Circuit Theory/Transients Summary and Study guide

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

RC or LC Circuits[edit]

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


  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[edit]

  • 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[edit]

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}