Physics Formulae/Electric Circuits Formulae

Lead Article: Tables of Physics Formulae

This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Electric Circuits, Electronics.

DC Quantities

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Electrical Resistance	$R$	$R=V/I$	Ω = V A-1 = J s C-2	[M][L]2 [T]-3 [I]-2
Resistivity, Scalar	$\rho$	$\rho =\frac{RA}{l}$	Ω m	[M]2 [L]2 [T]-3 [I]-2
Resistivity Temperature Coefficient, Linear Temperature Dependance	$\alpha$	$\rho -{\rho }_0={\rho }_0\alpha (T-T_0)$	K-1	[Θ]-1
Terminal Voltage for Power Supply	$V_{\mathrm{t}\mathrm{e}\mathrm{r}}$		V = J C-1	[M] [L]2 [T]-3 [I]-1
Load Voltage for Circuit	$V_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$		V = J C-1	[M] [L]2 [T]-3 [I]-1
Internal Resistance of Power Supply	$R_{\mathrm{i}\mathrm{n}\mathrm{t}}$	$R_{\mathrm{i}\mathrm{n}\mathrm{t}}=\frac{V_{\mathrm{t}\mathrm{e}\mathrm{r}}}{I}$	Ω = V A-1 = J s C-2	[M][L]2 [T]-3 [I]-2
Load Resistance of Circuit	$R_{\mathrm{e}\mathrm{x}\mathrm{t}}$	$R_{\mathrm{e}\mathrm{x}\mathrm{t}}=\frac{V_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}}{I}$	Ω = V A-1 = J s C-2	[M][L]2 [T]-3 [I]-2
Electromotive Force (emf), Voltage across entire circuit including power supply, external components and conductors	$ℰ$	$ℰ=V_{\mathrm{t}\mathrm{e}\mathrm{r}}+V_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}$	V = J C-1	[M] [L]2 [T]-3 [I]-1
Electrical Conductance	$G$	$G=1/R$	S = Ω-1	[T]3 [I]2 [M]-1 [L]-2
Electrical Conductivity, Scalar	$\sigma$	$\sigma =1/\rho$	Ω-1 m-1	[I]2 [T]3 [M]-2 [L]-2
Electrical Conductivity, Tensor	$\mathit{\sigma },{\sigma }_{\mathrm{i}\mathrm{j}}$	${\sigma }_{\mathrm{i}\mathrm{j}}\left(\begin{array}{ccc}{\sigma }_{11}& {\sigma }_{12}& {\sigma }_{13}\\ {\sigma }_{21}& {\sigma }_{22}& {\sigma }_{23}\\ {\sigma }_{31}& {\sigma }_{32}& {\sigma }_{33}\end{array}\right)$	Ω-1 m-1	[I]2 [T]3 [M]-2 [L]-2
Electrical Power	$P$	$P=VI$	W = J s-1	[M] [L]2 [T]-3
emf Power	$P$	$P_{\mathrm{e}\mathrm{m}\mathrm{f}}=Iℰ$	W = J s-1	[M] [L]2 [T]-3
Resistor Power Dissipation	$P$	$P=I^{2}R=V^2/R$	W = J s-1	[M] [L]2 [T]-3

Resistors in Series	$R_{\mathrm{n}\mathrm{e}\mathrm{t}}=\sum _i{R}_i$
Resistors in Parallel	$\frac{1}{R_{\mathrm{n}\mathrm{e}\mathrm{t}}}=\sum _i\frac{1}{R_i}$

Ohm's Law	Scalar form $V=IR$ Vector Form $𝐉=\sigma 𝐄$ Tensor Form, general applies to all points in a conductor ${𝐉}_i={\sigma }_{ij}{𝐄}_j$
Kirchoff's Laws	emf loop rule around any closed circuit $\sum _i{V}_i=\sum _i{I}_i{R}_i=0$ Current law at junctions $I_{\mathrm{i}\mathrm{n}}=I_{\mathrm{o}\mathrm{u}\mathrm{t}}$

AC Quantitites

Quantity (Common Name/s)	Common Name/s	Quantity (Common Name/s)	Quantity (Common Name/s)	Quantity (Common Name/s)
Resistive Load Voltage	$V_R$	$V_R=I_{R}R$	V = J C-1	[M] [L]2 [T]-3 [I]-1
Capacitive Load Voltage	$V_C$	$V_C=I_C{X}_C$	V = J C-1	[M] [L]2 [T]-3 [I]-1
Inductive Load Voltage	$V_L$	$V_L=I_L{X}_L$	V = J C-1	[M] [L]2 [T]-3 [I]-1
Capacitive Reactance	$X_C$	$X_C=\frac{1}{{\omega }_{\mathrm{d}}C}$	Ω-1 m-1	[I]2 [T]3 [M]-2 [L]-2
Inductive Reactance	$X_L$	$X_L={\omega }_{d}L$	Ω-1 m-1	[I]2 [T]3 [M]-2 [L]-2
AC Impedance	$Z$	$V=IZ$ $Z=\sqrt{R^2-{(X_L-X_C)}^2}$	Ω-1 m-1	[I]2 [T]3 [M]-2 [L]-2
Phase Constant	$\varphi$	$\tan \varphi =\frac{X_L-X_C}{R}$	dimensionless	dimensionless
AC Circuit Resonant Pulsatance	${\omega }_{\mathrm{r}\mathrm{e}\mathrm{s}}$	${\omega }_{\mathrm{d}}={\omega }_{\mathrm{r}\mathrm{e}\mathrm{s}}=\omega =\frac{1}{\sqrt{LC}}$	s-1	[T]-1
AC Peak Current	$I_0$	$I_0=I_{\mathrm{r}\mathrm{m}\mathrm{s}}\sqrt{2}$	A	[I]
AC Root Mean Square Current	$I_{\mathrm{r}\mathrm{m}\mathrm{s}},\sqrt{⟨I⟩}$	$I_{\mathrm{r}\mathrm{m}\mathrm{s}}=\sqrt{\frac{1}{T}\underset{0}{\overset{T}{\int }}{\left[I\left(t\right)\right]}^2\mathrm{d}t}$	A	[I]
AC Peak Voltage	$V_0$	$V_0=V_{\mathrm{r}\mathrm{m}\mathrm{s}}\sqrt{2}$	V = J C-1	[M] [L]2 [T]-3 [I]-1
AC Root Mean Square Voltage	$V_{\mathrm{r}\mathrm{m}\mathrm{s}},\sqrt{⟨V⟩}$	$V_{\mathrm{r}\mathrm{m}\mathrm{s}}=\sqrt{\frac{1}{T}\underset{0}{\overset{T}{\int }}{\left[V\left(t\right)\right]}^2\mathrm{d}t}$	V = J C-1	[M] [L]2 [T]-3 [I]-1
AC emf, Root Mean Square	${ℰ}_{\mathrm{r}\mathrm{m}\mathrm{s}},\sqrt{⟨ℰ⟩}$	${ℰ}_{\mathrm{r}\mathrm{m}\mathrm{s}}={ℰ}_{\mathrm{m}}/\sqrt{2}$	V = J C-1	[M] [L]2 [T]-3 [I]-1
AC Average Power	$⟨P⟩$	$⟨P⟩=ℰI_{\mathrm{r}\mathrm{m}\mathrm{s}}\cos \varphi$	W = J s-1	[M] [L]2 [T]-3
Capacitive Time Constant	${\tau }_C$	${\tau }_C=RC$	s	[T]
Inductive Time Constant	${\tau }_L$	${\tau }_L=L/R$	s	[T]

RC Circuits	RC Circuit Equation $R{q}^{\prime }+C^{-1}q=ℰ$ RC Circuit Capacitor Charging $q=Cℰ(1-e^{-t/RC})$
RL Circuits	RL Circuit Equation $L{i}^{″}+R{i}^{\prime }=ℰ$ RL Circuit Current Rise $I=\frac{ℰ}{R}(1-e^{-t/{\tau }_L})$ RL Circuit, Current Fall $I=\frac{ℰ}{R}{e}^{-t/{\tau }_L}=I_0{e}^{-t/{\tau }_L}$
LC Circuit	LC Circuit Equation $L{q}^{″}+q/C=ℰ$ LC Circuit Resonance $\omega =1/\sqrt{LC}$ LC Circuit Charge $q=Q\cos (\omega t+\varphi )$ LC Circuit Current $I=-\omega Q\sin (\omega t+\varphi )$ LC Circuit electrical potential energy $U_E=q^2/2C=Q^2{\cos }^2(\omega t+\varphi )/2C$ LC circuit magnetic potential energy $U_B=Q^2{\sin }^2(\omega t+\varphi )/2C$
RLC Circuits	RLC Circuit Equation $L{q}^{″}+R{q}^{\prime }+C^{-1}q=ℰ$ RLC Circuit Charge $q=Qe{T}^{-Rt/2L}\cos ({\omega }^{\prime }t+\varphi )$