SOLUTION: hi there, posted the problem below before but forgot a part of it, by equating real and imaginary parts find expressions for R6 and L in terms of R1, R2, R3, R4, R5 AND C given

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: hi there, posted the problem below before but forgot a part of it, by equating real and imaginary parts find expressions for R6 and L in terms of R1, R2, R3, R4, R5 AND C given      Log On


   

Question 88283: hi there, posted the problem below before but forgot a part of it,
by equating real and imaginary parts find expressions for R6 and L in terms of R1, R2, R3, R4, R5 AND C given that the freq is one rad per second. w=freq
(R1-j/wC)/(R4+R5) = (R2+R3)/(R6+jwL)
thanks
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Haven't had my caffeine fix for the day, so make sure you check my work step by step to
catch any mistakes that I might make.
.
Given:
.
%28R1+-+j%281%2F%28wC%29%29%29%2F%28R4+%2B+R5%29+=+%28R2+%2B+R3%29%2F%28R6+%2B+j%28wL%29%29
.
Since you are told that w = 1 radian per second, let's simplify the problem a little by
substituting 1 for w to get:
.
%28R1+-+j%281%2FC%29%29%2F%28R4+%2B+R5%29+=+%28R2+%2B+R3%29%2F%28R6+%2B+j%28L%29%29
.
Next, since you are interested in solving for R6 + jL, let's get that term into the numerator
by inverting both sides of the equation to get:
.
%28R4+%2B+R5%29%2F%28R1+-+j%281%2FC%29%29+=+%28R6+%2B+j%28L%29%29%2F%28R2+%2B+R3%29
.
Then multiply both sides by (R2 + R3) to eliminate the denominator on the right side and
get:
.
%28%28R4+%2B+R5%29%2A%28R2%2BR3%29%29%2F%28R1-j%281%2FC%29%29+=+%28R6+%2B+j%28L%29%29
.
Then multiply the left side by %28R1+%2B+j%281%2FC%29%29%2F%28R1+%2B+j%281%2FC%29%29. The numerator and the
denominator of this multiplier are the complex conjugate of the denominator of the left
side of the equation. This multiplication will change the denominator of the left side
of our equation from complex to real ... and note that the multiplier is a fraction
that is equal to 1 so it doesn't affect the equation we have.
.

.
Multiplying out the denominator on the left side results in it becoming:
.
R1%5E2+-+j%5E2%281%2FC%29%5E2 and since j%5E2+=+-1 the denominator is R1%5E2+%2B%281%2FC%29%5E2. This
makes the equation become:
.

.
Notice that the denominator is no longer a complex number. Next you can multiply out the
numerator by multiplying [(R1 + R5)*(R2 + R3)] times the two terms that make up the complex
number in the numerator. This multiplication changes the left side numerator and the equation
becomes:
.

.
Now divide the denominator (which is real) into the real and the reactive components of
the numerator to get:
.

.
Now you can set real and reactive components equal to find that:
.
R6+=+%28%28R4+%2B+R5%29%2A%28R2%2BR3%29%2A%28R1%29%29%2F%28R1%5E2%2B%281%2FC%29%5E2%29
.
and
.
L+=+%28%28R4%2BR5%29%2A%28R2%2BR3%29%2A%281%2FC%29%29%2F%28R1%5E2%2B%281%2FC%29%5E2%29
.
As I said at the "git-go", I suggest you track this step by step to catch any math errors.
The general approach is correct, but this is sort of complex to type out and I may have
introduced some glitches.
.
Hope this helps ...