Hey all,
I need help finding an error in a derivation I'm doing. The equation
Q = √(Rload/Rsource  1)
is used in impedance matching with L sections. Since I'm not one to blindly take an equation, like many students, I took to finding out where it comes from. I know that you're matching impedances, and in a circuit with two real loads and an L section, like this:
you have two RC pairs. A resistance and reactance pair will always have a Q value. That means we find:
Qsource = X1/R1
Qload = X2/R2
Okay, all is going well. This is all supported in the book (RF Circuit Design, Bowick).We set Qsource=Qload. But we can't solve for a definite solution unless we have one more equation. Thus comes in the given equation:
Q = √(R2/R1  1)
So I'm thinking "Ok that looks unlike other stuff where did it come from". I assume the loads are matched such that:
R1+X1 = R2  X2
and use the Q equations to replace the X1 and X2.
But that's where I stop and realize the Q equation for X1 and X2 assume that they are magnitudes only, when really they're complex values. The parallel combination of X2 and R2 will really come out to be complex as well, so replacing X2 with a real value might affect the answer.
I trudge on, with no idea of what effect this will have. I realized that the topic of Q being imaginary or real never came up. It must use the magnitudes instead.
So I did lots of algebra (due to mistakes, and trying again). I'm using Wolfram Alpha to check my work. But instead of
Q = √(R2/R1  1)
I get
Q = √(R2/R1)  1
where the  1 is outside of the square root. A far cry from the given equation. So I believe that my assumption in the parallel combination of a resistance and a reactance and then substituting out was faulty. In a way, I took [R2  something] as a function f(X2) and replaced X2. But obviously that's wrong, if only by a small factor.
How can I fix this up? Where does the equation come from?
Note: I also have qualms about adding a resistance and a reactance, since that does not work mathematically. But again, I let it go. These are likely places for mistakes, but I don't know how to handle them, with the Q being real, and all that. I guess I'm all out of brain energy. SOS!
I need help finding an error in a derivation I'm doing. The equation
Q = √(Rload/Rsource  1)
is used in impedance matching with L sections. Since I'm not one to blindly take an equation, like many students, I took to finding out where it comes from. I know that you're matching impedances, and in a circuit with two real loads and an L section, like this:
you have two RC pairs. A resistance and reactance pair will always have a Q value. That means we find:
Qsource = X1/R1
Qload = X2/R2
Okay, all is going well. This is all supported in the book (RF Circuit Design, Bowick).We set Qsource=Qload. But we can't solve for a definite solution unless we have one more equation. Thus comes in the given equation:
Q = √(R2/R1  1)
So I'm thinking "Ok that looks unlike other stuff where did it come from". I assume the loads are matched such that:
R1+X1 = R2  X2
and use the Q equations to replace the X1 and X2.
But that's where I stop and realize the Q equation for X1 and X2 assume that they are magnitudes only, when really they're complex values. The parallel combination of X2 and R2 will really come out to be complex as well, so replacing X2 with a real value might affect the answer.
I trudge on, with no idea of what effect this will have. I realized that the topic of Q being imaginary or real never came up. It must use the magnitudes instead.
So I did lots of algebra (due to mistakes, and trying again). I'm using Wolfram Alpha to check my work. But instead of
Q = √(R2/R1  1)
I get
Q = √(R2/R1)  1
where the  1 is outside of the square root. A far cry from the given equation. So I believe that my assumption in the parallel combination of a resistance and a reactance and then substituting out was faulty. In a way, I took [R2  something] as a function f(X2) and replaced X2. But obviously that's wrong, if only by a small factor.
How can I fix this up? Where does the equation come from?
Note: I also have qualms about adding a resistance and a reactance, since that does not work mathematically. But again, I let it go. These are likely places for mistakes, but I don't know how to handle them, with the Q being real, and all that. I guess I'm all out of brain energy. SOS!
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