RL and RC Circuits

1st order transient response

Response Types

• Natural Response
  • No sources, storage element has been energized and is supplying power
  • Fully charged at $t<0$
• Step Response
  • Storage element is energized with a time constant dependent on the circuit
  • Not fully charged at $t<0$
• Switched Circuit Response
  • Charges until given time and then switches to discharging
• Unbounded Response
  • When Thevenin equivalent resistance has an unbounded value
  • Keeps charging until the component breaks down

RC Circuits

• Resistors and Capacitors
• Time constant: $\tau =RC$
• Natural Response
  • 
  • $v(t)=V_0{e}^{-t/\tau },t\ge 0$
• Step Response
  • 
  • $v(t)=V_f+(V_o-V_f)e^{-t/\tau }$

RL Circuit

• Resistors and Inductors
• Time constant: $\tau =\frac{L}{R}$
• Natural Response
  • 
  • $i(t)=I_0{e}^{-t/\tau },t\ge 0$
• Step Response
  • 

$$\begin{array}{rl}i(t)& =I_f+(I_0-I_f)e^{-t/\tau }\\ & =\frac{V_s}{R}+\left(I_0-\frac{V_s}{R}\right)e^{-t/\tau }\end{array}$$

RLC Circuit

• Resistors and Inductors and Capacitors
• Analogous to a mass (inductor), spring (capacitor), and damper (resistor)
• Results
  • Under damped: moves quickly to equilibrium but will oscillate about the equilibrium as it does so (oscillations happen at the resonant frequency $w_o=\frac{l}{\sqrt{LC}}$)
  • Over damped: Moves slowly toward equilibrium
  • Critically damped: moves as quickly as possible toward equilibrium without oscillation about the equilibrium
• Natural Response
  • $v_c(t)=A_1{e}^{s_{1}t}+A_2{e}^{s_{2}t}$
  • Characteristic roots
    • $s_1=-\alpha +\sqrt{{\alpha }^2-{\omega }_0^2}$
    • $s_2=-\alpha -\sqrt{{\alpha }^2-{\omega }_0^2}$
  • Neper frequency: $\alpha =\frac{1}{2RC}$
  • Resonant radian frequency: ${\omega }_0=\frac{1}{\sqrt{LC}}$
  • Possibilities
    • $w_0^2<{\alpha }^2\to$ over damped
    • $w_0^2>{\alpha }^2\to$ under damped
    • $w_0^2={\alpha }^2\to$ critically damped
• Transient Responses
  • Parallel RLC
    • Natural → Voltage across the capacitor (same as others)
    • Step → Current through the inductor
  • Series RLC
    • Natural → Current through the inductor (same as others)
    • Step Response → Voltage across the capacitor