# I'm having difficulties finding an expression for the amplification of this OP-amp

#### Carllinderg

Joined Feb 22, 2016
2
Hi, in our assignment we have this circuit http://imgur.com/BiXFnit
and we are to derive an expression for the frequency response of the amplification (Uin/Uut) and write it in "Bode form" (don't know how it's known in english) so that we can draw a Bode plot by hand. Using nodal analysis I arrive at this expression for Uin/Uut http://imgur.com/rKZidQM, but I'm stuck at this stage. I don't know how to get this expression into Bode form and I would greatly appreciate any help.

#### crutschow

Joined Mar 14, 2008
23,783
The expression you have can be used to generate the Bode plot.
A Bode plot is simply the gain (output / input) and phase of the amp versus the input frequency.
It's typically plotted with a log scale for the frequency on the X-axis and gain plotted in dB on the Y-axis.
Phase is plotted with a linear Y-axis.
You calculate the gain/phase for a number of frequencies from your gain expression and use that to generate the plot.

#### Carllinderg

Joined Feb 22, 2016
2
Thank you, but the assignment calls for us to write it in what in Swedish is called "Bode normal form" where you need to express it as a product of these factors in image 1 http://imgur.com/a/xUqLO as well as constant unit-less amplification factors. A finished expression could look like the one in image 4, where the cut-off frequencies are calculated. (cut of frequency = 1/(2piR*C)

For example, the frequency response of a circuit would look like this http://imgur.com/8oRF9Id where Av is the amplification. It has to be expressed as a product of these factors so that they may be easily drawn by hand. Sorry if I'm not explaining myself well, English is not my first language.

#### crutschow

Joined Mar 14, 2008
23,783
Sorry, your links don't work for me. #### bertus

Joined Apr 5, 2008
20,180
Hello,

None of the links work. They all give a 404 error.

Bertus

#### MrAl

Joined Jun 17, 2014
6,811
Hi,

It sounds like you want to separate the factors in the numerator and denominator so that you can know what poles and zeros you have so that you can do the Bode plot. To start you would factor the numerator and denominator, then express each factor as something plus 1, such as:
(s/2+1)/((s/3+1)*(s/4+1)*(s/9+1))

Note this is factored but also each factor has a "+1" in it.

The starting point though is to factor numerator and denominator, then divide numerator and denominator by whatever is needed to get into that form.

Example:
(s+1)/(s^2+3*s+2)

The numerator is factored, so factoring the denominator we get:
(s+1)/((s+1)*(s+2))

and step 1 is complete, and the next step involves division.

Two of the factors are already in the form something plus 1, but we divide top and bottom by 1 anyway for clarity and get:
(s/1+1)/((s/1+1)*(s+2))

Now for the s+2 factor we divide top and bottom by 2 and get:
(1/2)*(s/1+1)/((s/1+1)*(s/2+1))

and that is in the correct form now, and we ended up with a constant of 1/2 and one zero and two poles and is now ready for graphing by hand without any further calculations.

Quadratic factors are a little more work but same idea. They also sometimes require a curved Bode line rather than a straight line. The curve depends on the damping factor.

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