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

${\mathcal {E}}\&{\mathcal {M}}$ (Vol. 2):    5:Electric Charges and Fields    6:Gauss's Law    7:Electric Potential    8:Capacitance    9:Current and Resistance    10:Direct-Current Circuits    11:Magnetic Forces and Fields    12:Sources of Magnetic Fields    13:Electromagnetic Induction    14:Inductance    15:Alternating-Current Circuits    16:Electromagnetic Waves

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Chapter 15

Alternating-Current Circuits

RLC circiut

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 ${\tfrac {V_{S}}{V_{P}}}={\tfrac {N_{S}}{N_{P}}}={\tfrac {I_{P}}{I_{S}}}$