Natural Response of Series RLC Circuits

• \[\begin{aligned} &\frac{d^{2} i}{d t^{2}}+\frac{R}{L} \frac{d i}{d t}+\frac{i}{L C}=0 \\ &\Rightarrow A s^{2} e^{s t}+\frac{A R}{L} s e^{s t}+\frac{A}{L C} e^{s t}=0 \\ & \Rightarrow A e^{s t}\left(s^{2}+\frac{R}{L} s+\frac{1}{L C}\right)=0 \end{aligned}\]
  in second-order D.E. Substituting
• \[s^{2}+\frac{R}{L} s+\frac{1}{L C}=0 \Leftarrow \color{red}{\text{characteristic equation of D.E.}}\]
  is the assumed solution we are trying to find, only the expression in parentheses can be zero: Since
• \[\begin{aligned} &s_{1}=-\frac{R}{2 L}+\sqrt{\left(\frac{R}{2 L}\right)^{2}-\frac{1}{L C}} \\ &s_{2}=-\frac{R}{2 L}-\sqrt{\left(\frac{R}{2 L}\right)^{2}-\frac{1}{L C}} \end{aligned}\]
  Roots of the characteristic equation

• \[\begin{aligned} &\boxed{s_{1}, s_{2}=-\alpha\pm\sqrt{\alpha^{2}-\omega_{0}^{2}}} \quad \boxed{\alpha=\frac{R}{2 L}} \quad \boxed{\omega_{0}=\frac{1}{\sqrt{L C}}} \end{aligned}\]
  Compact form

• \(s_{1}\) and \(s_{2}\) : natural frequencies, in nepers per second \((\mathrm{Np} / \mathrm{s})\), because they are associated with the natural response of the circuit

• \(\omega_{0}\) : resonant frequency or undamped natural frequency, in radians per second \((\mathrm{rad} / \mathrm{s})\)

• \(\alpha\) : neper frequency or damping factor, in nepers per second.

• \[s^{2}+\frac{R}{L} s+\frac{1}{L C}=0 \Rightarrow \boxed{s^{2}+2 \alpha s+\omega_{0}^{2}=0}\]
  Eq. can be written as and In terms of
• \[i(t) = A_1\cdot e^{s_1t} + A_2\cdot e^{s_2t}\]
  Natural response of the series RLC circuit

• Constants \({A_1,A_2}\) are determined using initial values \(i(0)\) and \(\dfrac{di(0)}{dt}\)

• Three types of solutions:

  1. If \(\alpha>\omega_{0} \Rightarrow\) over-damped case

  2. If \(\alpha=\omega_{0} \Rightarrow\) critically damped case

  3. If \(\alpha<\omega_{0} \Rightarrow\) under-damped case

Over-damped Case:

• \(\alpha > \omega_0 \Rightarrow C > 4L/R^2\)

• both roots \(s_1\) and \(s_2\) are negative and real

• \[i(t) = A_1\cdot e^{s_1t} + A_2\cdot e^{s_2t}\]
  Response :

• As \(t \uparrow\) response decays and approaches zero.

• \(\alpha = \omega_0 \Rightarrow C = 4L/R^2\)

• \[s_1 = s_2 = -\alpha = -\dfrac{R}{2L}\]
  Roots :
• \[i(t) = A_1\cdot e^{-\alpha t} + A_2\cdot e^{-\alpha t} = A_3 \cdot e^{-\alpha t} \quad (A_3 = A_1 + A_2)\]
  Response :

• Wrong solution, because the two initial conditions cannot be satisfied with the single constant \(A_3\) .

• What then could be wrong?

• When \(\alpha = \omega_0 = R/2L\)

• Under-damped case: \(\alpha < \omega_{0}\)

  \[\begin{aligned} s_{1} &=-\alpha+\sqrt{-\left(\omega_{0}^{2}-\alpha^{2}\right)}=-\alpha+j \omega_{d} \\ s_{2} &=-\alpha-\sqrt{-\left(\omega_{0}^{2}-\alpha^{2}\right)}=-\alpha-j \omega_{d} \\ \omega_{d} &=\sqrt{\omega_{0}^{2}-\alpha^{2}} \end{aligned}\]
• \[\begin{aligned} i(t) &=A_{1} e^{-\left(\alpha-j \omega_{d}\right) t}+A_{2} e^{-\left(\alpha+j \omega_{d}\right) t} \\ &=e^{-\alpha t}\left(A_{1} e^{j \omega_{d} t}+A_{2} e^{-j \omega_{d} t}\right) \end{aligned}\]
  Response:
• \[e^{j \theta}=\cos \theta+j \sin \theta, \quad e^{-j \theta}=\cos \theta-j \sin \theta\]
  Using Euler’s identities,

• \[\begin{aligned} i(t) &=e^{-\alpha t}\left[A_{1}\left(\cos \omega_{d} t+j \sin \omega_{d} t\right)+A_{2}\left(\cos \omega_{d} t-j \sin \omega_{d} t\right)\right] \\ &=e^{-\alpha t}\left[\left(A_{1}+A_{2}\right) \cos \omega_{d} t+j\left(A_{1}-A_{2}\right) \sin \omega_{d} t\right] \end{aligned}\]
  we get
• \[i(t)=e^{-\alpha t}\left(B_{1} \cos \omega_{d} t+B_{2} \sin \omega_{d} t\right)\]
  , and with and Replacing constants

• Presence of sine and cosine functions makes the natural response is exponentially damped and oscillatory in nature

• Response has time constant of \(1/\alpha\) and period \(T=2\pi/\omega_d\)

• Due to two types of storage elements oscillatory response is possible with flow of energy back and forth between the two

• Ringing: The damped oscillation ehibited by underdamped response

• Critically damped (decays fastest) is the borderline between underdamped and over-damped

• Over-damped has longest settling time i.e. to dissipate the intial stored energy

• Critically response is the best choice to led the response settle fastest without oscillation and ringing