7.1:
Instantaneous Power
For a circuit with a sinusoidal input, the power delivered to a passive element equals the product of the voltage and current across it.
Expressing voltage and current in a sinusoidal form and applying the trigonometric identity gives the expression for instantaneous power with two terms.
The first term is a constant dependent on the phase difference between the voltage and current, while the second term is a sinusoidal function with twice the frequency of the voltage or current.
For circuits with mixed resistance and reactance, the instantaneous power oscillates between positive and negative values.
In a purely resistive circuit, the voltage and current are in phase, resulting in a consistently positive instantaneous power dissipation.
For purely inductive and capacitive circuits, the phase shift between the voltage and current causes the instantaneous power to fluctuate between positive and negative values.
Inductors and capacitors cyclically store and release energy as the current through the inductor or voltage across the capacitor changes.
During the energy release cycle, the reactive elements act as a power source, supplying energy back to the circuit, leading to negative instantaneous power.
7.1:
Instantaneous Power
Instantaneous power is important in electrical circuits, mainly when dealing with sinusoidal input. Instantaneous power, denoted as p(t), results from the multiplication of the instantaneous voltage (v(t)) across an element and the instantaneous current (i(t)) flowing through it. This relationship adheres to the passive sign convention and represents a fundamental principle in electrical engineering.
When examining instantaneous power, it can be expressed as a combination of two distinct terms. The first term, which remains constant, is dependent on the phase difference between voltage and current. The second term, characterized by sinusoidal oscillation, exhibits a frequency twice that of either the voltage or current.
In circuits encompassing resistance and reactance elements, instantaneous power is dynamic, oscillating between positive and negative values. In cases of purely resistive circuits, where voltage and current are perfectly in phase, instantaneous power consistently dissipates positively.
Conversely, in circuits dominated by inductive and capacitive components, the phase shift between voltage and current leads to intermittent fluctuations in instantaneous power between positive and negative values. This phenomenon can be attributed to the cyclic storage and release of energy by inductors and capacitors as current traverses through the former and voltage appear across the latter. During the energy release phase, these reactive elements act as power sources, effectively supplying energy back into the circuit resulting in the manifestation of negative instantaneous power. This intricate interplay of voltage, current, and power dynamics underscores the fundamental principles governing electrical circuits. An understanding of these principles is critical for efficient energy management and the design of electrical systems.