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.

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) ${\displaystyle Y_{p}}$ (Table is at bottom of page)
3. The complete solution is the particular + the complementary.

${\displaystyle y(x){=}Y_{p}+y_{c}}$

${\displaystyle y_{c}(x){=}K_{1}+K_{2}e^{sx}}$

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

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

Value	Approximation
Cons.	A
${\displaystyle e^{t}}$	${\displaystyle Ke^{st}}$
sin(t)/cos(t)	${\displaystyle A\sin(pt)+B\cos(pt)}$
${\displaystyle t^{n}}$	${\displaystyle At^{n}+Bt^{n-1}+...+Ct+D}$
${\displaystyle t^{n}e^{t}}$	${\displaystyle At^{n}e^{pt}+Bt^{n-1}e^{pt}+...Ce^{pt}}$