Impedance and frequency-domain

  • Thread starter tandoorichicken
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In summary, the conversation discusses impedance in circuit analysis and how to find the total impedance of a circuit by transforming each element characteristic into the frequency domain and then summing up the quantities. The question posed is what happens if this transformation is reversed using an inverse transform, and whether a real resistance value can be obtained. The expert explains that the transformation does not have an inverse for the imaginary part, and that the resistance value is the real part of the impedance. The conversation also touches on the concept of equivalent resistance and how it applies to circuits with only resistors and capacitors or inductors. The expert concludes by mentioning that the impedance can only be calculated for specific frequencies in the time domain, and at the resonant frequency, a pure
  • #1
tandoorichicken
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Homework Statement


This is a general question regarding impedance in circuit analysis.
I know that in order to find the total impedance of a circuit, you transform each element characteristic into the frequency domain, i.e.,
[tex] R\rightarrow Z_R = R[/tex],
[tex] C\rightarrow Z_C = \frac{1}{Cs} [/tex],
[tex] L\rightarrow Z_L = Ls [/tex].

And then sum up the quantities properly to get an equivalent impedance.

My question is, what happens if you transform this quantity back using an inverse transform? Do you get a real resistance value?
 
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  • #2
This transformation doesn't have an inverse for the imaginary part, since the effect of capacitor and inductance cancel out each other...

The resistance value is the real part of the impedance... ie Re(Z) = R,
 
  • #3
what if its a circuit consisting of only a resistor and a capacitor?
 
  • #4
tandoorichicken said:
My question is, what happens if you transform this quantity back using an inverse transform? Do you get a real resistance value?

well, why would you expect that in general you get just a purely resistive circuit when you started with a circuit with a resistor, a capacitor and an inductor?

The property of the circuit won't change just because you have tranformed into a more convenient domain (the freq domain in this case) for calculation purposes and then changed it back. If you begin with a 1 ohms and a 2 ohms resistor, then you can say ok, the total R is 3 ohms, so I can replace them with just one 3 ohms resistor. But that's all you have really done in finding the equivalent resistance.

same applies in the case with capacitors and inductors included. you started with R , C, L. Since C and L make up the imagary part (in freq domain) they have a chance to cancel (fully or partially). If they cancel each other fully, then you will end up having just an R in your equivalent circuit. Now if they don't cancel fully you will have either one L or one C (of different value) left pending on the phase of the equivalent resistance in freq domain.
 
  • #5
In the time domain you can only calculate the impedance of the circuit for specific frequencies. At the resonant frequency, you have a pure resistance, below or over the resonant frequency, you have complex impedances, either RC or LC.
If you have a closed circuit, with sources and/or initial conditions, you can solve the resulting differential equation and have a frequency independent solution.
 

1. What is impedance?

Impedance is a measure of the opposition that a circuit or device presents to the flow of alternating current. It is a combination of resistance and reactance, and is typically measured in ohms.

2. How is impedance related to frequency?

Impedance and frequency are inversely proportional, meaning that as frequency increases, impedance decreases. This is due to the fact that at higher frequencies, the reactance component of impedance becomes dominant, causing the overall impedance to decrease.

3. What is the frequency-domain?

The frequency-domain is a mathematical representation of a signal or system in terms of its frequency components. It allows us to analyze the behavior of a signal as it varies with frequency, rather than with time as in the time-domain.

4. How is impedance measured in the frequency-domain?

Impedance can be measured in the frequency-domain using a variety of techniques, such as network analyzers or impedance analyzers. These instruments apply a known voltage or current to the circuit or device and measure the resulting response, allowing for the calculation of impedance.

5. What practical applications does impedance have in the frequency-domain?

Impedance is a crucial parameter in the design and analysis of electronic circuits and devices. It is used to optimize circuit performance, calculate power dissipation, and determine the frequency response of a system. It is also important in the design of communication systems, where impedance matching is necessary for efficient signal transmission.

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