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Help with Complex Frequency

  1. Jun 10, 2012 #1
    Help with "Complex Frequency"

    Hello all,

    I need some help with this concept. I don't really see how a frequency can be complex. I am using a vector network analyzer over a range of a few GHz, however the model I'm using requires as a complex frequency as input. How do I convert an angular frequency to complex frequency?

    By searching other posts, I found that complex frequency s = σ + jω, where σ is related to the decay rate. But how do I find this decay rate from a machine which is simply operating X Hz?

  2. jcsd
  3. Jun 10, 2012 #2
    Re: Help with "Complex Frequency"

    Are you really looking at a vector frequency like F(ωt) = Fo [cos(ωt) + j·sin(ωt)]?
  4. Jun 11, 2012 #3
    Re: Help with "Complex Frequency"

    I'm not entirely sure what you mean, but I don't think so.

    I simply have a model for the dielectric properties of a test material, and the model is explicit in s, the complex frequency. However I've only measured some reflection coeficients at specific frequencies, and I have until now only ever heard of real frequencies.
  5. Jun 11, 2012 #4

    Andy Resnick

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    Re: Help with "Complex Frequency"

    I've seen complex frequencies used when computing thermal properties of materials (specifically, the Hamaker constant).
  6. Jun 11, 2012 #5
    Re: Help with "Complex Frequency"

    Complex frequency may be misleading because it includes more information than just frequency. You have to consider it in the context of sinusoidal waves with a certain phase in time. Complex frequency is not a different or special kind of "imaginary frequency", the only frequency that is physical is real-number frequency; the phase and magnitude information is affected by this complex component. The s-plane includes magnitude and phase information in addition to frequency information, and complex numbers are a good way of representing the inter-relation of all these quantities. It is just given this name "complex" because complex numbers are how we represent the same values, magnitude - phase - frequency, in the frequency and time domain (see Euler's ... identity/formula I can't remember which is which).
    Last edited: Jun 11, 2012
  7. Jun 11, 2012 #6
    Re: Help with "Complex Frequency"

    Ok, that makes sense.

    So how do I construct the complex frequency from real frequency? Is there something like a fourier transform?
  8. Jun 11, 2012 #7
    Re: Help with "Complex Frequency"

    You should think of complex frequency as the s-plane or another complex domain. In fact, I don't really ever recall using the term complex frequency as a specific term during my studies, but I am not an expert. It seems very confusing to attach to that term. Let me be more specific, you will not find complex "imaginary frequency" information in the frequency domain that never shows up or hides itself from the time domain information that you experience as physically real.

    Are you familiar with phasors? http://en.wikipedia.org/wiki/Phasor That should help you to get to the bottom of your problem. Can you express a phasor as a complex number, and then encode the phase,magnitude, and frequency of your signal into complex form? My guess is that your network analyzer wants to know all of this information rather than just frequency.

    There is a guy on here called rbj that probably can give you an accurate and better answer, so maybe you should ask this in the EE forum.
    Last edited: Jun 11, 2012
  9. Jun 11, 2012 #8
    Re: Help with "Complex Frequency"

    Thanks for the help. At least now I know what to look for. So the article on S plane in wikipedia gives the following laplace transform:

    [itex]{F(s) = \int_0^{inf} f(t)e^{-st}dt}[/itex]​

    So if I have a constant frequency with time, I should get as F(s):

    [itex]{F(s) = \int_0^{inf} \omega e^{-st}dt} = \frac{\omega}{s}[/itex]​

    But now what is s?
  10. Jun 11, 2012 #9


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    Re: Help with "Complex Frequency"

    actually, i never thought that the problem new electrical engineers (and students) usually are having was with "complex frequency", but was with "complex signals" and with "negative frequency".

    i guess the concept of complex frequency might have some meaning in the context of exponentially damped (or increasing, in an unstable system) sinusoids. i guess you could say that this signal:

    [tex] x(t) = e^{-\alpha t} \cos( \omega t ) [/tex]

    has a complex frequency where [itex]\omega[/itex] might be considered to be the real part and [itex]\alpha[/itex] is the imaginary part.

    is this what you're having trouble with?
  11. Jun 11, 2012 #10
    Re: Help with "Complex Frequency"

    I think you have confused the laplace transform. f(t) is a function of time, and so w is just a constant in your attempt. You just transformed a DC signal (with amplitude of w). This will not give you the complex representation of a sine with a frequency w.

    Look at this, and find sin(wt) and cos(wt).
  12. Jun 11, 2012 #11
    Re: Help with "Complex Frequency"

    Can you provide the VNA model that you are using, and the specific setup item that you are trying to input. As mentioned above, you may be confusing complex frequency with complex frequency response. If you can show us exactly what you are trying to do we can help you sort it out.
  13. Jun 12, 2012 #12
    Re: Help with "Complex Frequency"

    Here is the VNA I'm using:


    And I'm trying to get the dielectric permitivities in relation to Y, the aperture admitance from this model.

    [itex]\begin{equation} Y = \frac{\sum_{n=1}^{4}
    \sum_{p=1}^{8}\hat{\alpha}(\sqrt{\epsilon_{r}})^{p}(sa)^{n}}{1+\sum_{m=1}^{4} \sum_{q=1}^{8}\hat{\beta}(\sqrt{\epsilon_{r}})^{q}(sa)^{m}}
    \end{equation} [/itex]​

    Where [itex]\ s = \sigma + i\omega[/itex] the complex frequency [itex]\ a[/itex] is the conductor radius, [itex]\hat{\alpha}[/itex] and [itex]\hat{\beta}[/itex] are modeling parameters.

    Thanks for the help.
  14. Jun 12, 2012 #13


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    Re: Help with "Complex Frequency"

    I would back off a bit from your "real" problem and start by figuring out how to get the admittance parameters for a simple RCL circult from your VNA.

    You need some sort of curve fitting process, to convert the measured "amplitude and phase response against frequency" into "poles, zeros and residuals" and thus turn the measurements into into an admittance function like Y(s) = s / (Ls^2 + Rs + 1/C).

    Just browing some of the analyser guides, it's not too clear whether the VNA has the software to do that built in, or you need some other signal processing software (e.g. simething supplied with the analyser that runs on a PC).

    The admittance function in your model is a ratio of low order polynomials, so it is of this general form, but it representing a network with several "resonant circuts", not just one.

    I don't think you want to start researching how to do the curve fitting yourself, especially if you are starting from the level of "what is a complex frequency" - you shoudn't need to understand all the details of how the software works to use it.

    Note, I'm more familiar with this in measuring mechanical vibrations, but the basic math is the same so the steps in the process must be similar.
  15. Jun 14, 2012 #14
    Re: Help with "Complex Frequency"

    Sounds like you are measuring the permittivity of some substance using the open ended coaxial probe method. Is this correct?
  16. Jun 16, 2012 #15
    Re: Help with "Complex Frequency"

    Yes, that's right. Here's the paper where I found the model I mentioned above. My measurements are not giving physical results, and I think the problem may be how I used the frequency.

  17. Jul 24, 2012 #16
    Re: Help with "Complex Frequency"

    So the transform according to what you sent is:
    [itex] \sin{\omega t} \implies \frac{s}{s^{2} + \omega ^2} = \frac{1/2}{s - jw} + \frac{1/2}{s + jw} [/itex]

    But I'm not really sure if the frequency generated by the VNA is a sine wave. Does someone know how to check this.

    Also, I'm still confused as to what 's' is. Being that it's in the exponent of the e^{-st} it appears to be the inverse of some time constant of decay. Where do I find a value for s? If my signal is of constant amplitude, can I assume that s = 0 ?
  18. Jul 24, 2012 #17
    Re: Help with "Complex Frequency"

    Do you have much experience with this method? If so, I'd really love to pick your brain for a bit.
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