1. 11/15/2006 Ch 7 System Consideration- Paul Lin 1 ECET 307 Analog Networks Signal Processing Ch 7 System Considerations 2 of 3 Fall 2006 http://www.etcs.ipfw.edu/~lin

2. 11/15/2006 Ch 7 System Consideration- Paul Lin 2 Ch 7: System Considerations Transfer Function Examples Transfer Function Examples Differential Equation and Transfer Function Differential Equation and Transfer Function Step and Impulse Responses Step and Impulse Responses

3. 11/15/2006 Ch 7 System Consideration- Paul Lin 3 Example 7-1 (a) Determine the transfer function and the impulse response of the circuit as shown below. The input excitation is v1(t) and the output response is v2(t). (a) Determine the transfer function and the impulse response of the circuit as shown below. The input excitation is v1(t) and the output response is v2(t).

4. 11/15/2006 Ch 7 System Consideration- Paul Lin 4 Example 7-1 Solution The transfer function is defined as output/input, or The transfer function is defined as output/input, or

5. 11/15/2006 Ch 7 System Consideration- Paul Lin 5 Example 7-1 Solution (cont.) The impulse response of the circuit is g(t) = L -1 G(s) The impulse response of the circuit is g(t) = L -1 G(s)

6. 11/15/2006 Ch 7 System Consideration- Paul Lin 6 Example 7-1 Solution (cont.) The response v2(t) The response v2(t) From the definition of G(s) = V2(s)/V1(s) From the definition of G(s) = V2(s)/V1(s)

7. 11/15/2006 Ch 7 System Consideration- Paul Lin 7 Example 7-1 Solution (cont.) Use the trick formula Use the trick formula The time function V(s) The time function V(s)

8. 11/15/2006 Ch 7 System Consideration- Paul Lin 8 Example 7-2 Determine the transfer function of the circuit. The input is v1(t) and the desired output is i2(t). Determine the transfer function of the circuit. The input is v1(t) and the desired output is i2(t).

9. 11/15/2006 Ch 7 System Consideration- Paul Lin 9 Example 7-2 Solution (cont.) 1. Draw the transformed circuit 2.Write a pair of simultaneous mesh equations

10. 11/15/2006 Ch 7 System Consideration- Paul Lin 10 Example 7-2 Solution (cont.) Rewrite the equations in the matrix form circuit 3.Find I2(s) using determinants

11. 11/15/2006 Ch 7 System Consideration- Paul Lin 11 Example 7-2 Solution (cont.) 4. Find the transfer function

12. 11/15/2006 Ch 7 System Consideration- Paul Lin 12 Example 7-3 Butterworth Low- Pass Filter Determine the transfer function for the following active filter: Determine the transfer function for the following active filter: A second-order Butterworth low-pass filter using an Op-amp, normalized to a cutoff frequency of 1 radian/sec with 1-Ω.A second-order Butterworth low-pass filter using an Op-amp, normalized to a cutoff frequency of 1 radian/sec with 1-Ω. An actual circuit is derived from the normalized circuit by scaling the frequency and resistance levels.An actual circuit is derived from the normalized circuit by scaling the frequency and resistance levels. v1(t) – inputv1(t) – input v2(t) - outputv2(t) - output

13. 11/15/2006 Ch 7 System Consideration- Paul Lin 13 Example 7-3 Solution (cont.) 1. Circuit Identification The op-amp is configured as a “voltage-follower”, Av = 1, or vout = vin. The op-amp is configured as a “voltage-follower”, Av = 1, or vout = vin. No-current is assumed to flow into the input + or non- inverting terminal: Zin = ∞. No-current is assumed to flow into the input + or non- inverting terminal: Zin = ∞. Four nodes: V1(s), V2(s), V3(s), V4(s) Four nodes: V1(s), V2(s), V3(s), V4(s) V2(s) = V4(s)V2(s) = V4(s)

14. 11/15/2006 Ch 7 System Consideration- Paul Lin 14 Example 7-3 Solution (cont.) 3. Write the simultaneous equation and solve for G(s)

15. 11/15/2006 Ch 7 System Consideration- Paul Lin 15 Example 7-3 Solution (cont.) Rearrange the equations, by substituting (7-22) into (7-21) and eliminate V4(s): Then substitute this equation and equation (7-22) into (7-20):

16. 11/15/2006 Ch 7 System Consideration- Paul Lin 16 Example 7-3 Solution (cont.) Simplify the above equation to obtain: And finally the transfer function G(s):

17. 11/15/2006 Ch 7 System Consideration- Paul Lin 17 Example 7-4 Determine the response resulting from an excitation x(t) = 5 sin t, assume that Determine the response resulting from an excitation x(t) = 5 sin t, assume that The input to a certain system is x(t) The input to a certain system is x(t) The output is y(t) The output is y(t) The impulse response of the system g(t) = 10e -t sin 2t The impulse response of the system g(t) = 10e -t sin 2t

18. 11/15/2006 Ch 7 System Consideration- Paul Lin 18 Example 7-4 (cont.) 1. Find the transfer function G(s)

19. 11/15/2006 Ch 7 System Consideration- Paul Lin 19 Example 7-4 (cont.) 2. Use MATLAB to find poles and zeros?? >> den1 = [1 2 5]; >> den2 = [1 0 1]; >> den = conv(den1, den2) den = 1 2 6 2 5 1 2 6 2 5 Factor out the Y(s) by using the reside function >> num =[0 0 0 100]; >> [r, p, k] = residue(num, den) r = 5.0000 + 2.5000i 5.0000 - 2.5000i 5.0000 + 2.5000i 5.0000 - 2.5000i -5.0000 -10.0000i -5.0000 +10.0000i -5.0000 -10.0000i -5.0000 +10.0000i

20. 11/15/2006 Ch 7 System Consideration- Paul Lin 20 Example 7-4 (cont.) p = -1.0000 + 2.0000i -1.0000 - 2.0000i -1.0000 + 2.0000i -1.0000 - 2.0000i -0.0000 + 1.0000i -0.0000 - 1.0000i -0.0000 + 1.0000i -0.0000 - 1.0000i The Y(s)?? Well! Try the Trick formula”

21. 11/15/2006 Ch 7 System Consideration- Paul Lin 21 Example 7-4 (cont.) Well! Try the Trick formula”

22. 11/15/2006 Ch 7 System Consideration- Paul Lin 22 Example 7-5 Determine the transfer function G(x) for the following system description: x(t) = 5 x(t) = 5 y(t) = 10e -2 t + 5e -t sin 2t y(t) = 10e -2 t + 5e -t sin 2tSolution

23. 11/15/2006 Ch 7 System Consideration- Paul Lin 23 7-2 Differential Equation and Transfer Function General differential equation description of the system General differential equation description of the system The highest derivative of the y determine the order of the system The highest derivative of the y determine the order of the system m ≥ n m ≥ n The transform of a highest-order derivative The transform of a highest-order derivative

24. 11/15/2006 Ch 7 System Consideration- Paul Lin 24 Example 7-6 (a). Determine a differential equation expressing the input-output relationship for the circuit below. (a). Determine a differential equation expressing the input-output relationship for the circuit below. (b). determine the transfer function for the circuit (b). determine the transfer function for the circuit

25. 11/15/2006 Ch 7 System Consideration- Paul Lin 25 Example 7-6 Solution This circuit will be solved using nodal equations This circuit will be solved using nodal equations Summing the current leaving the node, ∑ I = 0 Summing the current leaving the node, ∑ I = 0 Differentiate all terms to remove the integral sign, and move v1 terms to the right-hand side of the equation Differentiate all terms to remove the integral sign, and move v1 terms to the right-hand side of the equation

26. 11/15/2006 Ch 7 System Consideration- Paul Lin 26 Example 7-6 Solution (cont.) Assume all I.C.s are zero, and take Laplace transform Assume all I.C.s are zero, and take Laplace transform The transfer function is The transfer function is