Bode Plot Question..

[TecHNooB]

How do I graph the bode plot of this function:

H(jw) = (jw)^2 + jw + 1

I get imaginary zeroes which leaves me confuzzled.

 

[esun]

Find the magnitude and phase. Plot them.

H(jw) = (jw)^2 + jw + 1 = -w^2 + jw + 1
Re(H) = -w^2 + 1
Im(H) = w

|H| = sqrt(Re(H)^2 + Im(H)^2) = sqrt(w^4 - 2 w^2 + 1 + w^2) = sqrt(w^4 - w^2 + 1)
Arg(H) = arctan(Im(H)/Re(H)) = arctan(w/(1 - w^2))

Just plug those into a calculator and see what they spit out. You can do approximations (find the asymptotic shapes for small w and large w and the value at the inflection point) if you want.

 

[TecHNooB]

Originally posted by: esun
Find the magnitude and phase. Plot them.

H(jw) = (jw)^2 + jw + 1 = -w^2 + jw + 1
Re(H) = -w^2 + 1
Im(H) = w

|H| = sqrt(Re(H)^2 + Im(H)^2) = sqrt(w^4 - 2 w^2 + 1 + w^2) = sqrt(w^4 - w^2 + 1)
Arg(H) = arctan(Im(H)/Re(H)) = arctan(w/(1 - w^2))

Just plug those into a calculator and see what they spit out. You can do approximations (find the asymptotic shapes for small w and large w and the value at the inflection point) if you want.

Hmm.. so you can't do the normal bode approximations this? If the function ends up being 10*log(w^4-w^2+1), is that the same as 10*(w^4-w^2+1) on a log scale?

Btw, I have more questions for you 😛

 

[esun]

You can. If you want to do that, you'll have have to think about the asymptotic shape of things. For example:

For small w (w --> 0), we have |H| ~ 1 (or 0 dB), meaning the curve is relatively flat.

For large w (w --> infinity), we know that the w^2 term will dominate, so we'll have something like 20 log (w^2) = 40 log (w) (assuming H represents a voltage quantity; if a power quantity, it should be 10 log), meaning we'll get a +40 dB/dec slope in the magnitude plot.

For the break point, we can find the point where Re(H) = -w^2 + 1 = 0, giving w = 1. The value of |H| at w = 1 is |H(1)| = 1 = 0 dB.

Thus, we have a flat curve at 0 dB for w < 1 and a + 40 dB/decade curve for w > 1, with the value at w = 1 being 0 dB.