Maximizing Power in an Electric Circuit: Trigonometric Problem Solution

[bahramm]

Can anyone please give me some idea for this problem?
The instantaneous power, p, in an electric circuit is given by p = iv,
where v is the voltage and i is the current.
Calculate the maximum value of power in the circuit if v=0.02 sin(100 pi t)volts and i=0.6 sin(100pi t+ pi/4 ) amp
Calculate the first time that the power reaches a maximum value.[/I]

 

[Cyosis]

How do you determine the maximum of a function in general?

 

[Architeuthis Dux]

I would suggest using the equation for power, p = iv, to solve this problem. First, we can substitute the given values for v and i into the equation to get an expression for the instantaneous power in terms of time, t. This will give us a sinusoidal function for power, with a maximum value occurring at certain points in time.

To find the maximum value of power, we can use the trigonometric identity for the product of two sine functions, which is sin(a)sin(b) = 1/2[cos(a-b)-cos(a+b)]. Applying this identity to our expression for power, we get:

p = 0.6*0.02[cos(pi/4) - cos(100pi t + 5pi/4)]

We can see that the maximum value of power will occur when the cosine terms are equal to -1, which will happen when the argument of the cosine function is equal to pi. So, we can set 100pi t + 5pi/4 = pi and solve for t to find the time at which the maximum power occurs. This gives us t = 3/200 seconds.

Therefore, the maximum value of power in the circuit is 0.6*0.02 = 0.012 watts, and it occurs at t = 3/200 seconds. This information can be used to optimize the performance of the circuit and ensure that it is operating at its maximum power output. Additionally, this problem demonstrates the importance of understanding trigonometric functions and their applications in scientific calculations.