Lecture 02 Modeling (i) –Transfer function

1. Lecture 02 Modeling (i) –Transfer function 2.1Circuit Systems 2.2 Mechanical Systems 2.3 Transfer Function Modern Control Systems

2. 2.1 Circuit Systems Inductor Capacitor Resistor 應用的定律 ．克希荷夫電流定律(Kirchhoff Current Law) ．克希荷夫電壓定律(Kirchhoff Voltage Law) ．歐姆定律(Ohm’s Law) Modern Control Systems

3. 2.1 Circuit Systems Example 2.1： RC Series Circuit By Kirchhoff and Ohm’s Law Fig. 2.1 Example 2.2：By Kirchhoff Current Law (Node Analysis) Fig. 2.2 Modern Control Systems

4. 2.2 Mechanical Systems(Translational Motion) Spring(彈簧) : spring constant (彈簧常數) (張力) Damper(阻尼器) : Velocity (速率) : viscous friction Constant (黏滯摩擦係數) : Viscous Friction Force (黏滯摩擦力) Modern Control Systems

5. 2.2 Mechanical Systems(Translational Motion) Mass(質量) : Mass (質量) :inertia force (慣性力) ：acceleration (加速度) 應用的定律 牛頓定律： F：所有外力之和 (外力的方向與位移相同為正) Modern Control Systems

6. 2.2 Mechanical Systems(Translational Motion) Example 2.3: Mass-Spring-Damper Form Newton’s Second Law Under ZIC,take Laplace transform both sides (Transfer Function) Fig. 2.3 Note: ZIC=Zero Initial Condition Modern Control Systems

7. Fig. 2.4 Tire spring constant 2.2 Mechanical Systems(Translational Motion) Example 2.4：懸吊系統(Suspension system) Mathematical Model: Modern Control Systems

8. 2.2 Mechanical Systems(Rotational Motion) Spring(彈簧) T：torque轉矩 : angle 角度 Damper(阻尼器) ：angular velocity (角速度) Inertia(慣量) ：angular acceleration (角加速度) ：對轉動軸的慣量 Modern Control Systems

9. Gear train (1) (2) (3) no energy loss (4) Modern Control Systems

10. 2.2 Mechanical Systems(Rotational Motion) 應用的定律 =所有外加轉矩之和 (與角位移方向相同者為正) =物體的加速度 =轉動慣量 Modern Control Systems

11. 2.2 Mechanical Systems (Rotational Motion) Example2.5：Mass－Spring－Damper (Rotational System) Form Newton’s Second Law Under ZIC, take Laplace transform both sides Fig. 2.5 Modern Control Systems

12. 2.3 Transfer Function Definition ZIC: Zero Initial Condition Modern Control Systems

13. 2.3 Transfer Function Transfer Function: Gain that depends on the frequency of input signal When input Under ZIC, the steady state output (2.1) where is also called the DC gain. Special Case: Conclusion: Under ZIC, for sinusoidal input, the steady state Output is also a sinusoidal wave. Modern Control Systems

14. 2.3 Transfer Function Example 2.6 Find the output y(t) ? With reference to (2.1), we know and Fig. 2.6 A=0.707 <1, Attenuation! Modern Control Systems

15. 2.3 Transfer Function Derivation of T.F. from Differential Equation A Second-Order Example Set I.C. =0 and Take L.T. both sides (Transfer Function from r to y) Modern Control Systems

16. 2.3 Transfer Function Example 2.7: A second-order Circuit From Kirchihoff Voltage Law, we obtain Fig. 2.7 (Transfer Function from ) Modern Control Systems

17. 2.3 Transfer Function Example 2.3: Mass-Spring-Damper Form Newton’s Second Law Take Laplace Transfrom both sides Transfer Function Fig. 2.8 Modern Control Systems

18. 2.3 Transfer Function Transfer Function 轉移函數的相關名詞 p(s)=分子多項式， q(s)=分母多項式 (特性多項式, characteristic polynomial) ◆ q(s)=之階數稱為此系統之階數(order) ◆ 方程式 q(s)=0稱為 特性方程式(characteristic equation) ◆ q(s)之根稱為系統之極點 (pole) ◆ p(s)之根稱為系統之零點(zero) ◆ Modern Control Systems

19. 2.3 Transfer Function Example 2.8 Under ZIC, take L.T., we get the transfer function Poles： Zeros： Char. Equation: Fig. 2.9 Modern Control Systems

20. 2.3 Transfer Function Time Constant of a first-order system Consider : 稱為系統的時間常數(Time Constant) :稱為穩態直流增益 Modern Control Systems

21. 2.3 Transfer Function Example 2.9 Fig. 2.10 is called step-function. When A=1, it is called unit step function. The output voltage For RC=1 Modern Control Systems

22. 2.3 Transfer Function Time Constant: Measure of response time of a first-order system A 1 2 3 4 5 0 Fig. 2.11 pole-zero form gain-time constant form Modern Control Systems