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	$V_{\mathrm {ter} }\,\!$		V = J C^-1	[M] [L]² [T]^-3 [I]^-1
Load Voltage for Circuit	$V_{\mathrm {load} }\,\!$		V = J C^-1	[M] [L]² [T]^-3 [I]^-1
Internal Resistance of Power Supply	$R_{\mathrm {int} }\,\!$	$R_{\mathrm {int} }={\frac {V_{\mathrm {ter} }}{I}}\,\!$	Ω = V A^-1 = J s C^-2	[M][L]² [T]^-3 [I]^-2
Load Resistance of Circuit	$R_{\mathrm {ext} }\,\!$	$R_{\mathrm {ext} }={\frac {V_{\mathrm {load} }}{I}}\,\!$	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	${\mathcal {E}}\,\!$	${\mathcal {E}}=V_{\mathrm {ter} }+V_{\mathrm {load} }\,\!$	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	${\boldsymbol {\sigma }},\sigma _{\mathrm {ij} }\,\!$	$\sigma _{\mathrm {ij} }{\begin{pmatrix}\ \sigma _{11}&\sigma _{12}&\sigma _{13}\\\ \sigma _{21}&\sigma _{22}&\sigma _{23}\\\ \sigma _{31}&\sigma _{32}&\sigma _{33}\end{pmatrix}}\,\!$	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	$P\,\!$	$P_{\mathrm {emf} }=I{\mathcal {E}}\,\!$	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	$R_{\mathrm {net} }=\sum _{i}R_{i}\,\!$
Resistors in Parallel	${\frac {1}{R_{\mathrm {net} }}}=\sum _{i}{\frac {1}{R_{i}}}\,\!$

Ohm's Law	Scalar form $V=IR\,\!$ Vector Form $\mathbf {J} =\sigma \mathbf {E} \,\!$ Tensor Form, general applies to all points in a conductor $\mathbf {J} _{i}=\sigma _{ij}\mathbf {E} _{j}\,\!$
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 $I_{\mathrm {in} }=I_{\mathrm {out} }\,\!$

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	$Z\,\!$	$V=IZ\,\!$ $Z={\sqrt {R^{2}-\left(X_{L}-X_{C}\right)^{2}}}\,\!$	Ω^-1 m^-1	[I]² [T]³ [M]^-2 [L]^-2
Phase Constant	$\phi \,\!$	$\tan \phi ={\frac {X_{L}-X_{C}}{R}}\,\!$	dimensionless	dimensionless
AC Circuit Resonant Pulsatance	$\omega _{\mathrm {res} }\,\!$	$\omega _{\mathrm {d} }=\omega _{\mathrm {res} }=\omega ={\frac {1}{\sqrt {LC}}}\,\!$	s^-1	[T]^-1
AC Peak Current	$I_{0}\,\!$	$I_{0}=I_{\mathrm {rms} }{\sqrt {2}}\,\!$	A	[I]
AC Root Mean Square Current	$I_{\mathrm {rms} },{\sqrt {\langle I\rangle }}\,\!$	$I_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}\left[I\left(t\right)\right]^{2}\mathrm {d} t}}\,\!$	A	[I]
AC Peak Voltage	$V_{0}\,\!$	$V_{0}=V_{\mathrm {rms} }{\sqrt {2}}\,\!$	V = J C^-1	[M] [L]² [T]^-3 [I]^-1
AC Root Mean Square Voltage	$V_{\mathrm {rms} },{\sqrt {\langle V\rangle }}\,\!$	$V_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}\left[V\left(t\right)\right]^{2}\mathrm {d} t}}\,\!$	V = J C^-1	[M] [L]² [T]^-3 [I]^-1
AC emf, Root Mean Square	${\mathcal {E}}_{\mathrm {rms} },{\sqrt {\langle {\mathcal {E}}\rangle }}\,\!$	${\mathcal {E}}_{\mathrm {rms} }={\mathcal {E}}_{\mathrm {m} }/{\sqrt {2}}\,\!$	V = J C^-1	[M] [L]² [T]^-3 [I]^-1
AC Average Power	$\langle P\rangle \,\!$	$\langle P\rangle ={\mathcal {E}}I_{\mathrm {rms} }\cos \phi \,\!$	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 $Rq'+C^{-1}q={\mathcal {E}}\,\!$ RC Circuit Capacitor Charging $q=C{\mathcal {E}}(1-e^{-t/RC})\,\!$
RL Circuits	RL Circuit Equation $Li''+Ri'={\mathcal {E}}\,\!$ RL Circuit Current Rise $I={\frac {\mathcal {E}}{R}}\left(1-e^{-t/\tau _{L}}\right)\,\!$ RL Circuit, Current Fall $I={\frac {\mathcal {E}}{R}}e^{-t/\tau _{L}}=I_{0}e^{-t/\tau _{L}}\,\!$
LC Circuit	LC Circuit Equation $Lq''+q/C={\mathcal {E}}\,\!$ LC Circuit Resonance $\omega =1/{\sqrt {LC}}\,\!$ LC Circuit Charge $q=Q\cos(\omega t+\phi )\,\!$ LC Circuit Current $I=-\omega Q\sin(\omega t+\phi )\,\!$ LC Circuit electrical potential energy $U_{E}=q^{2}/2C=Q^{2}\cos ^{2}(\omega t+\phi )/2C\,\!$ LC circuit magnetic potential energy $U_{B}=Q^{2}\sin ^{2}(\omega t+\phi )/2C\,\!$
RLC Circuits	RLC Circuit Equation $Lq''+Rq'+C^{-1}q={\mathcal {E}}\,\!$ RLC Circuit Charge $q=QeT^{-Rt/2L}\cos(\omega 't+\phi )\,\!$