Question

На участке цепи с активным сопротивлением 4 Ом сила тока изменяется по закону i = 6.4*sin 314tОпределить действующее значение силы тока и активную мощность, выделяемую на этом участке.

Answer 1

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Answer 2

Calculation of the Root Mean Square (RMS) Current

To determine the root mean square (RMS) current and the active power dissipated in the circuit, we need to analyze the given equation for the current.

The equation for the current is given as: i = 6.4 * sin(314t)

To find the RMS current, we need to calculate the average value of the squared current over one complete cycle and then take the square root of that average value.

The average value of the squared current over one complete cycle can be calculated using the following formula:

I_rms = sqrt((1/T) * ∫[0 to T] (i^2) dt)

Where: - I_rms is the RMS current - T is the time period of the waveform (in this case, it is the time taken for one complete cycle) - i is the instantaneous current at time t

In this case, the time period T can be calculated as: T = 2π/ω

Where: - ω is the angular frequency of the waveform, given as 314 rad/s in this case.

Now, let's calculate the RMS current.

Calculation of the RMS Current

The RMS current can be calculated using the formula mentioned above. We need to find the average value of the squared current over one complete cycle.

The squared current can be calculated as: i^2 = (6.4 * sin(314t))^2

To find the average value of the squared current over one complete cycle, we need to integrate the squared current over one complete cycle and divide it by the time period T.

The integral of the squared current can be calculated as: ∫[0 to T] (i^2) dt = ∫[0 to T] (6.4^2 * sin^2(314t)) dt

To solve this integral, we can use trigonometric identities to simplify the expression.

The trigonometric identity we will use is: sin^2(x) = (1 - cos(2x))/2

Using this identity, the integral can be simplified as: ∫[0 to T] (6.4^2 * sin^2(314t)) dt = (6.4^2/2) * ∫[0 to T] (1 - cos(628t)) dt

Now, let's calculate the integral.

Calculation of the Integral

The integral can be calculated as follows:

∫[0 to T] (1 - cos(628t)) dt = [t - (1/628) * sin(628t)] [0 to T]

Substituting the values of T and simplifying the expression, we get:

∫[0 to T] (1 - cos(628t)) dt = [t - (1/628) * sin(628t)] [0 to 2π/ω]

Now, let's substitute the values and calculate the integral.

Calculation of the Active Power

To calculate the active power dissipated in the circuit, we need to multiply the RMS current by the resistance of the circuit.

The active power can be calculated using the formula: P = I_rms^2 * R

Where: - P is the active power - I_rms is the RMS current - R is the resistance of the circuit

In this case, the resistance is given as 4 Ω.

Now, let's calculate the active power.

Calculation Results

After performing the calculations, the following results are obtained:

- The RMS current (I_rms) is 4.525 A. - The active power (P) dissipated in the circuit is 81.45 W.

Please note that these calculations are based on the given equation for the current and the assumption that the circuit is purely resistive.