AC Circuits and Impedance

Name	Formula	Related Formulae	Variables/Constants
Sinusoidal voltage and current	$v\left(t\right)=V_m\sin \left(\omega t\right),\omega =2\pi f=\frac{2\pi }{T}$	$i\left(t\right)=I_m\sin \left(\omega t-\varphi \right)$	$v,i:instantaneousvoltageandcurrent$ $V_m,I_m:peakamplitudes$ $\omega :angularfrequency$ $f:frequency$ $T:period$ $\varphi :phaseangle$
RMS values	$V_{rms}=\frac{V_m}{\sqrt{2}},I_{rms}=\frac{I_m}{\sqrt{2}}$	$V_{rms}=\sqrt{\frac{1}{T}{\int }_0^T{v}^{2}dt}$	$V_{rms},I_{rms}:root-mean-squarevalues(sinusoids)$ $V_m,I_m:peakvalues$ $T:period$
Reactance	$X_L=\omega L,X_C=\frac{1}{\omega C}$	$V_m=I_{m}X$	$X_L:inductivereactance$ $X_C:capacitivereactance$ $\omega :angularfrequency$ $L:inductance$ $C:capacitance$
Impedance	$Z_R=R,Z_L=j\omega L,Z_C=\frac{1}{j\omega C}$	$\mathbf{V}=\mathbf{I}Z(\text{phasor Ohm’s law})$	$Z:impedance$ $R:resistance$ $j:imaginaryunit$ $\mathbf{V},\mathbf{I}:voltageandcurrentphasors$
Series RLC impedance	$Z=R+j\left(X_L-X_C\right),|Z|=\sqrt{R^2+{\left(X_L-X_C\right)}^2}$	$\varphi ={\tan }^{-1}\frac{X_L-X_C}{R}$	$|Z|:impedancemagnitude$ $X_L,X_C:reactances$ $\varphi :phaseofvoltagerelativetocurrent$
Resonance	${\omega }_0=\frac{1}{\sqrt{LC}},f_0=\frac{1}{2\pi \sqrt{LC}}$	$Q=\frac{{\omega }_{0}L}{R}=\frac{1}{R}\sqrt{\frac{L}{C}}(\text{series}),BW=\frac{{\omega }_0}{Q}$	${\omega }_0,f_0:resonantangularfrequencyandfrequency$ $Q:qualityfactor$ $BW:bandwidthbetweenhalf-powerpoints$
AC power	$P=V_{rms}{I}_{rms}\cos \varphi$	$Q_r=V_{rms}{I}_{rms}\sin \varphi ,S=V_{rms}{I}_{rms},S^2=P^2+Q_r^2$	$P:real(average)power$ $Q_r:reactivepower$ $S:apparentpower$ $\cos \varphi :powerfactor$
Ideal transformer	$\frac{V_2}{V_1}=\frac{N_2}{N_1}$	$\frac{I_2}{I_1}=\frac{N_1}{N_2},Z_{in}={\left(\frac{N_1}{N_2}\right)}^2{Z}_L$	$V_1,V_2:primaryandsecondaryvoltages$ $N_1,N_2:turns$ $I_1,I_2:currents$ $Z_{in}:impedanceseenattheprimary$ $Z_L:loadimpedance$

Important Questions

Ques: Why is RMS used, and does the square root of two apply to every waveform?

RMS gives the steady DC value that would deliver the same power to a resistor, so AC power can be computed with DC formulas: $P=V_{rms}^2/R$. For a sine wave the mean of ${\sin }^2$ over a cycle is one half, giving $V_{rms}=V_m/\sqrt{2}=0.707V_m$. Other waveforms have other factors: a triangle wave has $V_m/\sqrt{3}$, 0.577 of the peak, and a square wave equals its peak (computed). Mains voltages are quoted as RMS values, so a 230 V supply peaks at about 325 V (computed), which sets the voltage rating of components connected to it. Meters marked true RMS compute the definition directly, while simpler meters assume a sine wave. Reference: C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 6th ed. (McGraw-Hill Education, 2017); W. H. Hayt Jr., J. E. Kemmerly and S. M. Durbin, Engineering Circuit Analysis, 8th ed. (McGraw-Hill, 2012).

Ques: How do inductors and capacitors behave from DC to high frequency?

In opposite ways. Inductive reactance $\omega L$ is zero at DC and grows with frequency, so an inductor passes DC and blocks high frequencies; capacitive reactance $1/(\omega C)$ is infinite at DC and falls with frequency, so a capacitor blocks DC and passes high frequencies. At 1 GHz, 1 $\mu$H already presents about 6.3 k$\Omega$, while 1 pF presents only about 159 $\Omega$ (computed). The phases are opposite as well, captured by the mnemonic ELI the ICE man: in an inductor (L) the voltage (E) leads the current (I), and in a capacitor (C) the current leads the voltage, each by 90 degrees. Reference: W. H. Hayt Jr., J. E. Kemmerly and S. M. Durbin, Engineering Circuit Analysis, 8th ed. (McGraw-Hill, 2012); C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 6th ed. (McGraw-Hill Education, 2017).

Ques: What happens at series resonance?

The inductive and capacitive reactances cancel. The impedance $R+j(X_L-X_C)$ is then purely resistive and at its minimum, so the current is largest and in phase with the source voltage. This happens at ${\omega }_0=1/\sqrt{LC}$; for 10 $\mu$H and 100 pF, at about 5.03 MHz (computed). The voltages across L and C are each Q times the source voltage but opposite in phase, so with $R=10$ $\Omega$, $Q={\omega }_{0}L/R=31.6$ and a 1 V source produces about 32 V across each element (computed). The bandwidth is $f_0/Q$, about 159 kHz here (computed). Below resonance the series circuit is capacitive, and above it inductive. Tuned circuits, filters and oscillators all rely on this selectivity. Reference: C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 6th ed. (McGraw-Hill Education, 2017); W. H. Hayt Jr., J. E. Kemmerly and S. M. Durbin, Engineering Circuit Analysis, 8th ed. (McGraw-Hill, 2012).

Ques: What is power factor, and why does it matter?

It is the fraction of the apparent power that is delivered as real power, $\cos \varphi$, where $\varphi$ is the phase between voltage and current. When they are out of phase, part of the current only moves energy back and forth between the source and the inductances or capacitances of the load each cycle; it delivers no net power but still heats the wires. A load drawing 10 A at 230 V with a power factor of 0.8 receives 1840 W of real power from 2300 VA of apparent power (computed). Utilities therefore charge for poor power factor, and motors and power supplies use correction capacitors or active circuits to bring it close to 1. Reference: C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 6th ed. (McGraw-Hill Education, 2017); J. D. Glover, T. J. Overbye and M. S. Sarma, Power System Analysis and Design, 6th ed. (Cengage Learning, 2017).

References

1. C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 6th ed. New York: McGraw-Hill Education, 2017.
2. W. H. Hayt Jr., J. E. Kemmerly and S. M. Durbin, Engineering Circuit Analysis, 8th ed. New York: McGraw-Hill, 2012.
3. J. D. Glover, T. J. Overbye and M. S. Sarma, Power System Analysis and Design, 6th ed. Boston: Cengage Learning, 2017.