In a circuit, the equation for alternative voltage V is by sin(100 πt) volt. Draw the time voltage V = 40(t - V) graph with proper scale for one cycle. Calculate the root mean squa... In a circuit, the equation for alternative voltage V is by sin(100 πt) volt. Draw the time voltage V = 40(t - V) graph with proper scale for one cycle. Calculate the root mean square value of voltage.
Understand the Problem
The question is asking for two main tasks: first, to create a voltage-time graph based on the given sinusoidal equation for one cycle, and second, to calculate the root mean square (RMS) value of the voltage from the given equation. To draw the graph, we will analyze the sine function provided and set the appropriate scale. To calculate the RMS value, we apply the formula for RMS of a sinusoidal voltage.
Answer
The RMS value is $V_{rms} = \frac{V_m}{\sqrt{2}}$.
Answer for screen readers
The RMS value of the voltage is given by $V_{rms} = \frac{V_m}{\sqrt{2}}$.
Steps to Solve
- Identify the Sinusoidal Equation
Assuming the sinusoidal voltage equation is given as $V(t) = V_m \sin(\omega t + \phi)$ where $V_m$ is the maximum voltage, $\omega$ is the angular frequency, and $\phi$ is the phase angle.
- Define One Cycle for the Graph
Determine the period of the sinusoidal function using the formula $$ T = \frac{2\pi}{\omega} $$ This defines the duration of one complete cycle.
- Calculate Key Points for the Graph
To plot the graph, calculate key points at:
- $t = 0$
- $t = \frac{T}{4}$
- $t = \frac{T}{2}$
- $t = \frac{3T}{4}$
- $t = T$
Evaluate $V(t)$ at these points, which will help create the graph of the voltage over one cycle.
- Draw the Voltage-Time Graph
On a graph, plot the points calculated in the previous step on the y-axis (Voltage) against time on the x-axis. Connect the points with a smooth sinusoidal curve.
- Calculate the RMS Value
The RMS value of a sinusoidal voltage is calculated using the formula: $$ V_{rms} = \frac{V_m}{\sqrt{2}} $$ Substitute the maximum voltage $V_m$ to find the RMS voltage.
The RMS value of the voltage is given by $V_{rms} = \frac{V_m}{\sqrt{2}}$.
More Information
The root mean square (RMS) value represents the effective value of an AC voltage, which is important for determining the power delivered to electrical devices. In an ideal sinusoidal waveform, RMS is approximately 0.707 times the peak voltage.
Tips
- Forgetting to calculate the period correctly which could lead to incorrect plotting of the graph.
- Misidentifying the maximum voltage $V_m$ in the sinusoidal equation, leading to an incorrect RMS calculation.
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