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Impedance network and complex algebra

  1. Nov 6, 2014 #1
    1. The problem statement, all variables and given/known data
    This is one part of a wider question, I'm only posting the part I'm having trouble with.
    $$
    \begin{align}
    \text{Given an impedance network } B &= \frac{Z_1 \parallel Z_3}{Z_2 + Z_1 \parallel Z_3} \\
    \text{show that: } \frac{1}{B} &= 1 + \frac{R_2}{R_1} + j\frac{\omega CR_2}{1 + \omega CR_3}
    \end{align}
    $$
    2. Relevant equations
    $$\begin{align}
    Z_1 &= R_1 \\
    Z_2 &= R_2 \\
    Z_3 &= R_3 + C_1 \text{(in series)}
    \end{align}$$
    3. The attempt at a solution
    $$\begin{align}
    Z_3 &= R_3 - j\frac{1}{\omega C_1} \\
    Z_1 \parallel Z_3 &= \frac{Z_1 Z_3}{Z_1 + Z_3} \\
    &= \frac{R_1 \left( R_3 - j\frac{1}{\omega C_1} \right)}{R_1 + R_3 - j\frac{1}{\omega C_1}} \\
    \frac{1}{B} &= \frac{Z_1 \parallel Z_3}{Z_1 \parallel Z_3} + \frac{Z_2}{Z_1 \parallel Z_3} \\
    &= 1 + \frac{R_2}{\frac{R_1 \left( R_3 - j\frac{1}{\omega C_1} \right)}{R_1 + R_3 - j\frac{1}{\omega C_1}}} \\
    &= 1 + \frac{R_2 \left( R_1 + R_3 - j\frac{1}{\omega C_1} \right)}{R_1 \left( R_3 - j\frac{1}{\omega C_1} \right)}\\
    &= 1 + \frac{R_2 \left( R_3 - j\frac{1}{\omega C_1} \right)}{R_1 \left( R_3 - j\frac{1}{\omega C_1} \right)} + \frac{R_2 R_1}{R_1 \left( R_3 - j\frac{1}{\omega C_1} \right)}\\
    &= 1 + \frac{R_2}{R_1} + \frac{R_2}{R_3 - j\frac{1}{\omega C_1}} \\
    &= 1 + \frac{R_2}{R_1} + \frac{j\omega C_1 R_2}{1+ j\omega C_1 R_3}
    \end{align}$$
    I just can't get the right hand element to match, I've got an extra ##j##. What am I doing wrong?
     
  2. jcsd
  3. Nov 6, 2014 #2

    NascentOxygen

    User Avatar

    Staff: Mentor

    It's a fair bet that you are right, and the typesetter has dropped an italic j.

    I
    followed your working, nothing leapt out as being wrong.

    One way to check your own work is to assign the pronumerals some easy numeric values, and evaluate the expressions (manually, or using a calculator).
    e.g., let R1=1, R2=4, R3=3, C=5, w=1
     
  4. Nov 6, 2014 #3

    NascentOxygen

    User Avatar

    Staff: Mentor

  5. Nov 10, 2014 #4
    Thanks for the replies. I've just had a response from the tutor, the question is wrong and my answer is correct. :)
     
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