# transfer function derivation

#### suzuki

Joined Aug 10, 2011
119
Hello

I have a pretty simple circuit that I'd like to derive the transfer function for. It looks like

VIN----L----C--------------------VOUT
- -
- -
- -
GRD GRD

So i know that i could simply put the parallel branches together and then write a voltage divsion to get the transfer function Vout/Vin. However, when I compare it to transfer functions that are derived in IEEE papers, other factors such as the Quality factor Q are included. How can I also factor in these parameters? Would I also need to consider the resistances of the nonideal inductors in my transfer function?

thanks

edit: sorry, the L2 and Rload branches are between C and Vout

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#### hgmjr

Joined Jan 28, 2005
9,027
I am afraid that your attempt to depict the circuit ended up unintelligible.

hgmjr

#### suzuki

Joined Aug 10, 2011
119

#### steveb

Joined Jul 3, 2008
2,436
So i know that i could simply put the parallel branches together and then write a voltage divsion to get the transfer function Vout/Vin. However, when I compare it to transfer functions that are derived in IEEE papers, other factors such as the Quality factor Q are included. How can I also factor in these parameters? Would I also need to consider the resistances of the nonideal inductors in my transfer function?

thanks

edit: sorry, the L2 and Rload branches are between C and Vout
The decision to include parasitic effects like coil resistance and capacitor ESR (or even ESL) is based on too many unknown factors for us to answer. It all depends on what you are doing and on what the actual component and parasitic values are. I recommend you first do the analysis without the parasitics and then after you understand the basic circuit, consider real world values so you can decide if they are significant. Often, you have to do the analysis with the parasitics included (as a second step) in order to determine if they affect things you care about.

As far as the transfer function, please post the transfer function using the method you mentioned, since it is required that you show some work before we can help you with homework. Then, we can try to help you put it in a standard form which makes it easier to identify interesting properties such as resonant frequencies, cutoff frequencies, Q-factors, damping factors etc.

#### suzuki

Joined Aug 10, 2011
119
As far as the transfer function, please post the transfer function using the method you mentioned, since it is required that you show some work before we can help you with homework. Then, we can try to help you put it in a standard form which makes it easier to identify interesting properties such as resonant frequencies, cutoff frequencies, Q-factors, damping factors etc.
ok great. here is the transfer function that i found so far. i have tried to simplify it as much as i could.

http://i.imgur.com/ys4tg.gif

#### steveb

Joined Jul 3, 2008
2,436
ok great. here is the transfer function that i found so far. i have tried to simplify it as much as i could.

http://i.imgur.com/ys4tg.gif
OK, so this is a third order system. Generally, third order systems are harder to understand than first and second order systems, so usually people like to try and simplify the transfer funtion to be either a sum or concatanation of first and second order systems if possible.

Often, this is done numerically by first plugging in the actual circuit values and then identifying the poles and zeros. Other times, it's possible to make some algebraic approximations. On rare occations, the exact transfer funtion may simplify into first and second order systems.

Let's take your example. In your case, the general transfer function is too complex to simplify exactly. So you need to look at typical component values to see if a simplification can be made. For example, let's say you find (or decide) that 1/(L C) is much greater than R^2/(L L2). Note that at the end of this post you'll see that this physically means that the bandwidth of the bandpass filter times the lower cutoff frequency of the highpass filter is is much less than the center frequncy of the bandpass filter squared. This is physically necessary when you have a high-Q bandpass filter in a typical application.

Start with your transfer function, manipulated a little to make things clearer. Note that I was able to identify a parallel combination of L and L2 (I saw the product over sum form) to simplify the formula.

$$T={{s^2{{R}\over{L}}}\over{s^3+s^2{{R}/{(L//L_2)}}+{{s}/{(LC)}}+{{R}/{(L_2LC)}}}}$$

From there, the denominator can be approximately factored to obtain the following. This factoring is approximate and depends on 1/(L C) being much greater than R^2/(L L2).

$$T={{s^2{{R}\over{L}}}\over{(s+R/L_2)(s^2+s{{R}/{L}}+{{1}/{(LC)}}}}$$

This can then be rewritten as the following.

$$T={{s{{R}\over{L}}}\over{(s^2+s{{R}/{L}}+{{1}/{(LC)}}}} \ \ \ {{s}\over{(s+R/L_2) }}$$

This transfer function is obviously a concatanation of a standard bandpass filter and a standard highpass filter. If this isn't obvious, then you need to learn the standard forms of typical first and second order systems. Once you know the standard form, you can identify center frequency, cutoff frequency, bandwidth, Q-factors, damping factors, gains, etc.

Note that the standard forms are avalilable in system books and on-line as well. In your case, it's clear that the highpass cutoff frequency (in rad/s) is R/L2, the bandpass center frequency is sqrt(1/LC) and the bandwith, Q and damping factors are related to R/L.

The above will either look very straightforward, or very confusing, depending on your experience level. The bottom line for dealing with these situations is that you need some background knowledge of standard system types, which just takes memorization, and you need experience in the math and problem solving, which takes time and practice.

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