Transfer function is a ratio between the input u(t) and output y(t). This evaluates the zero initial condition of the system. The system in figure 1. has : 𝑅1 = 60𝑘𝛺, 𝑅2 = 100𝑘𝛺, 𝑅3 = 200𝑘𝛺 𝑎𝑛𝑑 𝐶1 = 𝐶2 = 0.2µ𝐹. For the given system shown in Figure 2, we are required to solve for its state space equation in terms of 𝑅1, 𝑅2, 𝑅3, 𝐶1, 𝐶2, with numerical values. In this text, we derived the transfer function in phase domain.

2. Transfer function

In this text, we derived the transfer function of the given system in phase domain.

Get the equivalent resistance for parallel combinations:

Apply simultaneous voltage division theorem (VDT) for point a and b in the circuit. For point a, get the value of V_a.

Equate value of V_a as source voltage for the series combination R_2+1/(C_2 s) , and obtain voltage Y(s) by equating (eq.1) to (eq.2).

Thus, the transfer function G(s) for the modelled circuit is:

Simplifying the modelled transfer function as:

3. Conclusion

In this text, we analytical derived the transfer function of the simple system as shown in Figure 1. We used a circuit reduction and voltage division theorem to model the differential equation of the system. After which, we applied Laplace transform and derived the transfer function in phase domain.

4. References

[1] Norman S. Nise. Control Systems Engineering (7th. ed.). 2015. John Wiley & Sons, Inc., USA.

[2] DiStefano, J. J., Stubberud, A. R., & Williams, I. J. (2013). Schaum's outline of theory and problems of feedback and control systems. New York: McGraw-Hill.

[3] Bishop, Robert H., and Richard C. Dorf. "Modern control systems." (2017).

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