Frequency response function

[discombobulated]

Homework Statement

The relationship between the input voltage V(t) and the output voltage across a resistor V_R(t) is:-

CR dV_R /dt + V_R = CR dV/dt

Circuit diagram of a capacitor and resistor with V_R(t) in series across V(t)

1. Show that the frequency response function G(iw) (w= omega) is given by G (iw) = iwCR/ (1 + iwCR)

2.Sketch the locus of the frequency response function, G(iw) (w= omega) on the argand diagram.

Homework Equations

CR dV_R /dt + V_R = CR dV/dt

G (iw) = iwCR/ (1 + iwCR)

The Attempt at a Solution

1. I got this G (iw) = iwCR/ (1 + iwCR) by voltage division, from the circuit, but how could I do that from the given equation?

2. By taking the real and imaginary parts; G(iw) = x + iy

x = (wcr)² /(1 + (wcr)²)
y = wcr/ ( 1 + (wcr)²)

G(iw) = (wcr)² /(1 + (wcr)²) + iwcr/ ( 1 + (wcr)²)

I'm having trouble eliminating wcr now...

 

[unplebeian]

I thought you'd apply Laplace transform and then subst s= jw to get the steady state solution. Then proceed to find the frequency response. But you arrived at the answer in one step, so why are you simplifying it further?

 

[piercas]

I would like to clarify that the frequency response function is a mathematical representation of the relationship between the input and output signals in a system. It is a transfer function that describes how a system responds to different frequencies of the input signal.

In this case, the frequency response function is G(iw) = iwCR/ (1 + iwCR), where w is the angular frequency and CR is the product of the capacitance and resistance in the circuit. This function shows that the output voltage VR(t) is dependent on the input voltage V(t) and the frequency of the input signal.

To derive this function from the given equation, we can use the Laplace transform. By taking the Laplace transform of both sides of the equation, we get sVR(s) = sV(s) + V(s)/CR. Solving for VR(s)/V(s) gives us G(s) = s/ (s + 1/CR). By substituting s = iw, we get the desired frequency response function G(iw) = iwCR/ (1 + iwCR).

As for the locus of the frequency response function on the argand diagram, it represents the complex plane where the real part is represented on the x-axis and the imaginary part on the y-axis. The frequency response function can be plotted as a point on this plane, with the real part being (wcr)^2/ (1 + (wcr)^2) and the imaginary part being wcr/ (1 + (wcr)^2). As the frequency w increases, the point moves towards the y-axis, indicating a higher response to higher frequencies.