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

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 $q=QeT^{-Rt/2L}\cos(\omega 't+\phi )\,\!$ 