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

[thomas49th]

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 capacitor only lags current. I'm confused

Thanks

 

[vela]

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^jϕ. How are the phases of these three complex quantities related?

 

[thomas49th]

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

 

[vela]

To get subscripts and superscript, go into advanced mode when posting, and you'll see options above the input area.

I'm not sure how far along you're into this network analysis stuff, but the transfer function is usually written as a function of s, that is, H(s) = V_o(s)/V_i(s). You talk about the s domain and the time domain, blah blah blah. When you're looking at a pure sine wave of frequency ω, that corresponds to the case when s=jω, so you often see quantities written as function of jω. It's just a convention. Maybe your class does it differently. (I'm going to use s instead of jω because it's easier to type. Just replace s with jω or simply ω everywhere you see it.)

What linear systems do is: 1) change the amplitude of a signal, and 2) change the phase of a signal. The factor by which the amplitude changes is called the gain. It corresponds to the magnitude of the transfer function. The shift in phase of the signal, as you can probably guess, is equal to the phase of the transfer function. Hopefully you can see this from the relationship V_o(s) = H(s) V_i(s), but I'll leave it to you to convince yourself that the statements are true.

So what the problem is asking you to find is the frequency at which the phase of the transfer function is -45 degrees and then to find the corresponding gain.

 

[DeeNos]

for your question! The argument of a transfer function, or arg(T(w)), represents the phase shift of the output signal compared to the input signal. In real life, this can be seen in various electronic systems such as filters, amplifiers, and control systems.

In your example of a low pass filter, the transfer function T(w) is a complex number that includes both magnitude (|T(w)|) and phase (arg(T(w))) components. The magnitude represents the ratio of the output voltage (V2) to the input voltage (V1), while the argument represents the phase shift between the two signals.

To calculate the magnitude of T(w) when V2 lags V1 by 45°, you can use the Pythagorean theorem to find the magnitude of the complex number. The magnitude will be the square root of the sum of the squares of the real and imaginary parts of T(w). In this case, the real part is 1 and the imaginary part is -RwC. So, the magnitude will be |T(w)| = √(1 + (RwC)^2).

As for the lag between V2 and V1, in this case, it is due to the phase shift caused by the capacitor in the filter. The capacitor does not only lag current, but it also causes a phase shift between voltage and current. This is why the phase shift is included in the transfer function.

I hope this helps to clarify the concept of the argument of a transfer function and how it relates to the magnitude and phase shift in a real-life system. Remember, the transfer function is a powerful tool in analyzing and designing electronic systems, and understanding its components is essential for any scientist or engineer.