complex equation

[pebbert9]

I have an equation that I'm finding very difficult to solve.

(R1 * x) / ( R + R1 * x) = R1 / ( R + R1 ) + V1 / V

I know everything but R1

R = 1000
x = 1.6468
V1 = 0.1
V = 12

substituting:

( R1 * 1.6468 ) / ( 1000 + R1 * 1.6468 ) = R1 / ( 1000 + R1 ) + 0.1 / 12

What is the best way for me to solve for R1?

Thank you,

Phil

 

[stapel]

One method might be to multiply through by the various denominators, getting R₁ "on top" where you can get at it.

. . . . .\(\displaystyle \large{\frac{xR_1}{R\,+\,xR_1}\,=\,\frac{R_1}{R\,+\,R_1}\,+\,\frac{V_1}{V}}\)


. . . . .\(\displaystyle \large{V(xR_1)(R\,+\,R_1)\,=\,VR_1(R\,+\,xR_1)\,+\,V_1(R\,+\,xR_1)(R\,+\,R_1)}\)


. . . . .\(\displaystyle \large{xVRR_1\,+\,xV{R_1}^2\,=\,VRR_1\,+\,xV{R_1}^2\,+\,V_1 R^2\,+\,xV_1 RR_1\,+\,V_1 RR_1\,+\,xV_1 {R_1}^2}\)


. . . . .\(\displaystyle \large{0\,=\,(V_1\,+\,xV_1){R_1}^2\,+\,(VR\,+\,xV_1 R\,+\,V_1 R\,-\,xVR)R_1}\)

Solving the two factors, we get:

. . . . .\(\displaystyle \large{0\,=\,R_1}\)

...or:

. . . . .\(\displaystyle \large{R_1\,=\,\frac{xVR\,-\,VR\,-\,xV_1 R\,-\,V_1 R}{V_1\,+\,xV_1}}\)

Plug in the values you've got, and see what you come up with. (It might be wise to check my work, too; this is fairly messy stuff.)

Eliz.

 

[pebbert9]

Hi Eliz,

Thanks for the quick reply.

I tried it out and it didn't work.
I did notice errors in your 4th line.

In the third line there is a xVR1^2 term on both sides, so they should cancel.
This leaves only one R1^2 term, xV1R1^2.
Also the V1R^2 term seam to have disappeared.

I think the 4th line should read:

0 = xV1R1^2 + ( VR + xV1R + V1R - xVR ) R1 + V1R^2

Can you solve this equation for me?

Thank you very much,

Phil

 

[Denis]

(R1 * x) / ( R + R1 * x) = R1 / ( R + R1 ) + V1 / V
All I can say is OUTCH!

Anyhow, let's clear the fog a bit by letting r = R, a = R1 and y = V1 / V:
so equation now is:
ax / (ax + r) = a / (a + r) + y
lcd right side:
ax / (ax + r) = [a + y(a + r)] / (a + r)
crisscross multiply:
ax(a + r) = (ax + r)[a + y(a + r)]
go nuts to get:
xya^2 + a(rxy + ry + r - rx) + r^2y = 0
Thassa quadratic; let k = (rxy + ry + r - rx) :
xya^2 + ka + r^2y = 0
solve that quadratic:
a = [-k +- sqrt(k^2 - 4xy^2r^2)] / (2xy)

With your data, solutions = 45510.83~ and 13.34~

 

[pebbert9]

Wow, the answer turned out to be complex.

I checked it and 13.34 turns out to be correct.

Thank you very much for your help.

I am trying to find out the correct resistance value (RTD) for a wheatstone bridge.

Thanks again,

Phil

 

[stapel]

In the third line there is a xVR1^2 term on both sides, so they should cancel.

They did cancel. That's why they're not in the next line.

This leaves only one R1^2 term, xV1R1^2.

Right; thanks. I'll edit to correct:


. . . . .\(\displaystyle \large{\frac{xR_1}{R\,+\,xR_1}\,=\,\frac{R_1}{R\,+\,R_1}\,+\,\frac{V_1}{V}}\)


. . . . .\(\displaystyle \large{V(xR_1)(R\,+\,R_1)\,=\,VR_1(R\,+\,xR_1)\,+\,V_1(R\,+\,xR_1)(R\,+\,R_1)}\)


. . . . .\(\displaystyle \large{xVRR_1\,+\,xV{R_1}^2\,=\,VRR_1\,+\,xV{R_1}^2\,+\,V_1 R^2\,+\,xV_1 RR_1\,+\,V_1 RR_1\,+\,xV_1 {R_1}^2}\)


. . . . .\(\displaystyle \large{xVRR_1\,=\,VRR_1\,+\,V_1 R^2\,+\,xV_1 RR_1\,+\,V_1 RR_1\,+\,xV_1 {R_1}^2}\)


. . . . .\(\displaystyle \large{0\,=\,xV_1{R_1}^2\,+\,VRR_1\,+\,xV_1 RR_1\,+\,V_1 RR_1\,-\,xVRR_1\,+\,V_1 R^2}\)


. . . . .\(\displaystyle \large{0\,=\,(xV_1){R_1}^2\,+\,(VR\,+\,xV_1 R\,+\,V_1 R\,-\,xVR)R_1\,+\,(V_1 R^2)}\)


Now apply the Quadratic Formula:


. . . . .\(\displaystyle \large{R_1\,=\,\frac{xVR\,-\,VR\,-\,xV_1 R\,-\,V_1 R\,\pm\,\sqrt{(xVR\,-\,VR\,-\,xV_1 R\,-\,V_1 R)^2\,-\,4(xV_1)(V_1 R^2)}}{2(xV_1)}}\)


Have fun. :shock:

Eliz.