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Reactive Energy
In electrical systems, reactive energy is associated with the storage and release of energy in inductors and capacitors. Inductors store energy in magnetic fields, and capacitors in electric fields. Unlike resistors, which consume energy as heat, inductors and capacitors do not consume energy; they temporarily store it and then return it to the system. This back-and-forth flow of energy is known as reactive energy.
For an inductor, the energy stored is given by: $$E_L = \frac{1}{2} L I^2$$
For a capacitor, the energy stored is: $$E_C = \frac{1}{2} C V^2$$
Resonance in Electrical Circuits
Resonance occurs in an electrical circuit when the inductive reactance (from inductors) and capacitive reactance (from capacitors) are balanced, resulting in the circuit's impedance being purely resistive (Ohmic). At the resonance frequency, energy oscillates between the inductor and the capacitor, manifesting as a continuous reactive energy exchange.
The inductive reactance is: $$X_L = \omega L$$
The capacitive reactance is: $$X_C = \frac{1}{\omega C}$$
At resonance, these reactances are equal:
$$\omega L = \frac{1}{\omega C} \quad \Longrightarrow \quad \omega_0 = \frac{1}{\sqrt{LC}}$$
Why Reactive Energy at Resonance?
At resonance, the energy stored in the inductor's magnetic field is completely transferred to the capacitor and vice versa at a specific frequency. This creates a state where energy is not consumed (as it would be in a resistive load) but continuously oscillates between the two storage elements.
In reactive circuits, the reactive power for an inductor is $$Q_L = I^2 X_L,$$ and for a capacitor it is $$Q_C = I^2 X_C.$$ At resonance, these cancel out (i.e., $$Q_L - Q_C = 0$$), indicating that the net reactive power is zero even though energy is continuously exchanged.
Electromagnetic Fields
More generally, resonance can occur in any system where electromagnetic fields oscillate, such as in radio antennas or even in molecular structures. In these cases, the concept of reactive energy still applies, as it involves the storage and release of energy in fields rather than its conversion into heat or work.
Capacitors in Delta Connection
When capacitors are connected in a delta configuration to a three-phase network, each capacitor is connected across two phases. This arrangement allows for a continuous flow of current through the capacitors even in the absence of a neutral conductor. In such systems, no current flows through the neutral; instead, the balancing current circulates through the phases back to the transformer.
In delta connections, the relationship between the line current (\(I_L\)) and the phase current (\(I_\phi\)) is:
$$I_L = \sqrt{3}\,I_\phi$$
Transformer in Star Connection
In a transformer connected in a star (Y) configuration, the three windings are connected to individual phases and joined together at the star point. If the load does not utilize a neutral conductor, there is no direct path for current flow through the star point; the current flows through the outer phases.
The relationship between the line voltage (\(V_L\)) and the phase voltage (\(V_\phi\)) in a star connection is:
$$V_L = \sqrt{3}\,V_\phi$$
Negative Active Power
Negative active power, or negative active current, arises when the phase of the current is inverted relative to the voltage. This situation can occur when capacitors introduce a phase shift in the current relative to the voltage.
The active power in an AC circuit is defined as:
$$P = V I \cos(\phi)$$
When the phase angle \(\phi\) results in a negative cosine, the active power becomes negative.
Transformer Behavior under Negative Active Power
In this state, the transformer must handle not only the usual magnetic and electrical stresses but also the effects of phase displacement. Negative active power on the secondary side leads to a reduced load, meaning that less electromagnetic force acts on the primary winding. Consequently, this can potentially reduce losses in the transformer.
The apparent power (\(S\)), which combines active power (\(P\)) and reactive power (\(Q\)), is given by:
$$S = \sqrt{P^2 + Q^2}$$
Summary
In summary, resonance is associated with reactive energy because it creates a state where energy continuously oscillates between the magnetic field of an inductor and the electric field of a capacitor without being dissipated. Additionally, when considering transformer systems, negative active power can reduce the effective load on the primary side, potentially improving efficiency. However, the overall impact must be evaluated within the context of the entire electrical system.
Three-Phase Power Measurement and Calculation for Unbalanced Loads
In practice, three-phase systems often experience unbalanced load conditions. Under these conditions, standard balanced formulas are insufficient, and additional analysis is required. The following formulas and methods provide a comprehensive approach.
Instantaneous and Average Active Power
For each phase (denoted as \(a\), \(b\), and \(c\)), the instantaneous power is given by:
$$p_k(t) = v_k(t) \, i_k(t) \quad \text{for } k = a,\,b,\,c$$
The average active power over a cycle \(T\) for each phase is:
$$P_k = \frac{1}{T}\int_0^T v_k(t) \, i_k(t) \, dt$$
The total active power is then the sum of the individual phase powers:
$$P_{total} = P_a + P_b + P_c$$
Complex Power
The complex power for each phase can be expressed as:
$$S_k = V_k I_k^*$$
where \(I_k^*\) denotes the complex conjugate of the current in phase \(k\). The total complex power of the system is the sum over all phases:
$$S_{total} = S_a + S_b + S_c$$
This complex power can be decomposed into its real (active) and imaginary (reactive) components.
Symmetrical Components Analysis
A powerful tool for analyzing unbalanced systems is the symmetrical components method. This method decomposes unbalanced phasors into three balanced sets: zero sequence, positive sequence, and negative sequence.
The symmetrical component currents are defined as:
$$I_0 = \frac{1}{3}(I_a + I_b + I_c)$$
$$I_1 = \frac{1}{3}(I_a + a\,I_b + a^2\,I_c)$$
$$I_2 = \frac{1}{3}(I_a + a^2\,I_b + a\,I_c)$$
Similarly, the symmetrical component voltages are:
$$V_0 = \frac{1}{3}(V_a + V_b + V_c)$$
$$V_1 = \frac{1}{3}(V_a + a\,V_b + a^2\,V_c)$$
$$V_2 = \frac{1}{3}(V_a + a^2\,V_b + a\,V_c)$$
Here, the operator \(a\) is the phase shift operator defined as:
$$a = e^{j120^\circ} = -\frac{1}{2} + j\,\frac{\sqrt{3}}{2}$$
This decomposition simplifies the analysis of unbalanced systems by separating the contributions of balanced components, allowing for targeted mitigation or correction strategies.
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