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| learn:courses:real-analog-chapter-8:start [2017/05/02 17:56] – [1. Overdamped System] Martha | learn:courses:real-analog-chapter-8:start [2022/10/04 11:38] (current) – external edit 127.0.0.1 | ||
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| ====== Real Analog: Chapter 8 ====== | ====== Real Analog: Chapter 8 ====== | ||
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| + | * Chapter 8 Video: | ||
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| ====== 8. Introduction and Chapter Objectives ====== | ====== 8. Introduction and Chapter Objectives ====== | ||
| - | Second order systems are, by definition, systems whose input-output relationship is a //second order// differential equation. A second order differential equation contains a //second order derivative// | + | Second order systems are, by definition, systems whose input-output relationship is a //second order// differential equation. A second order differential equation contains a //second order derivative// |
| Second order systems, like first order systems, are an extremely important class of systems. In previous chapters, we saw that the natural response of first order systems decays exponentially with time – the natural response decays monotonically to zero. The natural response of second order systems can, however, oscillate with time – we will see that a second order systems response can contain sinusoidal components. The motion of a pendulum, for example, can be modeled by a second order system. These oscillations are due to the transfer of energy between the two energy storage mechanisms; a pendulum, for example, oscillates because of the cyclic exchange of potential and kinetic energy of the mass. | Second order systems, like first order systems, are an extremely important class of systems. In previous chapters, we saw that the natural response of first order systems decays exponentially with time – the natural response decays monotonically to zero. The natural response of second order systems can, however, oscillate with time – we will see that a second order systems response can contain sinusoidal components. The motion of a pendulum, for example, can be modeled by a second order system. These oscillations are due to the transfer of energy between the two energy storage mechanisms; a pendulum, for example, oscillates because of the cyclic exchange of potential and kinetic energy of the mass. | ||
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| **d.)** In order for the system to be critically damped, the damping ratio $\zeta = 1$. From equation (8.47) with $\zeta = 1$, we obtain: | **d.)** In order for the system to be critically damped, the damping ratio $\zeta = 1$. From equation (8.47) with $\zeta = 1$, we obtain: | ||
| - | $$2 \zeta \omega_n = \frac{1R}{L} \Rightarrow 2(1)(1000) = \frac{2R}{1H} \Rightarrow = 1000 \Omega$$ | + | $$2 \zeta \omega_n = \frac{1R}{L} \Rightarrow 2(1)(1000) = \frac{2R}{1H} \Rightarrow |
| **e.)** Initial conditions on $v_C(t)$ are $v_C(0)$ and $\frac{dv_C(t)}{dt} |_{t=0}$ We are given $v_C(-)=1V$ in the problem statement, but we need to determine $\frac{dv_C(t)}{dt} |_{t=0}$; the current through the inductor, $i_L(0)$ can be used to determine this. The current through the inductor is related to the capacitor voltage via equation (8.43) above: | **e.)** Initial conditions on $v_C(t)$ are $v_C(0)$ and $\frac{dv_C(t)}{dt} |_{t=0}$ We are given $v_C(-)=1V$ in the problem statement, but we need to determine $\frac{dv_C(t)}{dt} |_{t=0}$; the current through the inductor, $i_L(0)$ can be used to determine this. The current through the inductor is related to the capacitor voltage via equation (8.43) above: | ||
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| $$y(t)=y_h(t) + y_p(t)$$ | $$y(t)=y_h(t) + y_p(t)$$ | ||
| - | The homogeneous solution of second order differential equations has been discussed in sections 8.1 and 8.4 and will not be repeated here. The particular solution of the differential equation (8.51) can be obtained by examining the solution to the equation after the homogeneous solution has died out. Letting $t \rightarrow \infty$ in equation (8.51) and noting that the forcing function is a constant as $t \rightarrow \infty$ allows us to set $\frac{d^2y(t \rightarrow \infty}{dt^2} = \frac{dy(t \rightarrow \infty)}{dt} = 0$ amd thus, | + | The homogeneous solution of second order differential equations has been discussed in sections 8.1 and 8.4 and will not be repeated here. The particular solution of the differential equation (8.51) can be obtained by examining the solution to the equation after the homogeneous solution has died out. Letting $t \rightarrow \infty$ in equation (8.51) and noting that the forcing function is a constant as $t \rightarrow \infty$ allows us to set $\frac{d^2y(t \rightarrow \infty)}{dt^2} = \frac{dy(t \rightarrow \infty)}{dt} = 0$ and thus, |
| $$\omega_{n}^{2} y_P(t) = A \Rightarrow y_P(t) = \frac{A}{\omega_{n}^{2}} | $$\omega_{n}^{2} y_P(t) = A \Rightarrow y_P(t) = \frac{A}{\omega_{n}^{2}} | ||
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