I would like to ask how the following is calculated:

sunboy25

I would like to ask how the following is calculated: [Copy link]

These are two examples about active filters in Chapter 3 of "Circuit Design Based on Operational Amplifier Design and Analog Integrated Circuits". Since my theoretical foundation is not very good, I don't know how the drawn lines and circles are calculated. Can anyone tell me the calculation process?

blade365

1. j/(2+j) is obtained by adding j10^3 to the above formula;

2. Multiply the upper and lower parts of the equation in 1 by 2-j, and the result is 1/5+2j/5;

3. Convert the equation in 2 into polar coordinate expression:

C=sqrt(a^2+b^2)=1/sqrt(5) acrtan(b/a)=63.43

blade365

H(j10^3)=C∠θ =1/sqrt(5)∠63.43

sunboy25

blade365 published on 2020-8-31 10:54 1. j/(2+j) is obtained by substituting j10^3 into the above formula; 2. Multiply the upper and lower parts of the formula in 1 by 2-j, and the result is 1/5+2j/5; 3. ...

Also, could you explain how the circled 1000+j500 in Example 3.2 was calculated?

blade365

blade365 posted on 2020-9-1 22:53 This involves partial division operations. Find A1 in equation 3.6. Let a=-1+j2 (1) b=-1-j2 ...

The last formula is wrong. It should be 1/2+j/4

blade365

sunboy25 posted on 2020-8-31 23:10 blade365 posted on 2020-8-31 10:54 1. j/(2+j) is obtained by adding j10^3 to the above formula; 2. Multiply the upper and lower parts of the formula in 1 by 2 ...

In Example 3.1, the final equation for H(s) uses equation (3.6) in the book and uses partial division to find A1.

First decompose the fraction H(s) into partial fractions. For specific operations, please refer to the "Rational Number Decomposition" section in advanced mathematics.

Let a = -1 +2j (1)

b = -1 -2j (2)

Find s*2*10^3=A1(sb)+A2(sa)

Let s = a, then A1 = a*2*10^3/(ab)=2*10^3(1/2 + j/4)= 1000+500j