SOLUTION: Can someone please help. I cannot figure out the correct formula. Using a Nonlinear System. Modeling Circuit Gain. In electronics, cicuit gain is modeled by G = Bt/R + rt

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Question 252428: Can someone please help. I cannot figure out the correct formula.
Using a Nonlinear System.
Modeling Circuit Gain. In electronics, cicuit gain is modeled by G = Bt/R + rt, where “R” is the value of a resister, “t” is temperature, Rt is the value of R at temperature “t”, and B is a constant. The sensitivity of the circuit to temperature is modeled by S = BR/(R + Rt)^2. If B = 3.7 and "t" is 90K (Kelvin), find the values of R and Rt that will make G = .4 and S = .001.
Answer: R= 187 , Rt = 645
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the 2 formulas you have to work with are:

G+=+%28%28B%2At%29+%2F+%28R+%2B+R%5Bt%5D%29%29

and:

S+=+%28B%2AR+%2F+%28%28R+%2B+R%5Bt%5D%29%5E2%29%29

R is the nominal value of a resistor.
R[t] is the value of the resistor at a certain temperature.
B is a constant.
t is the temperature using the Kelvin Scale.
S is the sensitivity of the circuit to temperature.

Your problem is:

"If B = 3.7 and "t" is 90K (Kelvin), find the values of R and Rt that will make G = .4 and S = .001. (Answer: R= 187 , Rt = 645)"

We have:

B = 3.7
t = 90
G = .4
S = .001

We want to find R and R[t].

The first formula is:

G+=+%28%28B%2At%29+%2F+%28R+%2B+R%5Bt%5D%29%29

Substituting in this formula gets:

.4+=+%28%283.7%2A90%29+%2F+%28R+%2B+R%5Bt%5D%29%29

We divide both sides of this equation by .4 and we multiply both sides of this equation by (R + R[t]) to get:

%28R+%2B+R%5Bt%5D%29+=+%28%283.7%2A90%29+%2F+.4%29

We solve for (R + R[t] to get:

%28R+%2B+R%5Bt%5D%29+=+832.5

The second formula is:

S+=+%28%28B%2AR%29+%2F+%28%28R+%2B+R%5Bt%5D%29%5E2%29%29%29

We substitute in this formula to get:

.001+=+%28%283.7%2AR%29+%2F+%28832.5%5E2%29%29

S and B were given.
R + R[t] was calculated from the first equation.

We multiply both sides of this equation by (832.5)^2 and we divide both sides of this equation by 3.7 to get:

R+=+%28.001+%2A+%28832.5%29%5E2%29%2F3.7%29

We solve for R to get R = 187.325

Since R + R[t] = 832.5, this means that R[t] = 832.5 - 187.325 = 645.1875

we round these out to get:

R = 187
R[t] = 645

This agrees with the answers you provided.

The key was solving for R + R[t] together in the first equation and then using that value to solve for R in the second equation.

Once we knew R, getting R[t] was a simple matter of subtraction.

I think that's what you were looking for.

Let me know if otherwise.