The rms output voltage of an AC source is 210 V, and the operating frequency is 90 Hz. Write the equation giving the output voltage as a function of time t.
U₀=U√2 =210•1.41 = 297 V
ω=2πf =2π•90=180π rad/s
U=U₀ cos ωt= 297 cos 180πt
The rms output voltage of an AC source is 200 V and the operating frequency is 100 Hz.
a. Calculate the maximum value of the AC voltage.
b. Write the equation giving the output voltage as a function of time
The equation giving the output voltage as a function of time t for an AC source can be given by:
V(t) = V_m * sin(2π * f * t + φ)
where:
V(t) is the instantaneous output voltage at time t,
V_m is the peak voltage (amplitude) of the AC source,
f is the frequency of the AC source in hertz (Hz), and
φ is the phase angle (constant).
In this case, the rms output voltage is given as 210 V, which is equivalent to the peak voltage (V_m) divided by the square root of 2 (√2).
So, V_m = 210 V * √2 = 297.2 V
The operating frequency is given as 90 Hz.
Assuming no phase shift (φ = 0), the equation becomes:
V(t) = 297.2 * sin(2π * 90 * t)
Therefore, the equation giving the output voltage as a function of time t is:
V(t) = 297.2 * sin(180πt)
To write the equation for the output voltage as a function of time, we need to consider the form of an AC voltage. AC voltage can be represented by a sine wave, which can be expressed as:
V(t) = V_peak * sin(2πft + φ)
Where:
- V(t) represents the voltage as a function of time
- V_peak is the peak voltage
- f is the frequency of the AC source in Hz
- t is the time
- φ is the phase angle
In the given question, we are given the root mean square (rms) voltage, which is 210 V. The rms voltage is related to the peak voltage (V_peak) by the equation:
V_rms = V_peak / √2
From this equation, we can calculate the peak voltage:
V_peak = V_rms * √2
= 210 V * √2
Now, substituting the values into the equation of an AC voltage, we have:
V(t) = (210 V * √2) * sin(2π * 90 Hz * t + φ)
The phase angle (φ) is not given in the question, so it is not possible to provide a specific equation without knowing φ.