The transfer function of the RL circuit is H(s) = 1 - (R1 + R2 + L1s)/R2. The magnitude of the transfer function represents the gain of the circuit at different frequencies.
To determine the transfer function and plot its magnitude, follow these steps:
Identify the input and output terminals
In the given circuit, the input is the voltage source, Vin, and the output is the voltage across the load, Vo.
Apply Kirchhoff's Voltage Law (KVL) to the circuit
Starting from the input voltage source, follow the loop around the circuit, applying KVL:
Vin - R1I1 - L1dI1/dt - R2*I2 = 0
Apply Kirchhoff's Current Law (KCL) at the junction
I1 = I2 + Io
Express the current through the load resistor (R2) in terms of the output voltage (Vo)
Io = Vo/R2
Substitute the current expression into the KVL equation
Vin - R1I1 - L1dI1/dt - R2*Io = 0
Vin - R1I1 - L1dI1/dt - R2*(Vo/R2) = 0
Substitute the current relationship (I1 = I2 + Io) into the modified KVL equation
Vin - R1*(I2 + Io) - L1*d(I2 + Io)/dt - Vo = 0
Combine like terms and rearrange the equation
Vin - R1I2 - R1Io - L1dI2/dt - L1dIo/dt - Vo = 0
Vin - (R1 + R2)Io - L1dIo/dt - Vo = 0
Express the output voltage (Vo) in terms of the input voltage (Vin)
Re-arrange the equation to solve for Vo:
Vo = Vin - (R1 + R2)Io - L1dIo/dt
Take the Laplace transform of both sides of the equation
Vo(s) = Vin(s) - (R1 + R2)Io(s) - L1sIo(s)
Express Io(s) in terms of Vo(s) using the relationship between Io and Vo
Io(s) = Vo(s)/R2
Substitute Io(s) into the Laplace transformed equation:
Vo(s) = Vin(s) - (R1 + R2)(Vo(s)/R2) - L1s(Vo(s)/R2)
Solve for the transfer function H(s) = Vo(s)/Vin(s)
H(s) = Vo(s)/Vin(s) = 1 - (R1 + R2 + L1s)/R2
Plot the magnitude of the transfer function (Bode plot)
Using a Bode plotter or other tools, plot the magnitude of the transfer function |H(s)| as a function of frequency (f). The magnitude of the transfer function represents the gain of the circuit at different frequencies.