) = core volume [m\(^3\)], \(t\) = lamination thickness [m].

Chapter 7: Single-Phase Transformers

\[E_1 = 4.44 f N_1 \Phi_m, \quad E_2 = 4.44 f N_2 \Phi_m, \quad K = \frac{N_2}{N_1} = \frac{E_2}{E_1} = \frac{I_1}{I_2}\] \[\Phi_m = \frac{V_1}{4.44 f N_1} \implies \Phi_m = \text{Constant (independent of load)}\]
Referred Parameters: To refer resistance/reactance from secondary to primary, divide by \(K^2\): \(R_2' = R_2 / K^2\). To refer from primary to secondary, multiply by \(K^2\): \(R_1' = K^2 R_1\).
\[\text{OC Test (on LV): } \cos\phi_0 = \frac{W_0}{V_0 I_0}, \quad I_w = I_0 \cos\phi_0, \quad I_\mu = I_0 \sin\phi_0, \quad R_0 = \frac{V_0}{I_w}, \quad X_0 = \frac{V_0}{I_\mu}\] \[\text{SC Test (on HV): } R_{eq1} = \frac{W_{sc}}{I_{sc}^2}, \quad Z_{eq1} = \frac{V_{sc}}{I_{sc}}, \quad X_{eq1} = \sqrt{Z_{eq1}^2 - R_{eq1}^2}\]
Testing Standard: OC test yields core/iron loss \(P_i = W_0\) (constant). SC test yields full-load copper loss \(P_{cu,FL} = W_{sc}\).
\[\eta = \frac{x S \cos\phi}{x S \cos\phi + P_i + x^2 P_{cu,FL}} \times 100\%\] \[\text{Condition for Maximum Efficiency: } x^2 P_{cu,FL} = P_i \implies x = \sqrt{\frac{P_i}{P_{cu,FL}}}\] \[\text{Autotransformer Copper Saving: } \frac{W_{\text{auto}}}{W_{\text{2-wdg}}} = 1 - K\]
Loading Fraction: \(x = \frac{\text{Actual kVA}}{\text{Rated kVA}}\). For 25% overload, \(x = 1.25\).

Chapter 8: DC Machines

\[\text{EMF Equation: } E_g = \frac{P \Phi Z N}{60 A} \text{ [V]}, \quad \text{Torque: } T = \frac{P \Phi Z I_a}{2\pi A} = 0.159 \frac{P \Phi Z I_a}{A} \text{ [N}\cdot\text{m]}\] \[E_b = V - I_a R_a \text{ (Motor)}, \quad E_g = V_t + I_a R_a \text{ (Generator)}\]
Winding Constants: \(P\) = number of poles, \(\Phi\) = flux per pole [Wb], \(Z\) = total armature conductors, \(N\) = speed [rpm]. Parallel paths: \(A = P\) (Lap winding), \(A = 2\) (Wave winding).
\[N = \frac{V - I_a R_a}{k \Phi} \implies \frac{N_2}{N_1} = \frac{E_{b2}}{E_{b1}} \times \frac{\Phi_1}{\Phi_2}\] \[T \propto \Phi I_a \implies \frac{T_2}{T_1} = \frac{\Phi_2}{\Phi_1} \times \frac{I_{a2}}{I_{a1}}\]
Speed Control Rules:
1. Armature Resistance Control: Add series resistance \(R_{\text{ext}}\). \(E_b = V - I_a(R_a + R_{\text{ext}})\) drops, speed drops below base speed.
2. Field Control: Increase field rheostat resistance, \(I_{sh}\) decreases, \(\Phi\) decreases, speed increases above base speed.

Chapter 9: AC Induction & Special Motors

\[N_s = \frac{120 f}{P} \text{ [rpm]}, \quad s = \frac{N_s - N}{N_s}, \quad N = N_s (1 - s), \quad f_r = s f \text{ [Hz]}\] \[E_{2r} = s E_2, \quad X_{2r} = s X_2\]
Parameters: \(N_s\) = synchronous speed, \(N\) = actual rotor speed, \(s\) = fractional slip, \(f_r\) = rotor induced EMF frequency.
\[\text{Running Torque: } T = \frac{k \, s E_2^2 R_2}{R_2^2 + (s X_2)^2}\] \[\text{Slip at Maximum Torque: } s_{mT} = \frac{R_2}{X_2}\] \[\text{Maximum (Breakdown) Torque: } T_{\text{max}} = \frac{k \, E_2^2}{2 X_2} \implies T_{\text{max}} \text{ is independent of } R_2!\]
Rotor Resistance Effect: Adding external rotor resistance \(R_{\text{ext}}\) increases starting torque and shifts the maximum torque peak to a higher slip (lower speed), but does NOT change the peak value of \(T_{\text{max}}\).
\[\text{Stepper Motor Step Angle: } \beta = \frac{N_s - N_r}{N_s N_r} \times 360^\circ = \frac{360^\circ}{m N_r}\]
Variables: \(N_s\) = stator teeth/poles, \(N_r\) = rotor teeth, \(m\) = number of phases/stacks.