L network impedance-matching circuit

[hanhan]

Hi,
I am reading about L network in chapter impedance matching. Here is part of the book that I can't understand. Please help me? Thanks.
**broken link removed**

 

[MrAl]

Hi,

Where is figure 4.4 ?

 

[hanhan]

Hi,

Where is figure 4.4 ?

Hi,
It is the first picture in the image.

 

[MrAl]

Hi,

That is Fig. 4.5 isnt it? That's what it says it is anyway. Post 4.4 too.

 

[hanhan]

Thanks,
Maybe there is a problem in the site. I can see clearly the figure 4.4 at the beginning of my post.

And here is the picture illustrate the end of my post.

 

[MrAl]

Hello again,

Wow that's strange. I can see it now. I dont know why it is not visible in the first post. It's still not showing up there. I suspect it has something to do with the way the ads show when not logged in, but i do log in and so i cant be sure what it is.

Anyway, if we start with a full analysis of the impedance as seen by the source itself (with it's internal series R), we get:
Z=(s^2*L*C*Rp+Rp+s*L)/(s*C*Rp+1)

Now if we did the design right that would have to equal Rs (the source series resistance) so we set that equal to Z:
Z=Rs
which is:
(s^2*L*C*Rp+Rp+s*L)/(s*C*Rp+1)=Rs

and we can make Rp=a*Rs because the output resistance will be a multiple of Rs. In the case of Rs=100 and Rp=1000, that means a=10.

Now because s is a complex variable we can solve this for L and C, and we get as a result:
C=sqrt(a-1)/(a*w*Rs)
L=(sqrt(a-1)*Rs)/w

So when the impedances are properly matched L is related to C by:
C/L=1/(a*R1^2)

which is another way of saying:
C/L=1/(Rs*Rp)

Now the Q of the RL network is:
xL/Rs

so we have:
Qs=w*L/Rs

and the Q of the RC network is:
Qp=Rp/(1/(w*C))=Rp*w*C

and the impedances are matched when Qs=Qp so that means that:
Qs/Qp=1

So we divide and we get:
C/L=1/(Rs*Rp)

which is exactly what we got from a pure analysis with no assumptions.

 

[hanhan]

Thanks a lot, MrAl.
Your answer really makes sense. Before asking the question I tried a simple example by two method and that works. However, the second method that uses Q was confusing because I could not prove it.
Your proof bases on a specific circuit (low pass filter). But I think it also works for high pass filter and other topologies of L network.
I am following you method to prove for other arrangements of L network as in figure 4.4.

 

[MrAl]

Hello again,

Oh that's good, you'll probably find this interesting. They should all work out.

Also, the A network looks like it is for a low to high impedance transformation while the B network looks like it is for a high to low impedance transformation. Does it say anything about this in your reference material?
That probably means that the C and D networks have this same basic relationship, if true.