The transfer function of a high-pass filter circuit with a resistor and capacitor is found using impedance and the voltage divider rule. The transfer function in the Laplace domain is H(s) = Vout(s) / Vin(s), simplifying to H(s) = 1 / (1+sRC) for RC circuits.
To find the transfer function for a circuit considering Vi as the input and Vout as the output, you need to analyze the circuit component's relationships in the frequency domain. For a high-pass filter using a resistor (R) and a capacitor (C), the transfer function in the Laplace domain, H(s), is given by H(s) = Vout(s) / Vin(s), where s is the complex frequency variable from Laplace transform.
The transfer function of the high-pass filter, which is a form of the RC circuit, can be derived by first writing the impedance of the capacitor, Zc = 1/(sC), and then applying the voltage divider rule where Vout = Vi * (Zc / (Zc + R)) leading to the transfer function H(s) = 1 / (1+sRC). In the case of a circuit with a resistor (R) and an inductor (L), you would instead use the inductor's impedance, Zl = sL.
This is a simplified analysis, with more complex circuits requiring a more detailed approach, taking into account additional components and their configurations. Additionally, for sinusoidal steady-state analysis, s is replaced by jω (where j is the imaginary unit and ω is the angular frequency) to find the frequency response of the circuit.