How Can You Prove the Voltage-Time Relationship Using Logarithms and Graphs?

[lewisbedwell]

Homework Statement

It is thought that the law relating the voltage (v) across an inductor and the time(t0) in milliseconds (ms) is given by the relationship v=Ae^(t/B). Where “A” and “b” are constants and “e” is the exponential function. The results of the experimentation are below:
V (Volts) 908.4 394.8 171.6 32.4 14.1 6.12
T (ms) 10 20 30 50 60 70

Homework Equations

lnV=1/B.t+lnA

The Attempt at a Solution

t is thought that the law relating the voltage (v) across an inductor and the time(t0) in milliseconds (ms) is given by the relationship v=Ae^(t/B). Where “A” and “b” are constants and “e” is the exponential function. The results of the experimentation are below:

V (Volts) 908.4 394.8 171.6 32.4 14.1 6.12
T (ms) 10 20 30 50 60 70

Show that the voltage and time is actually true, and then determine the values of the constants “A” and “B”.
To show that the voltage is true, we need to plot it as a straight line graph, before we do this we need to take the log of the volts as shown below in the new table.

V (Volts) 908.4 394.8 171.6 32.4 14.1 6.12
Volts (log) 6.8117 5.9784 5.1452 3.4782 2.6462 1.8116
T (ms) 10 20 30 50 60 70

We can see that if we apply the log laws to the original formula, that the formula becomes:
lnV=1/B.t+lnA
y=mx+c
We can also see that this looks like the straight line formula. We can work from here to build up a graph so we can make sure that it is of a straight line

(Graph would be here)


We can now begin to analyse this graph and work out some of the unknowns. To begin with we can work out what the value would be if t=0 so the graph can be extended. We can work out how much the graph increases from the 20 seconds to the 10 seconds, therefore:
6.8117-5.9784=0.833
 

 

[haruspex]

So what values do you get for A and B?
Btw, the numbers provided you do happen to give a very straight line, but in general that will not be the case. Have you been taught linear regression? If you are going to obtain the slope and intercept from just two points on the graph, as you have done, it would be better to choose two that are well apart. If you take adjacent ones then a small error in a y-value could lead to a large error in slope.

 

[lewisbedwell]

Hi thanks for the reply, I have just worked out the values of A and B, was really struggling with them for some reason. I have: A= 2090.238 and B = -12. Thanks for the pointer about not using adjacent points, I will have a rethink and use further away ones. It makes sense to! Thanks again. Really helped.

 

[haruspex]

A= 2090.238 and B = -12.

That's what I get too.