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9.10: Deriving Response Equations for Overdamped Second Order Systems

  • Page ID
    8346
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    For \(\zeta>1\), we can consider the damped natural frequency to be an imaginary number:

    \[\omega_{d}=\omega_{n} \sqrt{1-\zeta^{2}}=j \omega_{n} \sqrt{\zeta^{2}-1} \equiv j \mu_{d} \quad \text { where } \quad \mu_{d} \equiv \omega_{n} \sqrt{\zeta^{2}-1} \text { is real }\label{eqn:9.44} \]

    The general method of deriving transient response equations for the overdamped case is to substitute Equation \(\ref{eqn:9.44}\) into the Laplace transform Equation 9.3.5, and then proceed to invert the resulting equation, leading to general expressions that include IC response terms and convolution integrals, analogous to Equations 9.3.8 and 9.3.9.

    There is an easier method for finding overdamped-system response equations if the comparable underdamped-system equations have already been derived. The method is to use Equation \(\ref{eqn:9.44}\) in order to convert trigonometric terms of the \(\zeta<1\) equations into hyperbolic terms for the \(\zeta>1\) equations. From homework Problem 2.13, we have the following conversions valid for \(\zeta>1\):

    \[\cos \omega_{d} t=\cosh \mu_{d} t \quad \text { and } \quad \frac{\sin \omega_{d} t}{\omega_{d}}=\frac{\sinh \mu_{d} t}{\mu_{d}}\label{eqn:9.45} \]

    An example of applying Equations \(\ref{eqn:9.44}\) and \(\ref{eqn:9.45}\) is conversion of IC response Equation 9.4.1 from the underdamped (\(\zeta<1\)) form into the overdamped (\(\zeta>1\)) form:

    \[x(t)=e^{-\zeta \omega_{n} t}\left[x_{0} \cos \omega_{d} t+\left(\frac{\dot{x}_{0}+\zeta \omega_{n} x_{0}}{\omega_{d}}\right) \sin \omega_{d} t\right] \text { for } 0 \leq \zeta<1 \nonumber \]

    \[\Rightarrow \quad x(t)=e^{-\zeta \omega_{n} t}\left[x_{0} \cosh \mu_{d} t+\left(\frac{\dot{x}_{0}+\zeta \omega_{n} x_{0}}{\mu_{d}}\right) \sinh \mu_{d} t\right] \text { for } \zeta>1\label{eqn:9.46a} \]

    in which \(\mu_{d}=\omega_{n} \sqrt{\zeta^{2}-1}\). This IC-response equation is valid for \(0 \leq t\) and \(\zeta>1\). The hyperbolic functions are defined in terms of exponential functions as

    \[\cosh \mu_{d} t=\frac{e^{\mu_{d} t}+e^{-\mu_{d} t}}{2} \quad \text { and } \quad \sinh \mu_{d} t=\frac{e^{\mu_{d} t}-e^{-\mu_{d} t}}{2} \nonumber \]

    Therefore, IC-response Equation \(\ref{eqn:9.46a}\) can be written a bit more clearly as

    \[x(t)=\frac{1}{2} e^{-\zeta \omega_{n} t}\left[e^{\mu_{d} t}\left(x_{0}+\frac{\dot{x}_{0}+\zeta \omega_{n} x_{0}}{\mu_{d}}\right)+e^{-\mu_{d} t}\left(x_{0}-\frac{\dot{x}_{0}+\zeta \omega_{n} x_{0}}{\mu_{d}}\right)\right]\label{eqn:9.46b} \]

    All terms in Equation \(\ref{eqn:9.46b}\) are exponentially damped. Even the first term within the square brackets decays away exponentially because \(-\zeta \omega_{n}+\mu_{d}=-\zeta \omega_{n}+\omega_{n} \sqrt{\zeta^{2}-1}=-(\zeta-\sqrt{\zeta^{2}-1}) \omega_{n}<0\) for \(\zeta>1\).

    Example \(\PageIndex{1}\): \(RC\) band-pass filter, an overdamped 2nd order system

    We re-visit Section 5.4, where the input voltage to the \(RC\) band-pass filter is defined as \(e_{i}(t)\), the mid-circuit voltage between low-pass and high-pass stages is \(e_{m}(t)\), and the output voltage is \(e_{o}(t)\); also, the 1st order time constants of the low-pass and high-pass stages are (\(\tau_{L}\) and \(\tau_{H}\), respectively, defined in terms of resistance and capacitance values in the circuit. The two coupled 1st order ODEs derived in Section 5.4 are

    \[\text{for the low-pass filter stage, }\tau_{L} \dot{e}_{m}+e_{m}=e_{i}\label{eqn:5.16} \]

    \[\text{for the high-pass filter stage, }\tau_{H} \dot{e}_{o}+e_{o}=\tau_{H} \dot{e}_{m}\label{eqn:5.17} \]

    We combine these coupled 1st order ODEs into a single 2nd order ODE with the following operations: differentiate Equations \(\ref{eqn:5.16}\) and \(\ref{eqn:5.17}\); in the differentiated Equation \(\ref{eqn:5.16}\), replace \(\dot{e}_{m}\) using the original Equation \(\ref{eqn:5.17}\), and replace \(\ddot{e}_{m}\) using the differentiated Equation \(\ref{eqn:5.17}\); rearrange and collect terms to find the ODE relating output \(e_{o}(t)\) to input \(e_{i}(t)\):

    \[\tau_{L} \tau_{H} \ddot{e}_{o}+\left(\tau_{L}+\tau_{H}\right) \dot{e}_{o}+e_{o}=\tau_{H} \dot{e}_{i}\label{eqn:9.47} \]

    Due to the right-hand-side dynamics (the presence of \(\dot{e}_{i}\) rather than just \(e_{i}\)), we cannot cast this entire equation into the standard form Equation 9.2.2; see homework Problem 9.15 for a modified standard form of ODE and for convolution-integral response solutions. However, the order of the circuit is determined only by the left-hand-side terms, and it is clearly 2nd order, with natural frequency and damping ratio defined as

    \[\omega_{n}=\frac{1}{\sqrt{\tau_{L} \tau_{H}}} \quad \text { and } \quad \zeta=\frac{1}{2 \omega_{n}} \frac{\tau_{L}+\tau_{H}}{\tau_{L} \tau_{H}}=\frac{1}{2} \frac{\tau_{L}+\tau_{H}}{\sqrt{\tau_{L} \tau_{H}}} \nonumber \]

    Suppose, for example, that a particular circuit has the high-pass and low-pass break frequencies \(f_{H}\) = 10 Hz and \(f_{L}\) = 500 Hz, respectively. Then the time constants are \(\tau_{H}=1 / \omega_{H}=1 /\left(2 \pi f_{H}\right)=1.592 \mathrm{e}-2\) s, and \(\tau_{L}=1 / \omega_{L}=1 /\left(2 \pi f_{L}\right)=3.183 \mathrm{e}-4\) s, and the undamped natural frequency is \(\omega_{n}=4.443 \mathrm{e} 2\) rad/s (\(f_{n}=\omega_{n} / 2 \pi=70.71\) Hz). Finally, the damping ratio is \(\zeta=3.606\), which means that this is a strongly overdamped system.


    This page titled 9.10: Deriving Response Equations for Overdamped Second Order Systems is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by William L. Hallauer Jr. (Virginia Tech Libraries' Open Education Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.