Step-by-Step Explanation: Current and Voltage in an Inductor
Step 1: Relationship Between Current and Voltage
The voltage \( V \) across an inductor is determined by the rate at which the current \( I \) through it changes, according to the formula: \[ V = L \frac{dI}{dt} \] Here, \( V \) represents the voltage across the inductor, \( L \) signifies the inductance, \( \frac{dI}{dt} \) denotes the instantaneous rate of change of current with respect to time.
When the current \( I(t) \) follows a triangular waveform over time, the resulting voltage waveform will be directly proportional to this rate of change. Essentially, the voltage's behavior is dictated by how rapidly the current is altering at any given moment.
Step 2: Analyzing the Waveform
For a triangular current waveform, the rate of change \( \frac{dI}{dt} \) remains constant during both the rising and falling segments of the triangle. Consequently, this leads to a constant voltage during these intervals. The voltage waveform will directly mirror the shape of the derivative of the triangular current waveform.
The current can be modeled as a triangular waveform. The corresponding voltage waveform is then the derivative of this current waveform. If the current increases linearly over time, the voltage will be proportional to this constant rate of change, resulting in a waveform that can be represented by the graph of \( \sqrt{t/T} \), where \( T \) is the period of the triangular current waveform.
Conclusion
The voltage waveform corresponding to a triangular current waveform is determined by the principle that voltage is proportional to the derivative of current. This relationship yields a voltage waveform that follows the graphical representation of \( \sqrt{t/T} \).
