91Ó°ÊÓ

In inductive circuits, understanding how voltage behaves is crucial. For an inductor, the voltage can be calculated using the formula: \[ v(t) = L \frac{di(t)}{dt} \]This means the voltage is determined by the rate of change of current through the inductor. For instance, if you have an inductance of 100 mH and a current given by \( i(t) = 0.5 \sin(1000t) \), you first need to differentiate the current with respect to time.

• The derivative of \( 0.5 \sin(1000t) \) equals \( 500 \cos(1000t) \).
• By multiplying this derivative by the inductance (0.1 H), you obtain the voltage expression: \( v(t) = 50 \cos(1000t) \).

The voltage waveform is sinusoidal, shifted in phase compared to the current. It peaks when the rate of change of the current is greatest, which happens at the zero crossings of the sine wave of the current. This phase difference is what causes reactive power in the system.

The power in an inductive circuit isn't constant over time due to the nature of both current and voltage. Power is calculated as the product of voltage and current:
\[ p(t) = v(t) \times i(t) \]With the voltage \( v(t) = 50 \cos(1000t) \) and current \( i(t) = 0.5 \sin(1000t) \), the power expression becomes:

• Plug in the values: \( p(t) = 50 \cos(1000t) \times 0.5 \sin(1000t) \)
• This simplifies to \( 25 \sin(1000t) \cos(1000t) \)

Using the trigonometric identity \( \sin(2x) = 2 \sin(x) \cos(x) \), we can further simplify this expression to \( 12.5 \sin(2000t) \). The power waveform has a frequency twice that of the voltage and current, creating a dynamic interaction where energy is exchanged between the inductor and the circuit.

Inductors, like capacitors, can store energy. The energy stored in an inductor is governed by its magnetic field and is given by the equation:
\[ W(t) = \frac{1}{2} L i(t)^2 \]In this exercise, with an inductance of 0.1 H and current \( i(t) = 0.5 \sin(1000t) \), the energy expression becomes:

• Substitute for current: \( W(t) = \frac{1}{2} \times 0.1 \times (0.5 \sin(1000t))^2 \)
• This simplifies to \( 0.0125 \sin^2(1000t) \)

The energy waveform reflects the storage and release cycle of the inductor. It's proportional to the square of the current, meaning it peaks when the current waveform reaches its maximum. Unlike voltage or power, the frequency of the energy waveform is the same as the current, reflecting a purely reactive component without direct energy transfer.

Analyzing waveforms provides a visual understanding of how each aspect—voltage, power, and energy—behaves over time. In this scenario:

• Voltage waveform: \( 50 \cos(1000t) \) shows a cosine function indicating the phase shift.
• Power waveform: \( 12.5 \sin(2000t) \) is a sine wave with double the frequency, fluctuating between positive and negative values.
• Energy waveform: \( 0.0125 \sin^2(1000t) \) comes out as always positive since energy doesn't go negative.

The frequency relationship in the waveforms offers insight into their interactions. For this case:
- Current and voltage are at the same frequency (1000 rad/s) but out of phase.- Power's higher frequency (2000 rad/s) shows the dynamic exchange of power.- Observing these plots over the range of 0 to \(3 \pi \) ms, reveals how energy storage impacts circuit behavior, visibly correlating peaks and zeroes with our mathematical analysis.