# Find L such that 70% of the noise power is dissipated in R

1. Feb 24, 2013

### Gweniiix

I've been given the following homework exercise and I really need some help with part C.

Question1.

In figure 1. (attachment), an LR-lowpass filter is given with L = 120mH and R = 18 kΩ.
This filter is characterized by the transfer function: H(f) = Vuit(F)/Vin(f)
Determine the following paramters for this filter:

A) The 3dB bandwidth.
I found the answer to this question as follows: R/2∏L = 23.9 kHz

B) The equivalent noise bandwidth.
The answer to this question is: R/4L = 37.5 kHz

Now let the input signal of the lowpass filter be a bandlimited noise signal with power spectral density (PSD):

Pn(f) = 1 for f < 140 kHz
0 for f > 140 kHz

C) Determine a new value for L such that 70% of the total available noise power is dissipated in R (100% of the total available noise power is dissipated in R for L =0).

The answer to this question should be: The value of L = 26.9 mH, but I can't figure out why. Can someone please help me?

#### Attached Files:

• ###### LR circuit.png
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2. Feb 25, 2013

### rude man

1. What is the relation between the input psd (power spectral density) and the output, with x the input and y the output, given your psd of x and G(s) = 1/(Ts+1)?

2. What is the integral that gives you E(y2), the mean-square value of the output (the expectation of the output y)?

3. Compute the integral for the original L and then take 70% of that and recompute the value of L. Hint: E(y2) will be a function only of T = L/R and the cutoff frequency.

(Alternatiely, use the table of mean-square expectation integrals formulated by R.S. Phillips. But in this case the integral is easily solved).

It should be apparent that E(y2) is the power across R. For one thing, it's tyhe only dissipative element in your network. For another, R forms the output of the transfer function G(s).

3. Feb 26, 2013

### Gweniiix

I'm sorry, I'm just a beginner in this course and I don't completely understand what you're saying. I tried some things, but that doesnt seem to work.
I also don't really know how to compute the integral for the original L (point 3). I tried things, but what I've got now cant be right (can't type it her.. my phone won't allow me).
Can you maybe show me how to do it? I already handed in my homework exercise, but my teacher hasnt discussed this question (yet?) and I really want to know how it is done! :)

4. Feb 26, 2013

### rude man

The input to your network is characterized by a power spectral density function rather than a voltage. The input is flat noise up to the cutoff frequency, then it's zero. Power ~ v2 and the units of your input is volts-square per Hz. This is called a power spectral density function because, in 1 Hz bandwidth, the power (1 ohm assumed) is V^2. (This is not quite precise in general but it is if the noise spectrum is flat as it is here.) So let's call the input psd Si(jw) and the output psd is So(jw).

To get the output the best way for you is to take G(s) = 1/(Ts+1), let s = jw, then
Si(jw)*|G(jw)|2 = So(jw). Then
the expression for power is (1/2π) times the integral from -wc to +wc of the output power spectral density times dw. You are just integrating the output psd in other words, with the 1/2π needed for mathematical reasons.

You already had to evaluate this integral to compute the noise-equivalent bandwidth of your network BTW, or you should have. The integral is the same except for the limits of integration.

I really cannot go any further. This is a problem that should not have been assigned without sufficient background in the exotic world of random signals, IMO.