Transfer function and magintude known, unsure how to appoarch

Click For Summary
SUMMARY

The discussion centers on calculating the capacitive reactance (XC) required to make a complex load impedance (ZLoad = 138Ω + 460jΩ) appear purely resistive when placed in parallel with a given line impedance (ZLine = 2Ω + 20jΩ). The participants analyze the relationship between the impedances using the formula for parallel impedance and explore the implications of the unknown frequency (omega). A critical point raised is the need to clarify whether ZLine represents the characteristic impedance or a series impedance, as this affects the calculations significantly.

PREREQUISITES
  • Understanding of complex impedance and phasor representation
  • Familiarity with parallel impedance calculations
  • Knowledge of Thevenin's theorem and its application in circuit analysis
  • Basic concepts of capacitive reactance and its role in AC circuits
NEXT STEPS
  • Study the application of Thevenin's theorem in AC circuit analysis
  • Learn about calculating equivalent impedance in parallel circuits
  • Research the significance of characteristic impedance in transmission lines
  • Explore the relationship between frequency (omega) and impedance in AC circuits
USEFUL FOR

Electrical engineering students, circuit designers, and professionals working with AC circuit analysis and impedance matching will benefit from this discussion.

cybhunter
Messages
24
Reaction score
0

Homework Statement



Vs=7200V 0 degrees
ZLine=2Ω+20jΩ
ZLoad=138Ω+460jΩ
XC=unknown capacitance

(omega not given)

Find the capacitive reactance when, place in parallel with ZLoad will make the load appear purely resistive


Homework Equations


conj(Zline) =2Ω-20jΩ (for max power transfer to the load)

Zparallel=[(Zload)(XC)]/[Zload+XC] =Zline


The Attempt at a Solution



I multiply the numberator and denomator by the conjugate of (138+j460-jXC) to get

[[(460XC-j138XC)*(138-i(460-XC)]/(138^2+(460-XC)^2)]=2-20j

when I rearrange, I end up with:
68XC^2+240XC^2-9800XCj+980XC+210447i=230644
as you can see, I end up with a complex number as a solution, which does not make sense

How should I approach these kinds of problems where the transfer function (sans a component value) and the resulting vector is known?
 
Physics news on Phys.org
cybhunter said:
ZLoad=138Ω+460jΩ
XC=unknown capacitance

(omega not given)

Find the capacitive reactance when, place in parallel with ZLoad will make the load appear purely resistive
I don't think maximum power transfer need be brought into this. You are just asked to change 138Ω+460jΩ into 138+j0

If your examiner intended you do more than this, then the question fails to show it, as far as I can see.
 
Is Zline the characteristic impedance Z0 of the line, or is it a series impedance at the source? It's a very strange value to be Z0 of the line. Typical Z0 of a line is 50 + j0 ohms all the way to 300 + j0 ohms.

BTW Ω is not a parameter, it stands for "Ohms"!
 
rude man said:
BTW Ω is not a parameter, it stands for "Ohms"!
I believe OP is saying ω is not specified.
 
NascentOxygen said:
I believe OP is saying ω is not specified.

I believe you're right!
 
after racking my brain, going through practically a ream of paper and almost pulling my hair out for a couple days, apparently someone placed the problem on yahoo answers over a year ago:

http://answers.yahoo.com/question/index?qid=20110523124504AA7phCg

When I product over sum the impedance values (by theoretically shorting the independent voltage source), I end with an interesting conundrum: the Zth value is
2.06816Ω+19.1948jΩ (from (2+20j)*(138+460j)/(140+480j)). By plugging in XC= -19.1948j into the load resistance I end up with a j value that equals:
(-19.1948j)(138+460j)/(138+440.8052j) equals 0.2383-19.95j. When added to the the 2+20j (Line impedance), the resulting value is 2.2383+0.044j (2.239∠1.13) . I may be splitting hairs here but that 1.13 is driving me nuts
 
Something strange here. You need to know the electrical length θ = βx where β = phase constant and x = actual line length.

β = ω/v = ωT = 2π/λ

so you need to know ω and also delay time/unit length T, or just wavelength λ. Or L and C, inductance & capacitance per unit length.

These parameters cannot be deduced from Z0 alone.
 
cybhunter said:
after racking my brain, going through practically a ream of paper and almost pulling my hair out for a couple days, apparently someone placed the problem on yahoo answers over a year ago:

http://answers.yahoo.com/question/index?qid=20110523124504AA7phCg
That's handy. :smile:
When I product over sum the impedance values (by theoretically shorting the independent voltage source), I end with an interesting conundrum: the Zth value is
2.06816Ω+19.1948jΩ (from (2+20j)*(138+460j)/(140+480j)).
I'm with you there.
By plugging in XC= -19.1948j into the load resistance I end up with a j value that equals:
(-19.1948j)(138+460j)/(138+440.8052j)
That's where you are going wrong. You have determined what XC must be connected in parallel with the Thevenin impedance, but now you have decided to connect that XC in parallel with only part of what comprises the Thevenin impedance. Connecting a capacitor in parallel with the load (viz., 138+j460) is not equivalent to connecting that same capacitor in parallel with the Thevenin impedance.