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How can I simplify the denominator of the Op Amp KVL formula?

chrisjrollin

May 3, 2015
5
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May 3, 2015
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Hello,

I've induced the KVL formula from the Fig 9.21e(page 6) below
(source: https://coefs.uncc.edu/dlsharer/files/2012/04/G6.pdf)

  • Fig.9.21e

    upload_2015-5-3_16-12-53.png
  • Induced Formula:
    Formula_1.png
And my question is that, I'd like to know how the denominator of the formula above could be simplified as
that of Equation 9.43, shown on the page 6(refer to the image below).

Equation_9_43.png


For the numerator, I could make the arrangement simplifying it.
Formula_2.png


But for the denominator, this is by far I could make the arrangement of it.
Formula_3.png

May I ask your help to understand how the denominator could be simplified as that of Equeation 9.43?
I would appreciate your help if you give me some guidance to correctly arrange the denominator again.

Thank you.
 

The Electrician

Jul 6, 2012
117
Joined
Jul 6, 2012
Messages
117
Before trying to get your denominator into the same form as Equation 9.43, you should make sure that all the rest of your algebra is correct.

When I have two relatively complicated symbolic expressions that I need to test for equality, I use a modern symbolic algebra software to expand them both. When fully expanded, it's easy to compare because both are in the same form.

Half way down the page of your post, you give your attempt to derive an expression for GoVd, and then immediately after you give Equation 9.43 from the book for reference.

I have expended both expressions, and they're not the same.

compare-png.19991


You were able to manipulate your numerator to agree with Equation 9.43, but since your expression for GoVd is not the same as Equation 9.43, it will not be possible to make your denominator agree with 9.43

Either Equation 9.43 is wrong, or your expression for GoVd is wrong.
 

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The Electrician

Jul 6, 2012
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I compare my solution derived from Figure 9.21c with yours and they are identical. I think your book has an error in Equation 9.43

govd-png.20001
 

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chrisjrollin

May 3, 2015
5
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May 3, 2015
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Thanks all, you helped me a lot to understand the error of Equation 9.43, shown on the original material attached.
And here's how I also arranged the equation to make it more simplified:

1.PNG

2.PNG

What I've learned from all of it is that, it's not just practicing calculation technique but understanding the mechanism
deducing such equations from circuits.

Thanks.
 

The Electrician

Jul 6, 2012
117
Joined
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Messages
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It's interesting to note that Equation 9.43, even though it's incorrect, has a limiting value for Go -> infinity which is the same as the correct expression for GoVd:

go-limit-png.20026
 

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