I went over the Vx problem twice and got the same result, so there must be
something wrong with my procedure...but it's rock solid as far as I know:
loop current = vector sum of Ir + IL2 + Ic (where Ireactive = IL2 - Ic) and
Vz = vector sum Vx + VL1, where VL1 = XL1 * I. Plug first into second,
switch component currents with expressions in terms of Vx, isolate Vx
outside of the radical and solve for Vx.
The way to do it is to first write the voltage transfer ratio in terms of the
complex variable s. Since it's just a voltage divider, that's pretty easy. The
three components, L2, R, and C form the bottom leg of the divider and L1 forms
the top leg. Let z1 be the impedance of the top leg (just L1), and let z2 be
the impedance of the 3 components of the bottom leg in parallel. Then the
voltage transfer ratio of the divider is given by z2/(z1+z2). Expanding, and
using the complex variable s, we get:
Vz = Vx * [(1/(1/(s L2) + 1/R + s C))/(s L1 +1/(1/(s L2) + 1/R + s C))]
or Vz = Vx * CVTR (CVTR stands for complex voltage transfer ratio).
Replace the variable s in the expression for CVTR with (2 Pi f j), so that
CVTR = (1/(1/(2 Pi f j L2) + 1/R + 2 Pi f j C))/(2 Pi f j L1 +1/(1/(2 Pi f j L2)
+ 1/R + 2 Pi f j C))
Let CONJ[CVTR] be the complex conjugate of CVTR; that is CONJ[CVTR] is just the
same expression as CVTR, but with j (= SQRT[-1]) replaced by -j. Then form the
product: (CVTR * CONJ[CVTR]) and take the square root, so that the (real
magnitude) voltage transfer ratio you want is given as:
Vz = Vx * SQRT[CVTR * CONJ[CVTR]]
This is only the magnitude (or modulus) of the CVTR; the phase information has
been lost with this procedure.
I haven't used the
symbols C, L1, and L2 to represent their reactances, but rather have
used them to represent the values of those components.
Fair enough, but makes for a messier result. I can just as well set Xc = 1
/ 2piFC and so forth when I graph it.
The expression for input Z is:
Sqrt[(4*f^2*Pi^2*((L1 + L2)^2*R^2 + 16*C^2*f^4*L1^2*L2^2*Pi^4*R^2 +
4*f^2*L1*L2*Pi^2*(L1*(L2 - 2*C*R^2) - 2*C*L2*R^2)))/
(4*f^2*L2^2*Pi^2 + R^2 - 8*C*f^2*L2*Pi^2*R^2 +
16*C^2*f^4*L2^2*Pi^4*R^2)]
Um.. I'll look at that later... why does it keep growing?...
and the voltage you have called Vx is given by:
Vx = Vz*Sqrt[(L2^2*R^2)/((L1 + L2)^2*R^2 +
16*C^2*f^4*L1^2*L2^2*Pi^4*R^2 +
4*f^2*L1*L2*Pi^2*(-2*C*L2*R^2 + L1*(L2 - 2*C*R^2)))]
So that,
You made a couple of errors in your algebra:
Vx = Vz*L2*R / sqrt of:
R^2 L1^2 + R^2 L1 L2 + R^2 L2^2 + 16 pi^4 f^4 L1^2 L2^2 R^2 C^2 +
should be ...+ 2 R^2 L1 L2 +...
4 pi^2 f^2 L1^2 L2^2 - 8 pi^2 f^2 L1^2 L2 R^2 C - 8 pi^2 f^2 L1 L2^2 R^2
And this should be ...- 8 pi^2 f^2 L1 L2^2 R^2 C
When I plot this with R=100, C=20uF, L1=50uH, L2=10uH, I get a peak at about
12328 Hz, with a value at the peak of about 25.82
I verify these expressions I've posted by plotting the simpler complex
variable version on the same plot with the complicated real variable version,
and seeing that the two plots are identical (actually, they're on top of each
other).