ECE 311: Circuits II

Cheat Sheet

Impedance/Admittance

$Z_R = R$
$Z_L = j \omega L$
$Z_C = \frac{1}{j \omega C}$

$Y_R = \frac{1}{R}$
$Y_L = \frac{1}{j \omega L}$
$Y_C = j \omega C$

$Z = R + jX\Omega$
$Y = G + jB\Omega^{-1}$

$Y = \frac{1}{Z}$
$Z_1 || Z_2 \rightarrow Y_1 + Y_2$
$Y_1 || Y_2 \rightarrow Z_1 + Z_2$

Complex Numbers

$\bold{D} = a + jb = re^{j \phi} = r \angle \phi$
$a = rcos(\phi)$
$b = rsin(\phi)$
$r = \sqrt{a^2 + b^2}$
$\phi = arctan(b/a)$

Relationships

$V = IR$
$v(t) = i(t) * r(t)$
$v(t) = V_p cos(\omega t + \theta) = V_p e^{j(\omega t + \theta)} = \bold{V}e^{j \omega t}$
$\bold{V} = V_p e^{j\theta}$
$\bold{I} = I_p e^{j\phi}$

Root Mean Square (rms)

$V_{rms} \equiv \sqrt{ \frac{1}{T} \int^{t0 + T}{t0} v^2(t) dt }$
$V{rms} = \frac{V}{\sqrt{2}}$
$I_{rms} = \frac{I}{\sqrt{2}}$

Power

$p(t) = v(t) * i(t)$

$P = V{rms} * I{rms} cos(\theta - \phi) = V{rms} * I{rms} pf$
$P = \frac{V_p I_p}{2} cos(\theta - \phi) = \frac{V_p I_p}{2} pf$
$P = \frac{ | \bold{V} | ^2 }{2} \text{Re}{ \frac{1}{Z \text{*} ( \omega )} }$
$P = \frac{ | \bold{I} | ^2 }{2} \text{Re}{ Z( \omega ) }$

Complex Power: $\bold{S} = \bold{V}{rms} * \bold{I}{rms} \text{} = \frac{1}{2} \bold{V} \bold{I} \text{} = P + jQ$ VA
Apparent Power: $S = | \bold{S} | = | \bold{V}_{rms} | | \bold{I}{rms} \text{*} | = V{rms} I{rms}$ VA
Average Power: $P = \text{Re}{ \bold{V}{rms} \bold{I}{rms} \text{} } = \frac{1}{2} \text{Re}{ \bold{V} \bold{I} \text{*} }$ W
Reactive Power: $Q = \text{Im}{ \bold{V}{rms} \bold{I}_{rms} \text{} } = \frac{1}{2} \text{Im} { \bold{V} \bold{I} \text{} }$ VAR

Power Factor

$pf = \frac{P}{S} = cos(\theta - \phi)$

Compensating:
$C = \frac{L}{R^2 + \omega ^ 2 L^2}$

Three-Phase

Power

$P{ph} = V{ph}I{ph}pf$
$P{total} = 3V{ph}I{ph}pf$
$P_{\Delta} = 3 * P_Y$

Y-Source

$\bold{V}_a = V_p \angle 0 \degree$
$\bold{V}_b = V_p \angle -120 \degree$
$\bold{V}_c = V_p \angle +120 \degree$

$\bold{V}_{ab} = \bold{V}_a - \bold{V}b$
$\bold{V}{ab} = \sqrt{3} * V_p \angle 30 \degree$

$\bold{V}{ab} = \bold{V}{L} = \sqrt{3} \bold{V}_{ph} = \sqrt{3} \bold{V}_{an}$

Delta ($\Delta$) Source

$\bold{V}_{ab} = \bold{V}_a = Vp \angle 0 \degree$
$\bold{V}{ab} = -\bold{V}_c - \bold{V}_b$

Y Source, Delta Load

$\bold{I}{\Delta,AB} = \frac{\bold{V}{AB}}{Z\Delta}$
$\bold{I}{L} = \sqrt{3} * \bold{I}_{ph(source)}$

Delta-Y Conversion

$RA = \frac{R{AB} R{AC}}{R{AB} + R{AC} + R{BC}}$
$RB = \frac{R{AB} R{BC}}{R{AB} + R{AC} + R{BC}}$
$RC = \frac{R{AC} R{BC}}{R{AB} + R{AC} + R{BC}}$

Filters

Parallel RLC Circuits

$Z(\omega)=\frac{R}{\sqrt{1+Q_0^2 (\frac{\omega}{\omega_0} - \frac{\omega_0}{\omega})^2}}$

$Q_0 = R\sqrt{\frac{C}{L}}$
$\zeta = \frac{1}{2Q_0}$

Fourier Transforms

Fourier Transform Pairs

$$ \text{Function} \rightarrow \text{Fourier Transform} \rightarrow \text{Laplace Transform}\ f(t) \rightarrow F(j\omega) = F{f(t)} = \int{-\infty}^{\infty} f(t)e^{-j \omega t} dt \rightarrow F(s) = L{f(t)} = \int{0}^{\infty} f(t) e^{-st} dt\ \text{Implulse} \rightarrow \delta (t) \leftrightarrow 1 \rightarrow \delta \leftrightarrow 1\ \text{Constant} \rightarrow A \leftrightarrow 2\pi A\delta(\omega) \rightarrow A \leftrightarrow \frac{A}{s}\ \text{Cosine} \rightarrow cos(\omega_0 t) \leftrightarrow \pi[\delta(\omega + \omega_0) + \delta(\omega - \omega_0)] \rightarrow cos(\omega t) \leftrightarrow \frac{s}{s^2 + \omega_0^2}\ \text{Sine} \rightarrow sin(\omega_0 t) \leftrightarrow j\pi[\delta(\omega + \omega_0) + \delta(\omega - \omega_0)] \rightarrow sin(\omega t) \leftrightarrow \frac{\omega_0}{s^2 + \omega_0^2} $$