Step-by-Step Explanation: Current and Voltage in an Inductor
Step 1: Relationship Between Current and Voltage
In an inductor, the voltage \( V \) across the inductor is related to the rate of change of current \( I \) through the following equation: \[ V = L \frac{dI}{dt} \] where: \( V \) is the voltage across the inductor, \( L \) is the inductance, \( \frac{dI}{dt} \) is the derivative of the current with respect to time.
When the current \( I(t) \) varies with time in a triangular waveform, the voltage waveform will be proportional to the rate of change of the current. This means that the voltage will depend on how quickly the current changes over time.
Step 2: Analyzing the Waveform
If the current waveform is triangular, its rate of change \( \frac{dI}{dt} \) is constant during the rise and fall of the triangle, which results in a **constant voltage** during these periods. The voltage will follow the shape of the derivative of the triangular waveform.
The current waveform can be represented by a triangular waveform, and the voltage waveform will correspond to the derivative of the current waveform. If the current increases linearly over time, the voltage will be proportional to the rate of change, which corresponds to the graph of \( \sqrt{t/T} \), where \( T \) is the period of the triangular waveform.
Conclusion
The correct representation of the voltage waveform for a triangular current waveform is derived from the fact that voltage is proportional to the derivative of current, and this gives a voltage waveform that follows the graph of \( \sqrt{t/T} \).
