# When Is the Voltage Across an Inductor Maximum?

• zealeth
In summary: You can also check by plugging in the time you found back into the original formula and looking for a maximum.In summary, the maximum voltage across the inductor occurs at t = 0.0028 ms and has a value of 72.3 V. This can be found by taking the second derivative of the current equation and setting it equal to 0 to find the time at which the maximum occurs.
zealeth

## Homework Statement

The current in a 28mH inductor is known to be −10A for t≤0and (−10cos(400t)−5sin(400t))e^(−200t) A for t≥0. Assume the passive sign convention. At what instant of time is the voltage across the inductor maximum? What is the maximum voltage?

v(t) = L*di/dt

## The Attempt at a Solution

v(t) is at a max when di/dt is max since L is constant.

Taking the derivative of the equation for t≥0, I have:

di/dt = 5000e^(-200t)*sin(400t)

Finding the maximum using my calculator, I found t = 0.0028 ms.

Plugging this back into di/dt, di/dt = 2569.8

v(0.0028) = 0.028 * 2569.8 = 72.3 V.

Now both of these answers are correct, but my question is there a better way to find t than just plugging it into my graphing calculator and using the built in function to find the maximum? Is there a way this can be solved on a basic/scientific calculator? I know I could take the 2nd derivative of i(t) and set it equal to 0 but there is an infinite number of roots for t<0 and a good number of them for t>0. My professor never solved a problem similar to this in class so I'm not sure what his method is.

Your ##{dI\over dt}## looks a little strange. Differentiating ## I = \left (A \cos(\omega t) + B \sin (\omega t) \right )e^{-\alpha t}## is differentiating a product. How come you end up with only one sin term ?

Never mind, I can reproduce, sorry. Filling in numbers is useful sometimes...

So now differentiate again and rewrite as ##A^\prime \sin(\omega t + \phi)\ \exp(-200t)##.

From the ##\exp(-200t)## that descends monotonically it is obvious the first maximum is the maximum.

Last edited:
1 person
I know I could take the 2nd derivative of i(t) and set it equal to 0 but there is an infinite number of roots for t<0 and a good number of them for t>0.
There is no easy out, that's what you'll have to do. As BvU pointed out, you are looking for the
local maximum that lies 0 <t< T/2 because as time goes on the oscillations get smaller (it's a decaying exponential you have there).

I checked your answer; and I agree with it.

1 person

## What is the maximum voltage across an inductor?

The maximum voltage across an inductor is determined by the inductance of the coil and the rate of change of current passing through it. It can be calculated using the formula Vmax = L(di/dt), where Vmax is the maximum voltage, L is the inductance in henries, and di/dt is the rate of change of current in amperes per second.

## What factors affect the maximum voltage across an inductor?

The maximum voltage across an inductor is affected by the inductance of the coil, the rate of change of current, and the resistance of the circuit. A higher inductance or faster rate of change of current will result in a higher maximum voltage, while a higher resistance will result in a lower maximum voltage.

## How does the maximum voltage across an inductor relate to the energy stored in the inductor?

The maximum voltage across an inductor is directly proportional to the energy stored in the inductor. This means that the higher the maximum voltage, the more energy is stored in the inductor.

## Can the maximum voltage across an inductor damage electronic components?

Yes, the maximum voltage across an inductor can potentially damage electronic components if it exceeds their voltage ratings. It is important to design circuits with appropriate voltage ratings to prevent damage from high voltage spikes from inductors.

## How can the maximum voltage across an inductor be controlled or limited?

The maximum voltage across an inductor can be controlled or limited by using a voltage regulator or by adding a diode or other components to the circuit to suppress high voltage spikes. Proper circuit design and component selection can also help to limit the maximum voltage across an inductor.

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