Circuit Theory/Transients Summary and Study guide

This cover the basics of transients, the analysis of circuit response that goes away after a long time.

RC or LC Circuits

General solution steps for RL and LC circuits with a voltage source (with out voltage source Vc=0):

1. Use KVL and KCL, get 1st order differential equation
2. Find particular solution (Forcing Function) $Y_p$ (Table is at bottom of page)
3. The complete solution is the particular + the complementary.

$y(x)=Y_p+y_c$

$y_c(x)=K_1+K_2{e}^{sx}$

1. Substitute solution into differential equation to find $K_1$ and s. (Or find $K_1$ by solving in steady state.)
2. Use the given initial conditions to find $K_2$
3. Write final solution

RLC Circuits

• DC circuits -> constant forcing functions
• AC circuits -> sinusoidal forcing functions
• Particular solution for VDc =>L-> SC, C-> OC

Concept	Formula	notes
Damping Coefficiant (series LC)	$\alpha =\frac{R}{2L}$
Damping Coefficiant (parallel LC)	$\alpha =\frac{1}{2RC}$
Undamped resonant frequency	${\omega }_0=\frac{1}{\sqrt{LC}}$
General Form	$f(t)=\frac{d^{2}i(t)}{d{t}^2}+2\alpha \frac{di(t)}{dt}+{\omega }_0^{2}i(t)$
Characteristic equation	$s^2+2\alpha s+{\omega }_0^2=0$
Roots Characteristic eqn	$s_{1,2}=-\alpha \pm \sqrt{{\alpha }^2-{\omega }_0^2}$
Damping ratio	$\zeta =\frac{\alpha }{{\omega }_0}$
Overdamped	$x_c(t)=K_1{e}^{s_{1}t}+K_2{e}^{s_{2}t}$	roots real and distinct $\zeta >1$ $\alpha >\omega$
Critically damped	$x_c(t)=K_1{e}^{s_{1}t}+K_{2}t{e}^{s_{1}t}$	roots real and equal $\zeta =1$ $\alpha =\omega$
Natural Frequency	${\omega }_n=\sqrt{{\omega }_0^2-{\alpha }^2}$
Underdamped	$x_c(t)=K_1{e}^{-\alpha t}\cos {\omega }_{n}t+K_2{e}^{-\alpha t}\sin {\omega }_{n}t$	roots complex $\zeta <1$ $\alpha <\omega$

Table of Forcing functions

Value	Approximation
Cons.	A
$e^t$	$K{e}^{st}$
sin(t)/cos(t)	$A\sin (pt)+B\cos (pt)$
$t^n$	$A{t}^n+B{t}^{n-1}+\mathrm{...}+Ct+D$
$t^n{e}^t$	$A{t}^n{e}^{pt}+B{t}^{n-1}{e}^{pt}+\mathrm{...}C{e}^{pt}$