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| ====== Real Analog: Chapter 8 ====== | ====== Real Analog: Chapter 8 ====== | ||
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| + | * Lecture Material: | ||
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| + | * Chapter 8 Video: | ||
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| + | <-- | ||
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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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| In equation (8.13), $y_h(t)$ is the solution to the homogeneous, | In equation (8.13), $y_h(t)$ is the solution to the homogeneous, | ||
| - | $$y(t=0)=y_0 | + | $$y(t=0)=y_0 |
| - | \frac{dy(t)}{dt} |_{t=0} = y^{' | + | \frac{dy(t)}{dt} |_{t=0} = y^{' |
| Our approach to the solution of equation (8.13) will be consistent with our previous approach to the solution of first order homogeneous differential equations: we will assume the __form__ of the differential equation (8.13) plug this assumed solution into equation (8.13) and then use our initial conditions to determine any unknown constants in the solution. | Our approach to the solution of equation (8.13) will be consistent with our previous approach to the solution of first order homogeneous differential equations: we will assume the __form__ of the differential equation (8.13) plug this assumed solution into equation (8.13) and then use our initial conditions to determine any unknown constants in the solution. | ||
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| Where $K_1$ and $K_2$ are constants which depend upon the system initial conditions. | Where $K_1$ and $K_2$ are constants which depend upon the system initial conditions. | ||
| - | * The form of the $e^{\pm \omega_n \sqrt{\zeta^2-1})t}$ terms in the system response depends strongly upon the damping ratio. If $\zeta > 1$, these terms become growing or decaying exponentials. If $\zeta< | + | * The form of the $e^{\pm |
| * The $e^{-\zeta \omega_nt}$ term in the natural response causes the overall solution to decay as time increases. Thus, the natural response goes to zero as $t \rightarrow \infty$. | * The $e^{-\zeta \omega_nt}$ term in the natural response causes the overall solution to decay as time increases. Thus, the natural response goes to zero as $t \rightarrow \infty$. | ||
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| Sinusoidal signals are represented in terms of sine and/or cosine functions. In general, we will represent sinusoids as cosine functions. Our general expression for a sinusoidal signal is: | Sinusoidal signals are represented in terms of sine and/or cosine functions. In general, we will represent sinusoids as cosine functions. Our general expression for a sinusoidal signal is: | ||
| - | $$v(t) = V_p COS(\omega t + \theta) | + | $$v(t) = V_p \cos (\omega t + \theta) |
| Where $V_P$ is the zero-to-peak amplitude of the sinusoid, $\omega$ is the radian frequency of the sinusoid (we will always use radians/ | Where $V_P$ is the zero-to-peak amplitude of the sinusoid, $\omega$ is the radian frequency of the sinusoid (we will always use radians/ | ||
| $$\omega = \frac{2\pi}{T} | $$\omega = \frac{2\pi}{T} | ||
| + | |||
| + | {{ : | ||
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| + | |||
| === Note: === | === Note: === | ||
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| $$|A| = \sqrt{a^2+b^2} | $$|A| = \sqrt{a^2+b^2} | ||
| - | $$\theta_A = tan^{-1} \left( \frac{b}{a} \right) | + | $$\theta_A = \tan^{-1} \left( \frac{b}{a} \right) |
| Notice that the phase angle is defined counterclockwise from the positive real axis. Conversely, we can determine the rectangular coordinates from the polar coordinates from: | Notice that the phase angle is defined counterclockwise from the positive real axis. Conversely, we can determine the rectangular coordinates from the polar coordinates from: | ||
| - | $$a=Re \left{ A \right} = |A| cos(\theta_A) | + | $$a=Re \{ A \} = |A| \cos(\theta_A) |
| - | $$b=Im \left{ A \right} = |A| sin(\theta_A) | + | $$b=Im \{ A \} = |A| \sin(\theta_A) |
| - | Where the notation $Re \left{ A \right}$ and $Im \left{ A \right} denote the real part of $A$ and the imaginary part of $A$, respectively. | + | Where the notation $Re \{ A \}$ and $Im \{ A \}$ denote the real part of $A$ and the imaginary part of $A$, respectively. |
| The polar coordinates of a complex number $A$ are often represented in the form: | The polar coordinates of a complex number $A$ are often represented in the form: | ||
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| Thus, $e^{j \theta}$ is a complex number with magnitude 1 and phase angle $\theta$. From Figure 8.6, it is easy to see that this definition of the complex exponential agrees with Euler' | Thus, $e^{j \theta}$ is a complex number with magnitude 1 and phase angle $\theta$. From Figure 8.6, it is easy to see that this definition of the complex exponential agrees with Euler' | ||
| - | $$e^{\pm j \theta} = cos \theta \pm j sin \theta | + | $$e^{\pm j \theta} = \cos \theta \pm j \sin \theta |
| With the definition of equation (8.30), we can define any arbitrary complex number in terms of complex exponentials. For example, our previous complex number $A$ can be represented as: | With the definition of equation (8.30), we can define any arbitrary complex number in terms of complex exponentials. For example, our previous complex number $A$ can be represented as: | ||
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| $$A=|A| e^{j \theta_A} | $$A=|A| e^{j \theta_A} | ||
| - | We can generalize our definition of the complex exponential to time-varying signals. If we define a time varying signal $e^{j \omega t}$$ , we can use equation (8.31) to write: | + | We can generalize our definition of the complex exponential to time-varying signals. If we define a time varying signal $e^{j \omega t}$ , we can use equation (8.31) to write: |
| - | $$e^{\pm j \omega t} = cos \omega t \pm j sin \omega t (Eq. 8.33)$$ | + | $$e^{\pm j \omega t} = \cos \omega t \pm j \sin \omega t (Eq. 8.33)$$ |
| - | The signal $e^{j \omega t}$ can be visualized as a unit vector rotating around the origin in the complex plane; the tip of the vector scribes a unit circle with its center at the origin of the complex plane. This is illustrated in Figure 8.7. The vector rotates at a rate defined by the quantity $\omega$ -- the vector makes one complete revolution every $\frac{2\pi}{\omega} seconds. | + | The signal $e^{j \omega t}$ can be visualized as a unit vector rotating around the origin in the complex plane; the tip of the vector scribes a unit circle with its center at the origin of the complex plane. This is illustrated in Figure 8.7. The vector rotates at a rate defined by the quantity $\omega$ -- the vector makes one complete revolution every $\frac{2\pi}{\omega}$ seconds. |
| - | The projection of this rotating vector on the real axis traces out the signal $\cos \omega t$ , as shown in Figure 8.7, while the projection of the rotating vector on the imaginary axis traces out the signal $sin \omega t$, also shown in Figure 8.7. | + | The projection of this rotating vector on the real axis traces out the signal $\cos \omega t$ , as shown in Figure 8.7, while the projection of the rotating vector on the imaginary axis traces out the signal $\sin \omega t$, also shown in Figure 8.7. |
| - | Thus, we interpret the complex exponential function $e^{j\omega t}$ as an alternate “type” of sinusoidal signal. The real part of this function is $\cos \omega t$ while the imaginary part of this function is $sin \omega t$. | + | Thus, we interpret the complex exponential function $e^{j\omega t}$ as an alternate “type” of sinusoidal signal. The real part of this function is $\cos \omega t$ while the imaginary part of this function is $\sin \omega t$. |
| {{ : | {{ : | ||
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| $$V_P \cos (\omega t + \theta) = Re \{ V_P e^{j(\omega t + \theta)} \}$$ | $$V_P \cos (\omega t + \theta) = Re \{ V_P e^{j(\omega t + \theta)} \}$$ | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Exercises ==== | ||
| + | - A complex valued signal $v(t)$ is given by $v(t)=e^{j(\pi t - 30^{\deg})}$. What is the real part of the signal? | ||
| + | - Represent the signal $v(t) = \cos (300t)$ in terms of complex exponentials. | ||
| + | - Represent the complex number $3-j3$ in polar coordinates. | ||
| + | - Represent the complex number $e^{j \pi}$ in rectangular coordinates. | ||
| + | - Represent the complex number $e^{\frac{-j \pi}{2}}$ in rectangular coordinates. | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ===== 8.4: Second Order System Natural Response - Part 2 ===== | ||
| + | In section 8.2, we developed the __form__ of the solution of the differential equation governing the natural response of second order systems. The form of the solution contained so-called complex exponentials; | ||
| + | |||
| + | In section 8.2, we noted that the form of the natural response of a second order system was strongly dependent upon the damping ratio, $\zeta$. If the damping ratio was greater than one, all terms in the response decay exponentially, | ||
| + | |||
| + | In section 8.1, the differential equation governing the natural response of a second order system was written as: | ||
| + | |||
| + | $$\frac{d^2y_h(t)}{dt^2} + 2 \zeta \omega_n \frac{dy_h(t)}{dt} + \omega_{n}^{2} y_h (t) = 0 (Eq. 8.34)$$ | ||
| + | |||
| + | Where $y(t)$ is any system parameter of interest, $\omega_n$ is the// undamped natural frequency// and $\zeta$ is the //damping ratio//. The initial conditions are the value of the function $y(t)$ at $t = 0$ and the derivative of the function $y(t)$ at $t = 0$: | ||
| + | |||
| + | $$y(t=0)=y_0$$ | ||
| + | $$\frac{dy(t)}{dt} |_{t=0} = y^{' | ||
| + | |||
| + | (Eq. 8.35)$$ | ||
| + | |||
| + | In section 8.2, we wrote the solution to equation (8.34) in the form: | ||
| + | |||
| + | $$y_h(t) = e^{-\zeta \omega_n t} \left[ K_1e^{(\omega_n \sqrt{\zeta^2-1})t} + K_2e^{-(\omega_n \sqrt{\zeta^2-1})t} \right] | ||
| + | |||
| + | Where $K_1$ and $K_2$ are unknown coefficients which can be determined by application of the initial conditions provided in equation (8.35). The form of the solution of equation (8.36) will fall into one of three categories, depending on the value of damping ratio. The three possible cases are: | ||
| + | - If $\zeta > 1$, all terms in the solution will be either growing or decaying exponentials and the solution will decay exponentially with time. If the damping ratio is large, this decay rate can be very slow. A system with $\zeta > 1$ is said to be // | ||
| + | - If $\zeta < 1$, the terms $e^{\pm (\omega_n \sqrt{\zeta^2-1})t}$ are //complex exponentials// | ||
| + | - $\zeta=1$; the form of the solution in this case is approximately that of case 1 above, in that the solution will decay exponentially. However, in this case, the response decay rate will be faster than the response of any overdamped system with the same natural frequency. Systems with $\zeta=1$ are said to be // | ||
| + | |||
| + | Details of the responses for each of the above three cases are provided in the subsections below. | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== 1. Overdamped System ==== | ||
| + | For an overdamped system, $\zeta> | ||
| + | |||
| + | $$y_h(t)=e^{-\zeta \omega_nt} \left[ \frac{\dot{y}_0 + (\zeta + \sqrt{\zeta^2-1}) \omega_ny_0}{2 \omega_n \sqrt{\zeta^2-1}} e^{\omega_nt \sqrt{\zeta^2-1}} + \frac{\dot{y}_0 - (\zeta- \sqrt{\zeta^2-1})\omega_ny_0}{2\omega_n \sqrt{\zeta^2-1}} e^{-\omega_n t \sqrt{\zeta^2-1}} \right] | ||
| + | |||
| + | In equation (8.37), the $e^{-\zeta \omega_n t}$ term is a decaying exponential with time constant $\frac{1}{\zeta \omega_n}$. The $e^{\omega_n t \sqrt{\zeta^2-1}}$ is a growing exponential with time constant $\frac{1}{\omega_n \sqrt{\zeta^2-1}}$. Thus, the overall system response is a sum of two decaying exponential signals, one which is proportional to $e^{-\zeta \omega_n t} \cdot e^{\omega_n t \sqrt{\zeta^2-1}}$ and the other which is proportional to $e^{-\zeta \omega_n t} \cdot e^{-\omega_n t \sqrt{\zeta^2-1}}$. | ||
| + | |||
| + | The term $e^{-\zeta \omega_n t} \cdot e^{\omega_n t \sqrt{\zeta^2-1}}$ is the product of two exponentials: | ||
| + | |||
| + | |||
| + | |||
| + | The term $e^{\zeta \omega_n t} \cdot e^{-\omega_n t \sqrt{\zeta^2-1}}$ is the product of two decaying exponentials; | ||
| + | |||
| + | An example of the response of an overdamped system is shown in Figure 8.8, for various values of damping ratio. The two system time constants are readily observable in this example. Note that as the damping ratio increases, the overall time required for the system response to decay to zero increases. The response of overdamped systems cannot oscillate, however, the response can change sign __once__ (e.g. the function is allowed one zero-crossing). | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== 2. Underdamped System ==== | ||
| + | For an underdamped system, $\zeta <1$, and equation (8.36) becomes as shown in equation (8.38). | ||
| + | |||
| + | $$y_h(t)=e^{-\zeta \omega_n t} \left[ \frac{\dot{y}_0 + \zeta \omega_n y_0}{\omega_n \sqrt{1-\zeta^2}} \sin \left( \omega_n t \sqrt{1-\zeta^2} \right) + y_0 \cos \left( \omega_n t \sqrt{1-\zeta^2} \right) \right] | ||
| + | |||
| + | The solution is a decaying sinusoid. The decay rate is set by the term $e^{-\zeta \omega_n t}$, while the oscillation frequency of the sinusoid is $\omega_n \sqrt{1-\zeta^2}$. The oscillation frequency seen in the natural response is thus not identically the natural frequency of the system; it is also influenced by the damping ratio. This leads to the definition of the //damped natural frequency//: | ||
| + | |||
| + | $$\omega_d = \omega_n \sqrt{1-\zeta^2} | ||
| + | |||
| + | Oscillations seen in the system response will have radian frequency $\omega_d$; thus, the period of the oscillations is $\frac{2 \pi}{\omega_d}$. | ||
| + | |||
| + | Example responses for underdamped systems are shown in Figure 8.9; the responses shown are all for the same natural frequency and initial conditions – only the damping ratio varies. Note that smaller damping ratios result in slower decay rates for the response, oscillations persist for longer and are more pronounced for smaller damping ratios. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== 3. Critically Damped System ==== | ||
| + | For a critically damped system, $\zeta=1$, and equation (8.36) becomes as shown in equation (8.40). | ||
| + | |||
| + | $$y_h(t) = e^{-\zeta \omega_n t} \left[ y_0 + (\dot{y}_0 + \omega_n y_0 t) \right] | ||
| + | |||
| + | The critically damped system response does not oscillate although, as with the overdamped case, one zero crossing of the function is allowed. The importance of the critically damped system response is that, for a particular natural frequency, it has the shortest decay time __without oscillation__ of any system. An example response of a critically damped system is shown in Figure 8.10. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Example 8.3 ==== | ||
| + | For the circuit shown below: | ||
| + | - Write the differential equation for $v_C(t)$. | ||
| + | - If $L = 1H$, $R = 200\Omega$, and $C = 1x10^{-6}F$, | ||
| + | - For the conditions in part (2), is the system underdamped, | ||
| + | - For the values of L and C in part (2), determine the value of R that makes the system critically damped. | ||
| + | - If $v_C(0) = 1V$ and $i_L(0) = 0.01A$, what are the appropriate initial conditions to solve the differential equation determined in part (1)? | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | **a.)** As usual, we define the voltage across the capacitor and the current through the inductor as our variables and write KVL and KCL in terms of these variables. The figure below shows these variables, along with the associated currents through capacitors and voltages across inductors. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | KCL at the indicated node results in: | ||
| + | |||
| + | $$i_L(t) + C \frac{dv_C(t)}{dt}=0 | ||
| + | |||
| + | KVL around the indicated loop provides: | ||
| + | |||
| + | $$L\frac{di_L(t)}{dt} + 2Ri_L(t)=v_C(t) | ||
| + | |||
| + | The above two equations can be combined to obtain an equation for $v_C(t)$. To do this, we use the first equation to obtain: | ||
| + | |||
| + | $$i_L(t)=-C \frac{dv_C(t)}{dt} | ||
| + | |||
| + | Differentiating equation (8.43) provides: | ||
| + | |||
| + | $$\frac{di_L(t)}{dt} = -C\frac{d^2v_C(t)}{dt^2} | ||
| + | |||
| + | Substituting equations (8.43) and (8.44) into equation (8.42) results in: | ||
| + | |||
| + | $$-LC \frac{d^2v_C(t)}{dt^2} -2RC \frac{dv_C(t)}{dt} = v_C(t)$$ | ||
| + | |||
| + | Dividing the above by LC and grouping terms gives our final result: | ||
| + | |||
| + | $$\frac{d^2v_C(t)}{dt^2} + \frac{2R}{L} \frac{dv_C(t)}{dt} + \frac{1}{LC} v_C(t) = 0 (Eq. 8.45)$$ | ||
| + | |||
| + | **b.)** Equation (5) is of the form: | ||
| + | |||
| + | $$\frac{d^2y(t)}{dt^2} + 2 \zeta \omega_n \frac{dy(t)}{dt} + \omega_{n}^{2} y(t) = 0 (Eq. 8.46)$$ | ||
| + | |||
| + | Equating coefficients in equations (8.45) and (8.46) and substituting $L = 1H$, $R = 200\Omega$, and $C = 1x10^{-6}F$ results in: | ||
| + | |||
| + | $$2 \zeta \omega_n = \frac{2R}{L} = 400 (Eq. 8.47)$$ | ||
| + | |||
| + | $$\omega_{n}^{2} = \frac{1}{LC} = 1x10^6 | ||
| + | |||
| + | Solving equation (8) for the natural frequency results in $\omega_n = 1000$ rad/sec. Substituting this result into equation (8.47) and solving for the damping ratio gives $\zeta = 0.2$. The damped natural frequency is: | ||
| + | |||
| + | $$\omega_d = \omega_n \sqrt{1-\zeta^2} = 979.8 rad/sec$$ | ||
| + | |||
| + | **c.)** The damping ratio determined in part (b) is $\zeta = 0.2$; since this is less than one, the system is underdamped. | ||
| + | |||
| + | **d.)** In order for the system to be critically damped, the damping ratio $\zeta = 1$. From equation (8.47) with $\zeta = 1$, we obtain: | ||
| + | |||
| + | $$2 \zeta \omega_n = \frac{1R}{L} \Rightarrow 2(1)(1000) = \frac{2R}{1H} \Rightarrow R = 1000 \Omega$$ | ||
| + | |||
| + | **e.)** Initial conditions on $v_C(t)$ are $v_C(0)$ and $\frac{dv_C(t)}{dt} |_{t=0}$ We are given $v_C(-)=1V$ in the problem statement, but we need to determine $\frac{dv_C(t)}{dt} |_{t=0}$; the current through the inductor, $i_L(0)$ can be used to determine this. The current through the inductor is related to the capacitor voltage via equation (8.43) above: | ||
| + | |||
| + | $$i_L(t)=-C \frac{dv_C(t)}{dt}$$ | ||
| + | |||
| + | so at time $t=0$, | ||
| + | |||
| + | $$i_L(0)=-(1x10^{-6}F) \frac{dv_C(t)}{dt} |_{t=0}=0.01A$$ | ||
| + | |||
| + | Solving for $\frac{dv_C(t)}{dt} |_{t=0}$, | ||
| + | |||
| + | $$\frac{dv_C(t)}{dt}|_{t=0} = \frac{1}{C}i_L(0) = -\frac{0.01A}{1x10^{-6}F} = -10,000 V/sec$$ | ||
| + | |||
| + | We conclude this example with plots of the system response for underdamped, | ||
| + | |||
| + | Figure 8.11 shows the response of the circuit described by the differential equation determined in part (a) above, for the circuit parameters provided in part (b), to the initial conditions of part (e). Thus, the governing differential equation is | ||
| + | |||
| + | $$\frac{d^2v_C(t)}{dt^2} + \frac{2R}{L} \frac{dv_C(t)}{dt} + \frac{1}{LC} v_C(t)=0$$ | ||
| + | |||
| + | With $L=1H$, $R=200 \Omega$, and $C=1x10^{-6}F$, | ||
| + | |||
| + | $$\frac{d^2v_C(t)}{dt^2} + 400 \frac{dv_C(t)}{dt} + 1x10^6 v_C(t)=0$$ | ||
| + | |||
| + | The initial conditions are, from part (e): | ||
| + | |||
| + | $$v_C(0)=1V$$ | ||
| + | |||
| + | $$\frac{dv_C(t)}{dt}|_{t=0} = -10,000 V/sec$$ | ||
| + | |||
| + | Using MATLAB to evaluate the differential equation results in Figure 8.11. Figure 8.11 agrees with our expectations based on the calculations of part (b). In part (b), we determined that the damping ratio $\zeta = 0.2$, so that the system is underdamped – Figure 8.11 exhibits the oscillations (multiple zero axis crossings) that we would expect from an underdamped system. Likewise, we determined in part (b) that the damped natural frequency of the system is approximately 980 rad/sec. The period of the oscillations we would expect to see in the response is therefore: | ||
| + | |||
| + | $$T=\frac{2 \pi}{\omega_d} = 0.0064 seconds$$ | ||
| + | |||
| + | This value is consistent with the period of the oscillations seen in Figure 8.11. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | In part (d) above, we determined that the value of R resulting in a critically damped system is $R = 1000\Omega$. Re- evaluating the above governing differential equation with this value for R results in | ||
| + | |||
| + | $$\frac{d^2v_C(t)}{dt^2} + 2000 \frac{dv_C(t)}{dt} + 1x10^6 v_C(t)=0$$ | ||
| + | |||
| + | the initial conditions are as in the above example: | ||
| + | |||
| + | $$v_C(0)=1V$$ | ||
| + | |||
| + | $$\frac{dv_C(t)(t)}{dt} |_{t=0} = -10,000 V/sec$$ | ||
| + | |||
| + | The resulting response is shown in Figure 8.12. This plot also matches our expectations, | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | In order to obtain a better understanding of critically damped vs. overdamped systems, we increase R to $3000\Omega$. The resulting damping ratio is $\zeta = 3$; increasing R above the critically damped value will result in an overdamped system since the damping ratio is proportional to R. We will expect the response shape to be somewhat like that shown in Figure 8.12 (it will still be composed of decaying exponential functions) but the overdamped system should decay more slowly. This overdamped system response is shown in Figure 8.13. This response agrees with our qualitative expectations – the response does not oscillate, and the decay time is longer than that shown in Figure 8.12. | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Section Summary ==== | ||
| + | * It is common to categorize second order systems by their damping ratio. This also characterizes the shape of their natural response. The three categories are: | ||
| + | * If $\zeta > 1$, the system is said to be // | ||
| + | * If $\zeta < 1$, the system is said to be // | ||
| + | * $\zeta = 1$, the solution is said to be critically damped. In this case, the response will not oscillate, and the decay rate of the response will be faster than the response of any overdamped system with the same natural frequency. | ||
| + | |||
| + | For an underdamped system, the oscillations observed in the response have a radian frequency $\omega_d$ defined as: | ||
| + | |||
| + | $$\omega_d= \omega_n \sqrt{1-\zeta^2}$$ | ||
| + | |||
| + | $\omega_d$ is called the //damped natural frequency// of the system. The period of the oscillations in the natural frequency (the time between successive peaks) is $\frac{2 \pi}{\omega_d}$. | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Exercises ==== | ||
| + | **1.** The differential equation governing a circuit with output $y(t)$ is given by: | ||
| + | |||
| + | $$\frac{d^2y(t)}{dt^2} + 6\frac{dy(t)}{dt} + 144y(t) = 0$$ | ||
| + | |||
| + | What are the damping ratio and natural frequency of the circuit? Is the circuit under-, over-, or critically damped? | ||
| + | |||
| + | **2.** The differential equation governing a circuit’s natural response is: | ||
| + | |||
| + | $$\frac{d^2y(t)}{dt^2} + 32\frac{dy(t)}{dt} + 64y(t) = 0$$ | ||
| + | |||
| + | Where $y(t)$ is the circuit response. What are: | ||
| + | * The circuit’s natural frequency, | ||
| + | * The circuit’s damping ratio, and | ||
| + | * The __two__ time constants governing the circuit | ||
| + | |||
| + | **3.** The differential equation governing a circuit’s natural response is: | ||
| + | |||
| + | $$\frac{d^2y(t)}{dt^2} + 8 \frac{dy(t)}{dt} + 64y(t)=0$$ | ||
| + | |||
| + | Where $y(t)$ is the circuit response. What are: | ||
| + | * The circuit’s natural frequency, | ||
| + | * The circuit’s damping ratio, and | ||
| + | * The circuit' | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ===== 8.5: Second Order System Step Response ===== | ||
| + | In this section, we address the case in which the input to a second order system consists of the sudden application of a constant voltage or current to the circuit; this type of input can be modeled as a step function. The response of a system to this type of input is called the //step response// of the system. | ||
| + | |||
| + | The material presented in this section will emphasize the development of __qualitative__ relationship between the damping ratio and natural frequency of a system and the system’s time-domain response. We will also see that we can __quantitatively__ relate several specific response parameters to the system’s damping ratio and natural frequency. This approach allows us to infer a great deal about the expected system response directly from the damping ratio and natural frequency of the system, without explicitly solving the differential equation governing the system. This approach is also useful in system design, since we can readily determine the damping ratio and natural frequency necessary to provide the desired response shape. Since the damping ratio and natural frequency are typically functions of resistances, | ||
| + | |||
| + | In section 8.1, we wrote a general differential equation governing a second order system as: | ||
| + | |||
| + | $$\frac{d^2y(t)}{dt^2} + 2 \zeta \omega_n \frac{dy(t)}{dt} + \omega_{n}^{2}y(t)=f(t) | ||
| + | |||
| + | Where $y(t)$ is any system parameter of interest (for example, a voltage or current in an electrical circuit), $\omega_n$ and $\zeta$ are the //undamped natural frequency// and the //damping ratio// of the system, respectively, | ||
| + | |||
| + | In this chapter section, we restrict our attention to the specific case in which $f(t)$ is a step function. Thus, the forcing function to the system can be written as: | ||
| + | |||
| + | $$f(t)= Au_0(t)= \begin{cases} | ||
| + | 0, t < 0 \\ | ||
| + | A, t > 0 \\ | ||
| + | | ||
| + | (Eq. 8.50)$$ | ||
| + | |||
| + | Thus, the differential equation governing the system becomes: | ||
| + | |||
| + | $$\frac{d^2y(t)}{dt^2} + 2 \zeta \omega_n \frac{dy(t)}{dt} + \omega_{n}^{2} y(t) = Au_0(t) | ||
| + | |||
| + | In addition to the above restriction on the forcing function, we will assume that the initial conditions are all zero (we sometimes say that the system is //initially relaxed//). Thus, for the second-order system above, our initial conditions will be: | ||
| + | |||
| + | $$y(t=0)=0 \\ | ||
| + | |||
| + | \frac{dy(t)}{dt} |_{t=0} =0 \\ | ||
| + | |||
| + | (Eq. 8.52)$$ | ||
| + | |||
| + | Solving equation (8.51) with the initial conditions provided in equations (8.52) results in the //step response// of the system. | ||
| + | |||
| + | As in our discussion of forced first order system responses in section 8.1, we write the overall solution of the differential equation of equation (8.51) as the sum of a particular solution and a homogeneous solution. Thus: | ||
| + | |||
| + | $$y(t)=y_h(t) + y_p(t)$$ | ||
| + | |||
| + | The homogeneous solution of second order differential equations has been discussed in sections 8.1 and 8.4 and will not be repeated here. The particular solution of the differential equation (8.51) can be obtained by examining the solution to the equation after the homogeneous solution has died out. Letting $t \rightarrow \infty$ in equation (8.51) and noting that the forcing function is a constant as $t \rightarrow \infty$ allows us to set $\frac{d^2y(t \rightarrow \infty)}{dt^2} = \frac{dy(t \rightarrow \infty)}{dt} = 0$ and thus, | ||
| + | |||
| + | $$\omega_{n}^{2} y_P(t) = A \Rightarrow y_P(t) = \frac{A}{\omega_{n}^{2}} | ||
| + | |||
| + | Combining the particular and homogeneous solutions, assuming the system is underdamped $(\zeta < 1)$, and employing the initial conditions results in our final expression for the __step response of an underdamped second order system__: | ||
| + | |||
| + | $$y(t) = \frac{A}{\omega_{n}^{2}} \{ 1-e^{- \zeta \omega_n t} [ \cos (\omega_d t) + \frac{\zeta}{\sqrt{1-\zeta^2}} \sin (\omega_d t) ] \} (Eq. 8.54)$$ | ||
| + | |||
| + | Where $\omega_d = \omega_n \sqrt{1-\zeta^2}$ is the damped natural frequency of the system, previously defined in chapter 8.4. | ||
| + | |||
| + | It is common to interpret an underdamped second order system’s response in terms of the damping ratio and the natural frequency, rather than direct evaluation of equation (8.54). Figure 8.14 shows a typical step response for an underdamped second order system. The system response overshoots to a maximum value $y_p$ and has steady- state response $y_{ss}$. The maximum overshoot is generally normalized by the steady-state response and is presented in terms of a variable $M_P$ defined as: | ||
| + | |||
| + | $$M_P = \frac{y_P-y_{ss}}{y_{ss}} | ||
| + | |||
| + | $M_P$ is often presented as a __percent__, | ||
| + | |||
| + | The time domain parameters $M_P$, $t_r$, and $T$ are readily related to the parameters $\xi$, $\omega_n$, and $\omega_d$. We provide the following relations here, without proof: | ||
| + | |||
| + | $$M_P = e^{\frac{-\pi \xi}{\sqrt{1-\xi^2}}} | ||
| + | |||
| + | $$t_r \approx \frac{1.8}{\omega_n} | ||
| + | |||
| + | Note: for small damping ratios, ($\xi < 0.6$), equation (8.56) is often approximated as: | ||
| + | |||
| + | $$M_P \approx - \frac{\xi}{0.06} | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | ==== Section Summary ==== | ||
| + | * Typical parameters used to characterize the step response of an underdamped system are the rise time, the maximum overshoot, and the frequency of the oscillations in the response. These parameters are defined as follows: | ||
| + | * The rise time, $t_r$, is the time required for the response to go from 10% to 90% of its steady-state response | ||
| + | * The maximum overshoot provides the maximum value achieved by the response. The maximum overshoot is a normalized value, defined as: | ||
| + | |||
| + | $$M_P = \frac{y_p-y_{ss}}{y_{ss}}$$ | ||
| + | |||
| + | Where $y_p$ is the peak (or maximum) value of the response and $y_{ss}$ is the steady state system response. $M_P$ is often expressed as a percent, by multiplying the above quantity by 100. | ||
| + | |||
| + | * The radian frequency of the oscillations in the step response is given by the damped natural frequency: | ||
| + | |||
| + | $$\omega_d = \omega_n \sqrt{1-\zeta^2}$$ | ||
| + | |||
| + | The period of the oscillations in the response is then given by: | ||
| + | |||
| + | $$T= \frac{2 \pi}{\omega_d}$$ | ||
| + | |||
| + | * The rise time and maximum overshoot can be related to the damping ratio and natural frequency of the system. The appropriate relations are provided below: | ||
| + | * $t_r \approx \frac{1.8}{\omega_n}$ | ||
| + | * $M_P=e^{\frac{- \pi \xi}{\sqrt{1-\xi^2}}}$ or $M_P - \frac{\xi}{0.6}$ (for small damping ratios, $\xi < 0.6$) | ||
| + | |||
| + | The period of the oscillations can be used to cross-check the above results, since it depends upon both the damping ratio and natural frequency. | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | ==== Exercises ==== | ||
| + | - The differential equation governing the circuit shown below is: | ||
| + | |||
| + | $$\frac{d^2v(t)}{dt^2} + 12 \frac{dv(t)}{dt} + 400v(t) = 400u(t)$$ | ||
| + | |||
| + | If $u(t)=2u_0(t)$, | ||
| + | * The circuit’s natural frequency, | ||
| + | * The circuit’s damping ratio, | ||
| + | * The percent overshoot ($M_P$), the rise time ($t_r$), and the steady-state response of the capacitor voltage | ||
| + | * What is the maximum value seen by the capacitor voltage? | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | [[{}/ | ||
| + | [[{}/ | ||