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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**
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.
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.
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.
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