Exam 1

Thevenin

$R_t=\frac{v_{oc}}{i_{sc}}$

RL Circuits

Time Constants

$\tau =RC$ is the time constant of an RC circuit and it is the time it takes the voltage magnitude to drop to $36.8\%$ of its initial value

$\tau =\frac{L}{R}$ is the time constant of an RL circuit and it is the time it takes the current magnitude to drop to $36.8\%$ of its initial value

General Charging Formulas

Capacitor

$v_c(t)=V_s+(v_c(0^{+})-V_s)e^{\frac{-1}{RC}t}$

Inductor

$i_L(t)=\frac{V_s}{R}+(i_L(0^{-})-\frac{V_s}{R})e^{\frac{-R}{L}t}$

RLC Circuits

Parallel Damping: $\iota =\frac{\frac{1}{2RC}}{\frac{1}{\sqrt{LC}}}$

Series Damping: $\iota =\frac{\frac{R}{2L}}{\frac{1}{\sqrt{LC}}}$

Undamped Resonant Frequency for Both: ${\omega }_0=\frac{1}{\sqrt{LC}}$

Damped Resonant Frequency (use in imaginary case): ${\omega }_d=\sqrt{\frac{1}{LC}-(\frac{R}{2L}{)}^2}$ or $\sqrt{{\omega }_0^2-{\alpha }^2}$

Damping factor/coefficient series: $\alpha =\frac{R}{2L}$

Damping factor/coefficient parallel: $\alpha =\frac{1}{2RC}$

Term	RLC Series	RLC Parallel
$\alpha$	$\frac{R}{2L}$	$\alpha =\frac{1}{2RC}$
${\omega }_0$ - Undamped Resonant Frequency	$\frac{1}{\sqrt{LC}}$	$\frac{1}{\sqrt{LC}}$
$\iota$	$\frac{\alpha }{{\omega }_0}=\frac{\frac{R}{2L}}{\frac{1}{\sqrt{LC}}}$	$\frac{\alpha }{{\omega }_0}=\frac{\frac{1}{2RC}}{\frac{1}{\sqrt{LC}}}$

Term	Equation
$x(t)$ - Overdamped	$x_s(\infty )+e^{-\alpha t}(K_1{e}^{t\sqrt{{\alpha }^2-{\omega }_0^2}}+K_2{e}^{-t\sqrt{{\alpha }^2-{\omega }_0^2}})$
$x(t)$ - Critically Damped	$x_s(\infty )+e^{-\alpha t}(K_{1}t+K_2)$
$x(t)$ - Underdamped Damped	$x_s(\infty )+e^{-\alpha t}(K_1\cos ({\omega }_{d}t)+K_2\sin ({\omega }_{d}t))$

$\frac{d{v}_c(0^{+})}{dt}$ for a pure parallel circuit with just current source, resistor, inductor and capacitor all in parallel: $\frac{d{v}_c(0^{+})}{dt}=\frac{1}{C}(I_S(0^{+})-\frac{v_c(0^{-})}{R}-i_L(0^{-}))$

$\frac{d{i}_L(0^{+})}{dt}$ for a pure parallel circuit with just current source, resistor, inductor and capacitor all in parallel: $\frac{d{i}_L(0^{+})}{dt}=\frac{1}{L}{v}_c(0^{-})$

$\frac{d{i}_l(0^{+})}{dt}$ for a pure series circuit with just current source, resistor, inductor and capacitor all in series: $\frac{d{i}_l(0^{+})}{dt}=\frac{1}{L}(V_s(0^{+})-I_L(0^{-})R-V_C(0^{-}))$