Obtain transfer function using KVL

[majdi]

Hello and good day all, i need some guide for answer this homework.

Given C=1F, R1,R2 = 1ohm

1.Obtain the laplace transform of every component.

My Answer :

2. Obtain the transfer function of this system. You may use KVL to analyse the network.
-Here im stuck.

KVL should be:
V1(s) = VR1(s) + VC(s)
VC(s)= VR2(s) + VL(s)
V0(s) = VL(s)

i dont know how to get the transfer function.. Any link and material will be great full

 

[Laplace]

First recommendation: drop the (s) notation; the extra (s)'s will be confusing.

The transfer function is Vout/Vin so you want to find V0/V1. Using KVL is problematic because the first step in using KVL is to designate loop currents and solve for those. But that does not get you V0 easily. If instead one uses KCL, then sum the currents at a node in terms of the node voltages and impedance between nodes. Here there are two nodes with unknown voltages: V0 and V@R1R2C. So write two node equations. Solve and substitute to eliminate V@R1R2C, leaving V0. Then find the ratio V0/V1 as the transfer function.

 

[LvW]

I think, we can say that the voltage divider rule is based on KVL. Hence, applying this rule twice (at the node in the middle and at the output) the problem can be solved using KVL. .

 

[majdi]

Thank Laplace and LvW, i used to find two step. First for V1(s) and second V0(s) the solve with V0(s)/V1(s) to get the answer.






Find center voltage between R1-C-R2 as Vc(s)

Vc(s) = z - 1/sC / z X V1(s)

z = 1/sC x sL / 1/sC + sL

V0(s) = sL / R2 + sL x Vc(s) x V1(s)

V0(s) / V1(s) = sL / R2 + sL x Vc(s)

 

[Laplace]

I am experiencing difficulty with recognizing what paradigm you have used to write the circuit equations. However, it is a certainty that the transfer function will be some combination of R, L, C & s. Nothing else. The transfer function given above has a V.

 

[majdi]

I follow this reference http://www.egr.unlv.edu/~eebag/Chapter14.pdf

on page 3.

 

[LvW]

It is nearly impossible to read and check your "equations".
I only can guess that your "z" is the same quantitty as the "Z" in the reference (page 3): Impedance at the node in the middle of the circuit, correct?
In this case, your eqation (z = 1/sC x sL / 1/sC + sL) looks like a parallel connection of 1/sC and sL. This is, obviously, wrong because you did forget R2..
(In general, please use brackets for writing correct equations).

 

[Laplace]

Let's forget about your particular homework problem and consider instead the general problem of a cascade voltage divider. As shown in the circuit below, each of the 'Z' boxes can represent either a simple resistance, an imaginary reactance, or a complex impedance. The first method of solution shown is to write the node equations. The blue arrows on the circuit show the currents being summed at each node. Current leaving the node is designated as positive. The second method of solution is to use the voltage divider rule twice. Each method yields the same general transfer function that will apply to any circuit having the general form of a cascade voltage divider.

You can take this as an example of how to write node equations, or as an example of how to calculate voltage dividers, or you can take the general transfer function for cascade voltage division and apply it to your particular problem. For instance, I took the problem from page 3 of your reference and substituted the values as shown - and got the same answer, although I would recommend using a symbolic algebra engine since I don't like doing math drudge work.