Consider the ideal transformer, where i1(t)=20cos(120πt) Amps and v1(t)=100cos(120πt−45∘) volts, and a = 1 and b = 10. What is the rms value of i2 in amps? what is the amplitude of i2 in amps?
10
2
To find the rms value and amplitude of i2 in amps, we first need to determine the expression for i2.
In an ideal transformer, the voltage across the primary and secondary coils are related by the turns ratio (a:b). The current is inversely proportional to the turns ratio due to conservation of power.
Given a turns ratio of a: b = 1:10, we have:
i2(t) = (a/b) * i1(t)
Substituting the given values, we have:
i2(t) = (1/10) * 20cos(120πt)
Now we will calculate the rms value and amplitude of i2.
1. RMS value of i2:
The RMS (Root Mean Square) value of a periodic function f(t) is obtained by taking the square root of the average of the square of the function over a period.
i2(t) = (1/10) * 20cos(120πt)
To calculate the RMS value, we will square the function and integrate it over one period (T), and then take the square root.
i2^2(t) = (1/10)^2 * 20^2 * cos^2(120πt)
Integrating over one period, from 0 to T:
∫[0-T] { (1/10)^2 * 20^2 * cos^2(120πt) } dt
Using the trigonometric identity cos^2(x) = (1/2)(1 + cos(2x)), we can simplify further:
= (1/10)^2 * 20^2 * (1/2) * [ 1 + cos(2 * 120πt) ] dt
= (1/20) * [ t + (1/2π)sin(240πt) + C ] evaluated from 0 to T
= (1/20) * [ T + (1/2π)sin(240πT) - (1/2π)sin(0) ]
Since sin(0) = 0, the expression simplifies to:
= (1/20) * [ T + (1/2π)sin(240πT) ]
To find the RMS value, we substitute T = 1/f, where f is the frequency of the function:
RMS value of i2 = sqrt[ (1/20) * [ (1/f) + (1/2π)sin(240π/f) ] ]
Since the frequency is given as 120π in this case, we substitute that value:
RMS value of i2 = sqrt[ (1/20) * [ (1/(120π)) + (1/2π)sin(240π/(120π)) ] ]
= sqrt[ (1/20) * [ (1/(120π)) + (1/2π)sin(2) ] ]
Simplifying further:
RMS value of i2 = sqrt[ (1/20) * [ (1/(120π)) + (1/2π)sin(2) ] ]
2. Amplitude of i2:
The amplitude of a periodic function is simply the maximum value of the function. For a cosine function, the amplitude is equal to the magnitude of the coefficient of cosine.
In this case, the amplitude of i2 is (1/10) * 20 = 2 amps.
Therefore, the RMS value of i2 is sqrt[ (1/20) * [ (1/(120π)) + (1/2π)sin(2) ] ] and the amplitude of i2 is 2 amps.
To find the rms value and amplitude of i2, we need to use the turns ratio and the relationship between primary and secondary currents in an ideal transformer.
The turns ratio (a) represents the ratio of the number of turns in the primary coil to the number of turns in the secondary coil. In this case, a = 1, indicating a 1:1 turns ratio.
The relationship between primary (i1) and secondary (i2) currents is given by:
i2 = (b / a) * i1
where b is the turns ratio.
Given that a = 1 and b = 10, we can calculate the secondary current as follows:
i2 = (10 / 1) * i1
Since i1(t) = 20cos(120πt) Amps, substituting this into the equation gives:
i2(t) = 200 * cos(120πt) Amps
To find the rms value of i2, we need to take the square root of the average of the square of i2 over one period:
rms value of i2 = √(∫[0,T] (i2(t))^2 dt / T)
where T is the time period of the signal.
In this case, the signal is a cosine wave with a frequency of 120π radians/second. The time period T is given by T = 1 / f where f is the frequency. Plugging in the values gives T = 1 / 120π = 1 / (120 * 3.14159) = 0.00265 seconds.
Now let's calculate the rms and amplitude of i2:
rms value of i2 = √(∫[0,0.00265] (200 * cos(120πt))^2 dt / 0.00265)
Integrating and simplifying the expression will give you the final value.
To find the amplitude of i2, we can simply take the maximum value of i2, which is 200 Amps.
Please note that I have explained the general method to find the rms value and amplitude, and you can apply this method to any other transformer problem with different input values.