# OpenStax University Physics/E&M/Alternating-Current Circuits

## Chapter 15

#### Alternating-Current Circuits

AC voltage and current $v=V_{0}\sin(\omega t-\phi )$  if $i=I_{0}\sin \omega t.$
▭ RMS values $I_{rms}={\tfrac {I_{0}}{\sqrt {2}}}$  and $V_{rms}={\tfrac {V_{0}}{\sqrt {2}}}$  ▭ Impedance $V_{0}=I_{0}X$

▭ Resistor $V_{0}=I_{0}X_{R},\;\phi =0,$  where $X_{R}=R$
▭ Capacitor $V_{0}=I_{0}X_{C},\;\phi =-{\tfrac {\pi }{2}},$  where $X_{C}={\tfrac {1}{\omega C}}$  ▭ Inductor $V_{0}=I_{0}X_{L},\;\phi =+{\tfrac {\pi }{2}},$  where $X_{L}=\omega L$
▭ RLC series circuit $V_{0}=I_{0}Z$  where $Z={\sqrt {R^{2}+\left(X_{L}-X_{C}\right)^{2}}}$  and $\phi =\tan ^{-1}{\frac {X_{L}-X_{C}}{R}}$
▭ Resonant angular frequency $\omega _{0}={\sqrt {\tfrac {1}{LC}}}$  ▭ Quality factor $Q={\tfrac {\omega _{0}}{\Delta \omega }}={\tfrac {\omega _{0}L}{R}}$
▭ Average power $P_{ave}={\frac {1}{2}}I_{0}V_{0}\cos \phi =I_{rms}V_{rms}\cos \phi$ , where $\phi =0$  for a resistor.
▭ Transformer voltages and currents ${\tfrac {V_{S}}{V_{P}}}={\tfrac {N_{S}}{N_{P}}}={\tfrac {I_{P}}{I_{S}}}$

#### For quiz at QB/d_cp2.15

AC voltage and current $v=V_{0}\sin(\omega t-\phi )$  if $i=I_{0}\sin \omega t.$
RMS values $I_{rms}={\tfrac {I_{0}}{\sqrt {2}}}$  and $V_{rms}={\tfrac {V_{0}}{\sqrt {2}}}$
Impedance $V_{0}=I_{0}X$
Resistor $V_{0}=I_{0}X_{R},\;\phi =0,$  where $X_{R}=R$
Capacitor $V_{0}=I_{0}X_{C},\;\phi =-{\tfrac {\pi }{2}},$  where $X_{C}={\tfrac {1}{\omega C}}$
Inductor $V_{0}=I_{0}X_{L},\;\phi =+{\tfrac {\pi }{2}},$  where $X_{L}=\omega L$
RLC series circuit $V_{0}=I_{0}Z$  where $Z={\sqrt {R^{2}+\left(X_{L}-X_{C}\right)^{2}}}$  and $\phi =\tan ^{-1}{\frac {X_{L}-X_{C}}{R}}$
Resonant angular frequency $\omega _{0}={\sqrt {\tfrac {1}{LC}}}$
Quality factor $Q={\tfrac {\omega _{0}}{\Delta \omega }}={\tfrac {\omega _{0}L}{R}}$
Average power $P_{ave}={\frac {1}{2}}I_{0}V_{0}\cos \phi =I_{rms}V_{rms}\cos \phi$
Transformer voltages and currents ${\tfrac {V_{S}}{V_{P}}}={\tfrac {N_{S}}{N_{P}}}={\tfrac {I_{P}}{I_{S}}}$