Circuit Fundamentals

Name	Formula	Related Formulae	Variables/Constants
Electric current	$I=\frac{dQ}{dt}$	$I=nq{v}_{d}A,J=\frac{I}{A}$	$I:current$ $Q:charge$ $t:time$ $n:carrierdensity$ $q:carriercharge$ $v_d:driftvelocity$ $A:cross-sectionalarea$ $J:currentdensity$
Ohm’s law	$V=IR$	$R=\frac{V}{I},G=\frac{1}{R},J=\sigma E$	$V:voltage$ $I:current$ $R:resistance$ $G:conductance$ $J:currentdensity$ $\sigma :conductivity$ $E:electricfield$
Resistance of a conductor	$R=\frac{\rho L}{A}$	$\rho =\frac{1}{\sigma },R_s=\frac{\rho }{t}$	$\rho :resistivity$ $L:length$ $A:cross-sectionalarea$ $\sigma :conductivity$ $R_s:sheetresistance$ $t:filmthickness$
Resistivity of a semiconductor	$\rho =\frac{1}{q\left(n{\mu }_n+p{\mu }_p\right)}$	$v_d=\mu E$	$q:elementarycharge$ $n,p:electronandholeconcentrations$ ${\mu }_n,{\mu }_p:electronandholemobilities$ $v_d:driftvelocity$ $E:electricfield$
Temperature dependence of resistance	$R=R_0\left[1+\alpha \left(T-T_0\right)\right]$	$\rho ={\rho }_0\left[1+\alpha \left(T-T_0\right)\right]$	$R_0,{\rho }_0:valuesatreferencetemperature{T}_0$ $\alpha :temperaturecoefficientofresistance$ $T:temperature$
Electrical power	$P=VI=I^{2}R=\frac{V^2}{R}$	$W=Pt$	$P:power$ $V:voltage$ $I:current$ $R:resistance$ $W:energy$ $t:time$
Kirchhoff’s current law	$\sum I_{in}=\sum I_{out}$	${\sum }_k{I}_k=0(\text{at a node})$	$I_{in},I_{out}:currentsenteringandleavinganode$
Kirchhoff’s voltage law	${\sum }_{loop}{V}_k=0$	$V_{terminal}=\epsilon -Ir$	$V_k:voltagedropsaroundaclosedloop$ $\epsilon :sourceemf$ $r:internalresistance$
Series and parallel resistors	$R_{eq}=R_1+R_2+\dots (\text{series})$	$\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\dots (\text{parallel})$	$R_{eq}:equivalentresistance$ $R_1,R_2:individualresistances$
Voltage and current dividers	$V_2=V_{in}\frac{R_2}{R_1+R_2}$	$I_2=I_{in}\frac{R_1}{R_1+R_2}$	$V_{in}:inputvoltage$ $V_2:voltageacross{R}_2$ $I_{in}:totalcurrent$ $I_2:currentthrough{R}_2$
Thevenin and Norton equivalents	$V_{th}=V_{oc},R_{th}=\frac{V_{oc}}{I_{sc}}$	$I_N=\frac{V_{th}}{R_{th}},P_{L,max}=\frac{V_{th}^2}{4R_{th}}$	$V_{th}:Theveninvoltage$ $V_{oc}:open-circuitvoltage$ $I_{sc}:short-circuitcurrent$ $R_{th}:Theveninresistance$ $I_N:Nortoncurrent$ $\begin{array}{r}{P}_{L,max}:maximumpowerdeliveredtoaload{R}_L=\\ R_{th}\end{array}$
RC charging and discharging	$v_C\left(t\right)=V_s\left(1-e^{-t/\tau }\right),\tau =RC$	$v_C\left(t\right)=V_0{e}^{-t/\tau }(\text{discharge})$	$v_C:capacitorvoltage$ $V_s:sourcevoltage$ $V_0:initialvoltage$ $\tau :timeconstant$ $R:resistance$ $C:capacitance$

Important Questions

Ques: Is Ohm’s law R = VI or V = IR?

It is $V=IR$, so the resistance is $R=V/I$, the voltage needed per ampere of current. The product $VI$ is a different quantity, the power in watts. A unit check catches the slip: an ohm is a volt per ampere, while a volt times an ampere is a watt. A simple way to remember all three forms is the triangle with V on top and I and R below; covering the wanted quantity leaves the formula for it. For example, 5 V across 1 k$\Omega$ drives 5 mA and dissipates 25 mW (computed). Ohm’s law describes ohmic materials, whose resistance does not depend on the applied voltage; diodes and transistors are not ohmic, which is why they are described by their own current-voltage equations. 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: How is the resistivity of a semiconductor related to its doping?

Through the carrier concentrations and mobilities, $\rho =1/[q(n{\mu }_n+p{\mu }_p)]$. In a doped semiconductor one carrier type dominates, so for n-type material $\rho \approx 1/(q{N}_D{\mu }_n)$. Silicon doped with $1\times 10^{16}$ donors per cubic centimetre, with an electron mobility of about 1200 cm^2/(V s), has a resistivity of about 0.52 $\Omega$ cm (computed). Mobility falls at high doping because of ionized-impurity scattering, so resistivity drops more slowly than the doping rises. Metals behave differently: their carrier density is fixed and huge, and their resistance rises with temperature through lattice scattering, while an intrinsic semiconductor’s resistance falls with temperature because many more carriers are created. This relation is also how measured resistivity is converted back into doping concentration. Reference: B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed. (Pearson, 2015); S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007).

Ques: How can the series and parallel rules for R, L and C be remembered?

Resistors and inductors combine the same way; capacitors combine the opposite way. Resistances add in series and add as reciprocals in parallel. Capacitances add in parallel and add as reciprocals in series, because parallel plates simply add area, while capacitors in series share the same charge and their voltages add. Uncoupled inductors follow the resistor pattern. A single rule covers all three: impedances add in series, and the impedance of a capacitor is $1/(j\omega C)$, so its capacitance combines inversely. For example, 1 and 2 k$\Omega$ give 3 k$\Omega$ in series and 667 $\Omega$ in parallel, while 1 and 2 $\mu$F give 3 $\mu$F in parallel and 0.667 $\mu$F in series (computed). 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 does the RC time constant mean physically?

It is the time the capacitor takes to cover about 63 per cent of the way to its final voltage. The charging voltage is $V_s(1-e^{-t/\tau })$ with $\tau =RC$, so after one time constant it reaches $1-e^{-1}=63.2$ per cent and after five 99.3 per cent (computed), which is why five time constants are taken as fully settled. At the start an uncharged capacitor behaves like a short circuit and the current is $V_s/R$; at the end it behaves like an open circuit and the current is zero. A 1 k$\Omega$ resistor with a 1 $\mu$F capacitor gives $\tau =1$ ms (computed). An inductor in series with a resistor follows the same exponential with $\tau =L/R$. Reference: C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 6th ed. (McGraw-Hill Education, 2017); D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 10th ed. (Wiley, 2014).

References

1. W. H. Hayt Jr., J. E. Kemmerly and S. M. Durbin, Engineering Circuit Analysis, 8th ed. New York: McGraw-Hill, 2012.
2. C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 6th ed. New York: McGraw-Hill Education, 2017.
3. B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed. Boston: Pearson, 2015.
4. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. Hoboken, NJ: Wiley, 2007.
5. D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 10th ed. Hoboken, NJ: Wiley, 2014.