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ECET 307

ECET 307. Analog Networks Signal Processing Ch 7 System Considerations 2 of 3 Fall 2006 http://www.etcs.ipfw.edu/~lin. Ch 7: System Considerations. Transfer Function Examples Differential Equation and Transfer Function Step and Impulse Responses. Example 7-1.

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ECET 307

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

  2. Ch 7: System Considerations • Transfer Function Examples • Differential Equation and Transfer Function • Step and Impulse Responses 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). Ch 7 System Consideration- Paul Lin

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

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

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

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

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

  12. Example 7-3 Butterworth Low-Pass 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. • An actual circuit is derived from the normalized circuit by scaling the frequency and resistance levels. • v1(t) – input • v2(t) - output 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. • No-current is assumed to flow into the input + or non-inverting terminal: Zin = ∞. • Four nodes: V1(s), V2(s), V3(s), V4(s) • V2(s) = V4(s) Ch 7 System Consideration- Paul Lin

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

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

  18. Example 7-4 (cont.) 1. Find the transfer function G(s) 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 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 -10.0000i -5.0000 +10.0000i Ch 7 System Consideration- Paul Lin

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

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

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

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

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

  25. Example 7-6 Solution • This circuit will be solved using nodal equations • 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 Ch 7 System Consideration- Paul Lin

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

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