The Designer's Guide Community Forum

The Designer's Guide Community Forum
https://designers-guide.org/forum/YaBB.pl
Design >> Analog Design >> Why do we plot magnitude vs. jw
https://designers-guide.org/forum/YaBB.pl?num=1169137041

Message started by Croaker on Jan 18^th, 2007, 8:17am

Title: Why do we plot magnitude vs. jw
Post by Croaker on Jan 18^th, 2007, 8:17am

Let's say we take an s-domain transfer function H(s). This is really a 3D surface where the x-axis is sigma, y-axis is j*omega (or jw) and the z-axis is the magnitude ( s=sigma + j*omega ). When making a Bode plot, we only look at the slice where sigma = 0, i.e. we are plotting the magnitude values of the z-axis vs. the y or jw axis.

Why are we only interested in this magnitude along the jw axis? (Of course this gives us our frequency response, but I never thought too deeply about the nature of the s-plane before).

What would be the significance of plotting a slice where sigma != 0?

Title: Re: Why do we plot magnitude vs. jw
Post by Ken Kundert on Jan 18^th, 2007, 8:36am

σ = 0 corresponds to the case of sinusoidal steady state. If you allow σ > 0 then you are choosing signals to represent both the stimulus and response that are growing exponentials. If σ < 0 then your signals are all decaying exponentials. When σ = 0 the signals are all simple sinusoids, and so the response you compute is the sinusoidal steady-state response. The case where σ = 0 is also referred to as phasor analysis.

-Ken

Title: Re: Why do we plot magnitude vs. jw
Post by Croaker on Jan 18^th, 2007, 10:00am

OK, that makes sense since you want your Bode plot to show the magnitude in response to sinusoids at various frequencies. :)

Is there any case where you'd want to look at a magnitude slice where say σ < 0 ? (I just noticed the Greek alphabet feature!) I guess this would be the magnitude plotted at various frequencies of a decaying sinusoid.

Title: Re: Why do we plot magnitude vs. jw
Post by mg777 on Jan 18^th, 2007, 12:27pm

The interesting thing (provided you assume nice functions that are absolutely integrable) is that both the representations are entire. That is, you could use either the Fourier kernel exp(jwt) or the Laplace kernel exp(st) for unique linear decompositions with arbitrarily bounded LMS error. In some sense it's like how a function can be expanded either as a Fourier series (sine and cosine) or as a Chebyshev series (T_n) with the latter converging much faster within a certain zone (what would be called the 'passband' in Chebyshev filters). Also gives some idea of why wavelet representations are more dense than Fourier or Laplace, and why the Karhunen-Loeve expansion of an AWGN contains the least number of eigenvalues.

Basis functions bear out George Orwell's inunction that "All animals are equal, but some animals are more equal than others"

M.G.Rajan
www.eecalc.com

Title: Re: Why do we plot magnitude vs. jw
Post by wuyuchun on Jan 18^th, 2007, 11:27pm

If you use a H(s) in which sigma/=0,then you get a response(including amplitude and phase) to not pure sine but attenuated or amplified increasingly . But our analysis of a system is base on sum of sine in different frequence,so we assume sigma=0,and deduce the bode plot.