# Rc filter transfer function

How do you know if a transfer function for a RC filter is correct?

For example how would you tell whether or not this is a correct transfer function: $\frac{\frac{R_1}{R_1R_2}}{1+j\omega R_1C}$

Or generally how would you tell if a function is indeed a correct transfer function?

Edit: If I have understood this correctly, then the below function cannot be a transfer function.

$\frac{1+ \frac{j\omega CR_1R_2}{R_1+R_2}}{1+j\omega R_1C}$

• You calculate it correctly, then it's correct. That's true for every model of a physical system. What is your question? Sep 3, 2017 at 10:03
• My question is this, is there a way of telling whether or not a function is a transfer function without having the circuit and only the function itself. Is there a property that all transfer functions share that would make you able to tell this? Sep 3, 2017 at 10:12
• Y = $X^2$ - is this correct? Sep 3, 2017 at 10:16
• @user3368747 define "all transfer functions": Of what kind of things? A transfer function is just a mapping from an input to an output function of time; every function that does that is a "correct" transfer function. Just as Andy said, you can't say "is $Y=X^2$ a correct function?", without defining correct. Sep 3, 2017 at 10:19
• I'd add a circuit diagram as otherwise we can't check that the transfer function you have is the correct one. There are a few sanity checks you can do as pointed out in the answer you have accepted but if we don't know what $R_1$, $R_2$ and $C$ are we can't check it for you. Sep 3, 2017 at 15:10

No way to be 100% sure of the correctness, but some validations can be done:

1) calculation of units/dimmension

You add 1 (adimensional) and $j\omega R_1C$ thus $\omega R_1C$ must be adimensional. It is, pass check 1.

The global function has unit $\Omega^{-1}$, this is not valid for a transfer function of type $V_{out}/V_{in}$ (that must be dimensionless) but could be valid for $I_{out}/V_{in}$. Suspicious. What kind of transfer function are you calculating? Should it be $\frac{R_1}{R_1+R_2}$ ?

2) Do some limit checks.

Your transfer function is $1/R_2$, independent of $R_1,$ when w=0. Is it in the circuit ? Suspicious.

Your transfer function is 0 when $R_2 = \infty$. Could you check that in the original circuit?

...

The second case that is included in the question is dimensionless and internally coherent, thus, it is a valid transfer function for a system Vout/Vin or similar. That doesn't means it is correct, but it has the correct dimensions.

• Yes, you are correct it should have been +. I have corrected the mistake. So, basically the transfer function needs to have correct dimensions if it is indeed a transfer function? Sep 3, 2017 at 14:28
• Please, do not "correct" it. As example of how to verify a transfer function, the original case, with an small error, is best. Sep 3, 2017 at 14:29
• Changed it back. I also added another function, and from what I understand this one cannot be a transfer function since it has dimensions $\Omega^{1}$ Sep 3, 2017 at 14:34
• All physic formulas must have the correct dimensions: if you evaluate an speed, it must be m/s; if you evaluate a transfer function that is Vout/Vin, it must be adimensional (warning: if the transfer is between, by example, volts to motor speed, it must be m/(s*v) ); etc. Moreover, all subparts of the expression must be coherent: you can not add/substract items with different dimension. Sep 3, 2017 at 14:34
• Kudos for doing the dimensional analysis! I just want to point out that $I_{out}/V_{in}$ is not totally invalid. For example, vacuum tubes and MOSFETs are best modeled as voltage-controlled current sources, and have transfer functions of exactly this form. This is precisely why the term "transconductance" (units of $\Omega^{-1}$) was coined. Sep 7, 2017 at 17:07

A simple RC filter might be high pass or low pass and it will have the transfer function form of: -

$\dfrac{j\omega CR}{1+j\omega CR}$ for a high pass

And

$\dfrac{1}{1+j\omega CR}$ for a low pass

If you know that your formula relates to a high pass or low pass simple RC filter then you can definitely say it is wrong by examining the units. Having effectively 1/R in the numerator makes it have the wrong units for a T.F. for this type of simple circuit.

So, unless you can categorically show the circuit, you cannot really proceed much.

• These forms are valid only for adimensional transfer functions (i.e. Vout/Vin or Iout/Iin), first order, and ignores the possible multiplier factor. Sep 3, 2017 at 12:13

I have no idea what this question means but if you just want to verify that the transfer function you have found is correct or not, one of the ways is to do the same thing via another method and see if you get the same answer . I guess you are doing the basics so what you must have been taught is use tools like VDR/CDR etc (as they work for linear circuits in phasor domain/frequency domain, the trick is especially used when you have just started AC circuits). You can use them to find the transfer function, the other way is to make the mathematical model of the system that is sort of make a differential equation (something you would have done when you would have studied RC/RL/RLC and then integrated to find unit step response because that is the most basic one). Once the equation has been made, you can transition to the frequency domain or S-domain by Laplace and put S=jw and see if you get the same answer.

If you have not studied that then you must atleast have a guess what the response would look like if plotted, write the transfer function down in MATLAB and make a bode plot.

These are the easiest secondary ways to verify, the primary way is to recheck while making sure that you have not made any calculation/logical mistake