AC, power & resonance formulas

Any periodic waveform; integrate over any one full period.

Any periodic waveform. Square first, then average, then take the root.

$v=V_m\sin (\omega t+\theta )$. The full-cycle average is zero; the rms value does not depend on the phase $\theta$.

$V_{\text{avg}}$ is the half-cycle average for a waveform whose full-cycle average is zero; $V_{\text{peak}}$ is the largest absolute value. For a sine, $k_f=\pi /(2\sqrt{2})\approx 1.11$ and $k_c=\sqrt{2}\approx 1.414$.

Only when the components have different frequencies (dc counts as frequency zero): products of different frequencies average to zero over the common period. Add same-frequency terms as phasors first.

Inductor: voltage leads current by $90^{\circ }$. Capacitor: current leads voltage by $90^{\circ }$. Resistor: in phase.

$X>0$: inductive, current lags. $X<0$: capacitive, current leads. $\varphi$ is the angle by which voltage leads current. The arctangent form assumes $R>0$ (a passive impedance); for $R=0$, $\varphi =+90^{\circ }$ or $-90^{\circ }$ with the sign of $X$.

$G=1/R$ only when $X=0$. Susceptances: capacitor $B_C=\omega C>0$, inductor $B_L=-1/(\omega L)<0$.

Same as dc, with complex numbers: the voltage divider is for elements in series, the current divider for two parallel branches. Magnitudes add only when the phasors are in phase.

rms values. $P$ is the average of $p(t)$. $Q$ is the amplitude of the part of $p(t)$ that flows back and forth between the source and the fields of L and C, with zero average.

rms phasors, power absorbed by the load. With peak phasors, $S=\frac{1}{2}{V}_m{I}_m^{\ast }$. The conjugate goes on the current; writing $V^{\ast }I$ instead flips the sign of $Q$.

$X_L=\omega L$ and $X_C=1/(\omega C)$ are positive magnitudes. An inductor absorbs reactive power, a capacitor delivers it, and neither absorbs average power.

Lagging when current lags voltage ($\varphi >0$, $Q>0$, inductive); leading when current leads ($\varphi <0$, $Q<0$, capacitive); unity at $\varphi =0$. Always state lagging or leading with the number.

Every element taken with the passive sign convention. $P$ and $Q$ of loads add whether they are in series or parallel; apparent powers $|S|$ do NOT add in general (only when every load has the same power-factor angle).

${\varphi }_1$, ${\varphi }_2$ = pf angles before and after (lagging taken positive), $|V|$ = rms voltage across the capacitor, $P$ in W. For a final leading pf take ${\varphi }_2$ negative.

$V_n$, $I_n$ = rms values of the $n$-th harmonic; ${\varphi }_n$ = angle by which $V_n$ leads $I_n$. A harmonic present in only one of $v$, $i$ contributes nothing. Power factor $=P/(V_{\text{rms}}{I}_{\text{rms}})$ using the full rms values. If $v$ is a pure sinusoid and only $i$ is distorted, this becomes $\text{pf}=(I_1/I_{\text{rms}})\cos {\varphi }_1$: distortion factor times displacement factor.

$V_{Th}$ rms. Start every restricted-load problem from this expression.

rms $V_{Th}$ (with a peak value $V_{Th,m}$ it is $V_{Th,m}^2/(8R_{Th})$). The Thevenin source then delivers twice this power, so the efficiency measured against the Thevenin equivalent is 50%.

When the load cannot have reactance. Gives less power than the conjugate match unless $X_{Th}=0$.

Ideal series RLC and ideal parallel RLC (frequency-independent R). The value of R does not shift it.

$R$ is the total series resistance, including the coil's own resistance and the source's internal resistance.

The L and C voltages are $180^{\circ }$ apart and cancel, but each is $Q$ times the supply voltage.

Constant-amplitude voltage source. At ${\omega }_1$ and ${\omega }_2$: $|I|=I_{\max }/\sqrt{2}$, $P=P_{\max }/2$, $|Z|=\sqrt{2}R$, impedance angle $-45^{\circ }$ at ${\omega }_1$ (capacitive) and $+45^{\circ }$ at ${\omega }_2$ (inductive). ${\omega }_0$ is their exact geometric mean.

Exact for any $Q$. For high $Q$ (roughly $Q\ge 10$), ${\omega }_{1,2}\approx {\omega }_0\mp \text{BW}/2$; the arithmetic symmetry about ${\omega }_0$ is only approximate, the geometric one is exact.

Ideal parallel RLC. Current magnification: at ${\omega }_0$, $|I_L|=|I_C|=Q|I_{\text{source}}|$. A current source with internal resistance $R_s$ puts $R_s$ in parallel with $R$: use $R\parallel R_s$ in $Q$ and BW.

A series $R_s+j{X}_s$ and a parallel combination of $R_p$ and $j{X}_p$ have the same impedance at the frequency where the values are computed, and only there. For $Q\gg 1$, $R_p\approx Q^2{R}_s$ and $X_p\approx X_s$, which is where the tank result $Z_D\approx Q^{2}R$ below comes from.

${\omega }_0$ = frequency of unity pf; $Z_D$ = dynamic impedance (purely resistive) at that frequency. A real ${\omega }_0$ exists only if $R<\sqrt{L/C}$. Exactly, $Z_D=R(1+Q^2)$ with the coil $Q={\omega }_{0}L/R$ taken at this ${\omega }_0$; for $Q\gg 1$, ${\omega }_0\approx 1/\sqrt{LC}$ and $Z_D\approx Q^{2}R$.

At ${\omega }_c$: $|H|=1/\sqrt{2}$ ($-3$ dB), phase $-45^{\circ }$. Asymptotic roll-off $-20$ dB/decade above ${\omega }_c$.

At ${\omega }_c$: $|H|=1/\sqrt{2}$, phase $+45^{\circ }$. Asymptotic slope $+20$ dB/decade below ${\omega }_c$.

Match the denominator to read ${\omega }_0$ and $Q$ directly: series RLC has ${\omega }_0/Q=R/L$, parallel RLC has ${\omega }_0/Q=1/(RC)$. Damping ratio $\zeta =1/(2Q)$. The band-pass and band-stop bandwidth is ${\omega }_0/Q$. Asymptotic slopes: low-pass $-40$ dB/decade above ${\omega }_0$, high-pass $+40$ dB/decade below it, band-pass $\pm 20$ dB/decade on its two skirts.