Consider a simple alternating current generator such as Hippolyte Pixii's dynamo. The current generated by such dynamo can be described as simple oscillating cosine signal. Assuming for simplicity that the max current generated each time a pole of Pixii's magnet passed the coil is (2/pi)0.5 amperes and that the magnet rotate with a characteristic angular frequency ù0.
Write the mathmatical expression for the time-domain signal (current vs. time) from such AC generator. For simplicity, assume that the magnet was already spinning at the beginning of the observation and the magnet's poled happened to be just below the coils at time zero.
I am not really sure how to start, I just need know that I need something relate current with time and equation should have cosine.
Thanks for any help
Well, let me just clown around with this math problem for a moment and see what I can come up with. Don't worry, I won't be too silly with it.
So, we know that the max current generated each time a pole of Pixii's magnet passed the coil is (2/π)^(0.5) amperes. Let's call this "Imax" for convenience.
Now, assuming the magnet was already spinning at the beginning of the observation, we can say that at time zero, the cosine function is at its max value of 1. So, our equation for the time-domain signal can be expressed as:
I(t) = Imax * cos(ω0t)
Here, I(t) represents the current at time "t," Imax is the maximum current, ω0 is the characteristic angular frequency, and "cos" stands for cosine.
So, to summarize, the mathematical expression for the time-domain signal for Hippolyte Pixii's dynamo can be written as:
I(t) = (2/π)^(0.5) * cos(ω0t)
Hope that puts a smile on your face!
To write the mathematical expression for the time-domain signal from Hippolyte Pixii's dynamo, we can start by considering the rotation of the magnet's poles passing the coil. Given that the magnet rotates with a characteristic angular frequency ω0, we can express the time variable as t = ω0 * time, where time represents the elapsed time.
Now, let's assume that the maximum current generated each time a pole of the magnet passes the coil is (2/π)^(0.5) amperes. Since the generated current can be described as a simple oscillating cosine signal, the mathematical expression for the current I(t) can be written as:
I(t) = I_max * cos(ω0 * t)
Where:
I(t) represents the current at time t,
I_max is the maximum current generated (2/π)^(0.5) amperes, and
cos(ω0 * t) represents the cosine function of the angular frequency ω0 multiplied by the time variable t.
This expression describes how the current generated by the Hippolyte Pixii's dynamo varies with time, with the oscillation pattern following a cosine waveform.
To find the mathematical expression for the time-domain signal (current vs. time) from the AC generator, we can start by understanding the basic principles of an AC generator like Hippolyte Pixii's dynamo.
In an AC generator, a coil of wire is placed near a magnet. As the magnet rotates, the changing magnetic field induces a voltage in the coil, which in turn generates an alternating current. This alternating current can be described as a simple oscillating cosine signal.
In this case, the maximum current generated each time a pole of the magnet passes the coil is (2/π)0.5 amperes. To express this in a mathematical equation, we need to consider the angular frequency (ω0) of the rotating magnet.
The angular frequency (ω0) represents the rate at which the magnet rotates and is related to the number of complete rotations per unit time. To find the relationship between the angular frequency (ω0) and time (t), we can use the formula:
θ = ω0t
Here, θ represents the angular position of the magnet at time t.
Now, let's consider that at time zero, the magnet's pole happens to be just below the coils. At this moment, the angular position (θ) is zero.
To express the time-domain signal mathematically, we can use the cosine function, which has a period of 2π. Since the magnet completes one full rotation (2π radians) in a certain time interval, we can write the equation as:
I(t) = (2/π)0.5 * cos(ω0t)
In this equation, I(t) represents the current as a function of time, (2/π)0.5 is the maximum current generated each time a pole passes the coil, and cos(ω0t) represents the oscillating behavior of the current.
So, the mathematical expression for the time-domain signal (current vs. time) from Hippolyte Pixii's AC generator is I(t) = (2/π)0.5 * cos(ω0t).