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)	$\alpha {=}{R \over 2L}$
Damping Coefficiant (parallel LC)	$\alpha {=}{1 \over 2RC}$
Undamped resonant frequency	$\omega _{0}{=}{1 \over {\sqrt {LC}}}$
General Form	$f(t){=}{d^{2}i(t) \over dt^{2}}+2\alpha {di(t) \over dt}+\omega _{0}^{2}i(t)$
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	$\zeta {=}{\alpha \over \omega _{0}}$
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	$x_{c}(t){=}K_{1}e^{s_{1}t}+K_{2}te^{s_{1}t}$	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
$e^{t}$	$Ke^{st}$
sin(t)/cos(t)	$A\sin(pt)+B\cos(pt)$
$t^{n}$	$At^{n}+Bt^{n-1}+...+Ct+D$
$t^{n}e^{t}$	$At^{n}e^{pt}+Bt^{n-1}e^{pt}+...Ce^{pt}$