EE 108 Midterm 1 Notes

$S = V I^{*} \space \text{(current conjugate)}$
$P=S \cdot cos(\theta)$ → $S = \frac{P}{cos(\theta)}$
$Q = S \cdot sin(\theta) = P \cdot tan(\theta)$

$\text{Power Factor} = cos(\theta) = \frac{P}{S}$

Work

$w_p=mgh \space \text{[J]}$

Ohm’s Law

$V=IR \hspace{15pt} I=\frac{V}{R}$

Power

$P=\frac{w_p}{t} \space \text{[W, kW]}$
$P = I^2R$
$P=\frac{v^2}{R}$ (v = voltage drop)

$1 W = 1 \frac{J}{s}$ (1 Watt equals 1 Joule/s)
$1kWh = 3.6 \cross 10^6 J$

Efficient Motor (80%)

$\frac{w_p}{10 sec} = \frac{10,000}{10} = 1000W = P_{mechanical}$
$1000W = 0.8 = P_{electrical}$
$P_E = \frac{1000}{0.8} = 1.25 kW$

$E_c = \frac{1}{2}CV^2 \hspace{30pt} E_L = \frac{1}{2}Li^2$

Voltage Divider

$\fbox{$ V_{out} = V_{in} \frac{R_2}{R_1 + R_2} $ }$

Parallel Resistance

$R_{tot} = R_1 || R_2$
$\frac{1}{R_{tot}} = \frac{1}{R_1} + \frac{1}{R_2}$
$R_{tot} = (\frac{1}{R_1} + \frac{1}{R_2})^{-1}$
$\fbox{$ R_{tot} = \frac{R_1R_2}{R_1 + R_2} $ }$

$P_{AC} = \frac{V^2_{AC}(t)}{R} \hspace{10pt}$ ¹
$P_{R} = \frac{V^2_{s}}{2R} \hspace{10pt}$ ²

$\frac{V_{DC}^2}{R} = \frac{V_{AC}^2}{2R}$
$V_{AC} =\sqrt{2} V_{DC}$

$\fbox{$ V_{rms} = \frac{V_{pk}}{\sqrt{2}} $ }$

$\fbox{$ V_{pk} = \sqrt{2} V_{rms} $ }$

Top: voltage waveform
Bottom: AC Power waveform

Instantaneous Power pulsates at 2w. But we care about average power. ↩︎
(AC power is half of DC power under the same peak voltage) ↩︎

ε = permittivity of free space $(8.854 \cross 10^{-12})$
A = surface area $(m^2)$
d = distance $(m)$

$C = \frac{\epsilon{} A}{d} \space [F] \hspace{25pt} C = \frac{Q}{V}$
$Q = CV$

$\frac{dQ}{dt} = C \frac{dv}{dt}$
$I = C \frac{dv}{dt}$

(current leads voltage by 90°)