R, L, C element laws
Formulas
\[ v = Ri,\quad p = vi = i^2 R = \frac{v^2}{R} \]
\[ v_L = L\frac{di}{dt},\quad i_L = \frac{1}{L}\int v\,dt + i(0) \]
\[ i_C = C\frac{dv}{dt},\quad v_C = \frac{1}{C}\int i\,dt + v(0) \]
\[ w_L = \tfrac12 L i^2,\quad w_C = \tfrac12 C v^2 \]
Notes
KCL: \(\sum i = 0\) at a node. KVL: \(\sum v = 0\) around a loop. Passive sign: current into + of \(v\).
Initial conditions & continuity
Formulas
\[ i_L(0^+) = i_L(0^-)\qquad\text{(inductor current continuous)} \]
\[ v_C(0^+) = v_C(0^-)\qquad\text{(capacitor voltage continuous)} \]
\[ \frac{di_L}{dt}\Big|_{0^+} = \frac{v_L(0^+)}{L},\qquad \frac{dv_C}{dt}\Big|_{0^+} = \frac{i_C(0^+)}{C} \]
Exam use
Find DC steady state before switch (\(L\) short, \(C\) open) → get \(i_L(0^-)\), \(v_C(0^-)\). After switch, write KVL/KCL at \(t=0^+\) to get \(v_L(0^+)\), \(i_C(0^+)\), then slopes.
Nodal analysis (matrix form)
Formulas
\[ \mathbf{G}\,\mathbf{V} = \mathbf{I} \]
\[ G_{ii} = \sum\text{conductances at node }i,\quad G_{ij} = -\sum\text{conductances between }i\text{ and }j \]
Notes
Unknowns = node voltages. Current sources enter RHS. Voltage sources → supernode or source transformation.
Mesh analysis (matrix form)
Formulas
\[ \mathbf{R}\,\mathbf{I} = \mathbf{V} \]
\[ R_{ii} = \sum\text{resistances in mesh }i,\quad R_{ij} = -\sum\text{shared resistances} \]
Notes
Unknowns = mesh currents. Voltage sources enter RHS. Current sources → supermesh.
Series / parallel & Thévenin–Norton
Formulas
\[ R_\mathrm{eq,s} = \sum R_k,\quad \frac{1}{R_\mathrm{eq,p}} = \sum\frac{1}{R_k} \]
\[ L_\mathrm{eq,s}=\sum L,\quad C_\mathrm{eq,s}^{-1}=\sum C^{-1};\quad L_\mathrm{eq,p}^{-1}=\sum L^{-1},\quad C_\mathrm{eq,p}=\sum C \]
\[ V_\mathrm{Th}=V_\mathrm{oc},\quad R_\mathrm{Th}=R_\mathrm{in}\text{ (sources off)},\quad I_N=\frac{V_\mathrm{Th}}{R_\mathrm{Th}} \]
First-order RL / RC — time constants & step
Formulas
\[ \tau_{RL} = \frac{L}{R_\mathrm{Th}},\qquad \tau_{RC} = R_\mathrm{Th}\,C \]
\[ x(t) = x(\infty) + \bigl[x(0^+) - x(\infty)\bigr]e^{-t/\tau},\quad t\ge 0 \]
Special cases
\[ i_L(t)=I_f+(I_0-I_f)e^{-(R/L)t};\quad v_C(t)=V_f+(V_0-V_f)e^{-t/(RC)} \]
\[ \text{zero-input: }x(\infty)=0;\quad\text{zero-state: }x(0^+)=0 \]
\(R_\mathrm{Th}\) = resistance seen by \(L\) or \(C\) with independent sources killed. At \(t=\tau\): ~63% of the way to final; \(5\tau\approx\) settled.
Series RLC — \(\alpha\), \(\omega_0\), cases
Formulas
\[ \alpha = \frac{R}{2L},\qquad \omega_0 = \frac{1}{\sqrt{LC}},\qquad \omega_d=\sqrt{\omega_0^2-\alpha^2} \]
\[ s^2 + 2\alpha s + \omega_0^2 = 0,\quad s_{1,2}=-\alpha\pm\sqrt{\alpha^2-\omega_0^2} \]
Cases
- Overdamped \(\alpha>\omega_0\): \(x(t)=A_1 e^{s_1 t}+A_2 e^{s_2 t}\)
- Critically damped \(\alpha=\omega_0\): \(x(t)=(A_1+A_2 t)e^{-\alpha t}\)
- Underdamped \(\alpha<\omega_0\): \(x(t)=e^{-\alpha t}(A_1\cos\omega_d t+A_2\sin\omega_d t)\)
Find \(A_1,A_2\) from \(x(0^+)\) and \(dx/dt|_{0^+}\). Steady \(x(\infty)\) from DC final circuit.
Parallel RLC
Formulas
\[ \alpha = \frac{1}{2RC},\qquad \omega_0 = \frac{1}{\sqrt{LC}} \]
Same three damping cases as series. Natural response usually written for \(v_C(t)\) (or \(i_L\)). Source-free: use IC only.
Complete response (classical)
Formulas
\[ x(t) = x_n(t) + x_f,\qquad x_f = x(\infty)\text{ for DC step} \]
\[ x_n\text{ uses same }\alpha,\omega_0\text{ as source-free DE} \]
Write DE from KVL/KCL → find roots → apply IC at \(0^+\) on the complete expression.
Common Laplace pairs
Formulas
\[ \mathcal{L}\{1\}=\frac{1}{s},\quad \mathcal{L}\{t\}=\frac{1}{s^2},\quad \mathcal{L}\{e^{-at}\}=\frac{1}{s+a} \]
\[ \mathcal{L}\{\sin\omega t\}=\frac{\omega}{s^2+\omega^2},\quad \mathcal{L}\{\cos\omega t\}=\frac{s}{s^2+\omega^2} \]
\[ \mathcal{L}\{e^{-at}\sin\omega t\}=\frac{\omega}{(s+a)^2+\omega^2},\quad \mathcal{L}\{e^{-at}\cos\omega t\}=\frac{s+a}{(s+a)^2+\omega^2} \]
\[ \mathcal{L}\{t\,e^{-at}\}=\frac{1}{(s+a)^2},\quad \mathcal{L}\{u(t-a)f(t-a)\}=e^{-as}F(s) \]
Properties & theorems
Formulas
\[ \mathcal{L}\{f'(t)\}=sF(s)-f(0^-),\quad \mathcal{L}\{f''(t)\}=s^2 F-sf(0^-)-f'(0^-) \]
\[ \mathcal{L}\!\left\{\!\int_0^t f(\tau)\,d\tau\right\}=\frac{F(s)}{s} \]
\[ f(0^+)=\lim_{s\to\infty}sF(s),\qquad f(\infty)=\lim_{s\to 0}sF(s)\ \text{(IVT/FVT)} \]
s-domain models (with IC)
Formulas
\[ Z_R=R,\quad Z_L=sL,\quad Z_C=\frac{1}{sC} \]
\[ V_L(s)=sL\,I(s)-L\,i(0),\quad I_C(s)=sC\,V(s)-C\,v(0) \]
\[ \text{series:}\; L\,i(0)\text{ voltage source};\quad \frac{v(0)}{s}\text{ voltage (C)} \]
Redraw circuit in \(s\)-domain → nodal/mesh algebra → partial fractions → inverse Laplace.
Partial fractions (quick)
Formulas
\[ \frac{N(s)}{(s+a)(s+b)}=\frac{A}{s+a}+\frac{B}{s+b} \]
\[ \text{complex: }\frac{\alpha s+\beta}{(s+a)^2+\omega^2}\ \to\ e^{-at}(A\cos\omega t+B\sin\omega t) \]
Transfer function
Formulas
\[ H(s)=\frac{Y(s)}{X(s)}=\frac{b_m s^m+\cdots+b_0}{a_n s^n+\cdots+a_0} \]
\[ H(j\omega)=|H|\,e^{j\angle H},\quad |H|_{\mathrm{dB}}=20\log_{10}|H(j\omega)| \]
Poles of \(H(s)\) = natural modes. For Bode: substitute \(s=j\omega\).
Bode asymptotic rules
Rules
- \(K/(j\omega)^n\): slope \(-20n\) dB/dec; phase \(-90°n\)
- \((1+j\omega/\omega_c)\): +20 dB/dec after \(\omega_c\); phase \(0\to +90°\)
- \(1/(1+j\omega/\omega_c)\): −20 dB/dec after \(\omega_c\); phase \(0\to -90°\)
- Quadratic \((s/\omega_0)^2+2\zeta s/\omega_0+1\): ±40 dB/dec; peak if \(\zeta<0.707\)
\[ \omega_c\text{ (corner)}=\text{break frequency from factor roots} \]
Resonance, \(Q\), bandwidth & filters
Formulas
\[ Q=\frac{\omega_0}{\mathrm{BW}},\qquad \mathrm{BW}=\omega_H-\omega_L\ \text{(rad/s half-power)} \]
\[ \text{series RLC: }Q=\frac{\omega_0 L}{R}=\frac{1}{R}\sqrt{\frac{L}{C}} \]
\[ \text{parallel RLC: }Q=R\sqrt{\frac{C}{L}}=R\omega_0 C \]
\[ f_c=\frac{1}{2\pi RC}\ \text{(1st-order RC LPF/HPF)};\quad f_0=\frac{1}{2\pi\sqrt{LC}} \]
LPF: pass \(\omega<\omega_c\). HPF: \(\omega>\omega_c\). BPF: pass near \(\omega_0\); BSF: reject near \(\omega_0\).
Z-parameters (open-circuit)
Formulas
\[ V_1=z_{11}I_1+z_{12}I_2,\quad V_2=z_{21}I_1+z_{22}I_2 \]
\[ z_{11}=\frac{V_1}{I_1}\Big|_{I_2=0},\ z_{21}=\frac{V_2}{I_1}\Big|_{I_2=0},\ z_{12}=\frac{V_1}{I_2}\Big|_{I_1=0},\ z_{22}=\frac{V_2}{I_2}\Big|_{I_1=0} \]
Y-parameters (short-circuit)
Formulas
\[ I_1=y_{11}V_1+y_{12}V_2,\quad I_2=y_{21}V_1+y_{22}V_2 \]
\[ y_{11}=\frac{I_1}{V_1}\Big|_{V_2=0},\ \ldots \]
ABCD (transmission) parameters
Formulas
\[ V_1=AV_2+BI_2,\quad I_1=CV_2+DI_2 \]
\[ A=\frac{V_1}{V_2}\Big|_{I_2=0},\ B=\frac{V_1}{I_2}\Big|_{V_2=0},\ C=\frac{I_1}{V_2}\Big|_{I_2=0},\ D=\frac{I_1}{I_2}\Big|_{V_2=0} \]
Cascade: \(\begin{bmatrix}A&B\\C&D\end{bmatrix}_\mathrm{eq}=\begin{bmatrix}A_1&B_1\\C_1&D_1\end{bmatrix}\begin{bmatrix}A_2&B_2\\C_2&D_2\end{bmatrix}\) (port-2 of first into port-1 of second; \(I_2\) polarity convention as taught).
h- and g-parameters
Formulas
\[ V_1=h_{11}I_1+h_{12}V_2,\quad I_2=h_{21}I_1+h_{22}V_2 \]
\[ I_1=g_{11}V_1+g_{12}I_2,\quad V_2=g_{21}V_1+g_{22}I_2 \]
Reciprocity & key conversions
Formulas
\[ \text{Reciprocal: }z_{12}=z_{21},\ y_{12}=y_{21},\ AD-BC=1,\ h_{12}=-h_{21} \]
\[ \mathbf{Y}=\mathbf{Z}^{-1};\quad A=\frac{z_{11}}{z_{21}},\ B=\frac{\Delta_z}{z_{21}},\ C=\frac{1}{z_{21}},\ D=\frac{z_{22}}{z_{21}} \]
\[ \Delta_z=z_{11}z_{22}-z_{12}z_{21} \]
Symmetric network: \(z_{11}=z_{22}\) (etc.). For reciprocal passive networks \(AD-BC=1\).
Magnetic circuit basics
Formulas
\[ \mathfrak{R}=\frac{l}{\mu A},\qquad \mathrm{MMF}=\mathcal{F}=NI=\Phi\,\mathfrak{R} \]
\[ \Phi=BA,\quad B=\mu H=\mu_0\mu_r H,\quad \mu_0=4\pi\times 10^{-7} \]
\[ H=\frac{NI}{l}\ \text{(mean path)},\quad \text{air gap: }\mathfrak{R}_g=\frac{l_g}{\mu_0 A_g} \]
Analogy: \(\mathcal{F}\leftrightarrow V\), \(\Phi\leftrightarrow I\), \(\mathfrak{R}\leftrightarrow R\). Series reluctances add.
Faraday’s law & inductance
Formulas
\[ e = -N\frac{d\Phi}{dt} = -\frac{d\lambda}{dt},\quad \lambda=N\Phi=Li \]
\[ L=\frac{N\Phi}{i}=\frac{N^2}{\mathfrak{R}} \]
Core losses
Formulas
\[ P_h = \eta\, B_m^{1.6}\, f\, V \quad\text{(Steinmetz; exponent ~1.5–2.5)} \]
\[ P_e = K_e\, B_m^2\, f^2\, t^2\, V \]
\[ P_\mathrm{core}=P_h+P_e \]
\(V\) = volume, \(t\) = lamination thickness. Eddy ∝ \(f^2\); hysteresis ∝ \(f\).
Force on current-carrying conductor
Formulas
\[ F = B\,I\,l\quad (\perp),\qquad \vec F=I\,\vec\ell\times\vec B \]
EMF equation & turns ratio
Formulas
\[ E = 4.44\, f\, N\, \Phi_m = 4.44\, f\, N\, B_m A \]
\[ a=\frac{N_1}{N_2}=\frac{E_1}{E_2}\approx\frac{V_1}{V_2}=\frac{I_2}{I_1} \]
Efficiency & voltage regulation
Formulas
\[ \eta=\frac{\text{output}}{\text{output}+\text{losses}}=\frac{V_2 I_2\cos\phi}{V_2 I_2\cos\phi+P_i+P_\mathrm{cu}} \]
\[ \%\mathrm{VR}=\frac{V_{\mathrm{no\text{-}load}}-V_{\mathrm{full\text{-}load}}}{V_{\mathrm{full\text{-}load}}}\times 100 \]
\[ \text{max }\eta:\ P_\mathrm{cu}=P_i\ \Rightarrow\ \text{kVA}_\mathrm{opt}=S_\mathrm{rated}\sqrt{\frac{P_i}{P_{\mathrm{cu,fl}}}} \]
\(P_i\) ≈ constant (OC); \(P_\mathrm{cu}=I^2 R\) ∝ (load)². At max \(\eta\), iron loss = copper loss.
OC & SC tests
Formulas
\[ \text{OC (LV): }P_0=P_i,\quad I_0,\ V_0;\quad R_0=\frac{V_0}{I_w},\ X_0=\frac{V_0}{I_\mu} \]
\[ I_w=\frac{P_0}{V_0},\ I_\mu=\sqrt{I_0^2-I_w^2} \]
\[ \text{SC (HV): }P_\mathrm{sc}=P_{\mathrm{cu,fl}},\ Z_{01}=\frac{V_\mathrm{sc}}{I_\mathrm{sc}},\ R_{01}=\frac{P_\mathrm{sc}}{I_\mathrm{sc}^2} \]
\[ X_{01}=\sqrt{Z_{01}^2-R_{01}^2} \]
Referred quantities
Formulas
\[ R_2'=a^2 R_2,\quad X_2'=a^2 X_2,\quad Z_2'=a^2 Z_2 \]
\[ R_{01}=R_1+R_2',\quad X_{01}=X_1+X_2' \]
Generated EMF (DC machine)
Formulas
\[ E_g = \frac{\Phi Z N P}{60 A} \]
\(\Phi\) Wb/pole, \(Z\) armature conductors, \(N\) rpm, \(P\) poles, \(A\) parallel paths (\(A=P\) lap; \(A=2\) wave).
Torque
Formulas
\[ T = 0.159\,\Phi\, Z\, I_a\,\frac{P}{A}\ \text{(N·m)}\approx\frac{PZ\Phi I_a}{2\pi A} \]
\[ T\propto\Phi I_a \]
Voltage equations
Formulas
\[ \text{Motor: }E_b=V-I_a R_a\quad(\text{shunt: }I_L=I_a+I_f) \]
\[ \text{Generator: }E_g=V+I_a R_a \]
\[ \text{Series motor: }E_b=V-I_a(R_a+R_\mathrm{se}) \]
Speed & speed control
Formulas
\[ N\propto\frac{E_b}{\Phi}\propto\frac{V-I_a R_a}{\Phi} \]
- Flux control (field rheostat): above base speed (weaken \(\Phi\))
- Armature voltage / rheostat: below base speed
- \(N_2/N_1=(E_{b2}/E_{b1})(\Phi_1/\Phi_2)\)
Synchronous speed & slip
Formulas
\[ N_s=\frac{120 f}{P}\ \text{(rpm)},\qquad s=\frac{N_s-N_r}{N_s} \]
\[ N_r=N_s(1-s),\qquad f_r=s f \]
\(f_r\) = rotor frequency. At standstill \(s=1\); sync. speed \(s=0\) (ideal, zero torque for induction motor).
Torque (induction motor)
Formulas
\[ T=\frac{3}{2\pi N_s}\, \frac{s E_2^2 R_2}{R_2^2+(s X_2)^2}\quad (N_s\text{ in rps}) \]
\[ T=\frac{3}{\omega_s}\frac{V^2(R_2/s)}{(R_1+R_2/s)^2+(X_1+X_2)^2}\ \text{(approx. Thévenin forms vary)} \]
Maximum torque
Formulas
\[ s_m=\frac{R_2}{X_2}\quad\text{(approx., neglected stator)},\quad T_\mathrm{max}\propto\frac{1}{2X_2} \]
\[ \text{with stator: }s_m=\frac{R_2}{\sqrt{R_\mathrm{Th}^2+(X_\mathrm{Th}+X_2)^2}} \]
\(T_\mathrm{max}\) independent of \(R_2\) (approx.); \(s_m\) rises if \(R_2\) increased (wound-rotor starters).
Starting & power
Formulas
\[ T_\mathrm{st}=T\big|_{s=1}\propto\frac{E_2^2 R_2}{R_2^2+X_2^2} \]
\[ P_g=(1-s)P_g+s P_g=\underbrace{P_m}_{\text{mech}}+\underbrace{P_\mathrm{cu2}}_{\text{rotor Cu}} \]
\[ \eta\approx\frac{(1-s)P_g-\text{mech losses}}{\text{input}} \]