# Argument Of Transfer Function - What does arg(T(w)) mean in real life

Hi, I have a question something along the lines of

Here is a low pass filter, when the resistor = R and the capacitor = C. There is a sinusoidal voltage input source (V1) and a voltage across the capacitor (V2). The transfer function is T(w) = V2/V1.
Determine the |T(w)| and arg(T(w)) where T(w) is the transfer function. Calculate the magnitude of T(w) when V2 lags V1 by 45°

OKAY! So w = 2*pi*f.

T(w) = $$\frac{1-jRwC} {-(RWC)^2 - 1}$$

|T(w)| = $$\frac{\sqrt{1+(RwC)^2}}{(RWC)^2 - 1}$$

so am I right in thinking arg(T(w)) is tan(x) = -RwC???

BUT how on earth do I determine the magnitude when V2 lags V1 by 45°. I thought the capcitor only lags current. I'm confused

Thanks

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vela
Staff Emeritus
Homework Helper
Hi, I have a question something along the lines of

Here is a low pass filter, when the resistor = R and the capacitor = C. There is a sinusoidal voltage input source (V1) and a voltage across the capacitor (V2). The transfer function is T(w) = V2/V1.
Determine the |T(w)| and arg(T(w)) where T(w) is the transfer function. Calculate the magnitude of T(w) when V2 lags V1 by 45°

OKAY! So w = 2*pi*f.

T(w) = $$\frac{1-jRwC} {-(RWC)^2 - 1}$$

|T(w)| = $$\frac{\sqrt{1+(RwC)^2}}{(RWC)^2 - 1}$$

so am I right in thinking arg(T(w)) is tan(x) = -RwC?
Yes, that's right.
BUT how on earth do I determine the magnitude when V2 lags V1 by 45°. I thought the capacitor only lags current. I'm confused.
V1(jω), V2(jω), and T(jω) are all complex quantities, so you can write each in polar form, i.e., re. How are the phases of these three complex quantities related?

$$r = \sqrt{1+(RwC)^2}$$
ϕ = arctan(-RwC)

V2/V1 = T is that what you were suggesting? so can you say that something like T(jw) = V1(jw)/V2(j(w-pi/2))?

P.S How did you get to use super and sub script without LaTex?

EDIT: Is any of this useful
V1 = cos(w) + jsin(w)
V1 = e^(jϕ)
V2 = V1/(100jwC + 1)

or am I just spouting obvious facts?

Is there like some magical property such that the phases add up to 90°?

I thought about adding phasers together but If you wanted me to do that you wouldn't of suggested converting into polar notation as that is useful for multiplication and division?

Why have you written j's in front of the w's? Are you talking about the complex part only?

Thanks
Thomas

Last edited:
vela
Staff Emeritus
Homework Helper