Chapter 26: Power in AC Circuits
Date: 23-06-19
Definitions
Power is the rate of flow of energy past a given point in the circuit. In ac circuits, inductors and capacitors may result in the periodic reversal of the direction of energy flow.
Real Power
The portion of power averaged over a complete cycle of the ac waveform that results in net transfer of energy in one direction; this is known as the active power.
Reactive Power
The portion of stored power due to stored energy, which returns to the source in each cycle is known as the reactive power.
In this chapter we are introduced to the concepts of active, reactive and apparent power. These are basic tools needed to understand and calculate flows of energy from generators to loads without which we would need to perform complex calculations involving network analysis.
In all ac systems except the very smallest, power is transported in three-phase form.
Power Factor
This is important to consider as a power factor less than 1 means an ac circuit’s wiring has to carry more current than would be necessary with zero reactance, to deliver the same amount of true power to the resistive load. Calculations of power in ac circuits together with the advantages of power factor improvement are included in this chapter.
26.1 Introduction
The instantaneous power is the product of the changing voltage v and changing current i. Since v and i are both sinusoidal, so is their product p = v . i WATTS.
26.2 Determination of Power in AC Circuits
(a) Purely Resistive AC Circuit
Let v = Vm sin(ω t). The resulting current is i = Im sin(ω t). The corresponding power is given by:
From the double angle formulae cos(2A) = 1 – 2sin2(A), which gives sin2(A) = (1/2)(1 – cos(2A)). Thus:
Given that the cosine function is periodic, it will have an average of 0 over a complete cycle. We know that V = (1/√2) Vm and I = (1/√2) Im. The average power developed P is:
Also, P = I2 R = V2 / R as for a dc circuit V = IR. The power in a purely resistive ac circuit is given by:
(b) Power In Purely Inductive AC Circuit
Voltage leads current by 90° in a purely inductive circuit:
The instantaneous power p = vi = (Vm Im sin(ω t)) . (sin(ω t – π/2)). However, sin(ω t – π/2) = -cos(ω t).
Rearranging gives p = -(1/2) Vm Im (2 sin(ω t) cos(ω t)). From the double ∠ formula 2sin(ω t)cos(ω t) = sin(2ω t), thus:
The power is sinusoidal; the average power is 0 over a complete cycle. The symbol for average power is P.
Power Delivery and Return: When v and i are both positive, power is positive. In general, an increase in current through an inductor causes energy to be transferred from the source to the magnetic field. This energy is returned when the current is decreasing. Since power is taken from the inductor and delivered to the source, the average power in a purely inductive circuit is zero.
(c) Purely Capacitive Circuits
Let v = Vm sin(ω t). The current leads voltage by 90° in a purely capacitive circuit, so i = Im sin(ω t + π/2).
This gives the equation p = vi = Vm Im sin(ω t) sin(ω t + π/2). Since sin(ω t + π/2) = cos(ω t):
It shows similar results as inductive circuits, being zero over a complete cycle, but it is worth noting that when voltage is decreasing, energy is transferred from the charged capacitor back to the source.
R-L and R-C AC Circuits
For an R-L or R-C circuit, let v = Vm sin(ω t). The current i will then equal Im sin(ω t + φ). The phase angle φ will be positive for an R-C circuit and negative for an R-L circuit. The instantaneous power p is given by:
The rule sin(A)sin(B) = (1/2)[cos(A-B) – cos(A+B)] can be used. Substituting A = ω t and B = ω t + φ gives:
Since cos(-φ) = cos(φ) and the term cos(2ω t + φ) has a mean value of 0 over a cycle:
P = (1/2) Vm Im cos(φ) =
(1/2) (√2 V) (√2 I) cos(φ) =
The p waveform is at twice the frequency of the v and i waveforms. For an R-L circuit, the areas above the horizontal axis represent the power supplied to the load. The areas below the axis represent the power returned from the inductor when the magnetic field collapses. For a capacitor, the area below the axis represents the power being returned from the charged capacitor. The difference in areas above and below the x-axis represents the heat loss due to circuit resistance.
If IR is the rms current flowing through the resistance, the average power can only be dissipated in the resistance, so P = IR2 R. The average power equations P = VI cos(φ) and P = IR2 R can be used as an equation for power in a circuit containing resistance and inductance or capacitance, where V, I, and IR are rms values.
26.3 Power Triangle and Power Factor
The power triangle consists of the True or Active power (P = VI cos(φ)), Apparent power (S = VI in voltamperes or VA), and Reactive power (Q = VI sin(φ) in var).
The power tri∠ is not a phasor diagram since P, S, and Q are mean values, not rms values of sinusoidally varying quantities.
- Apparent Power (S): S = VI
- Active Power (P): P = VI cos(φ)
- Reactive Power (Q): Q = VI sin(φ)
Using complex notation, S = P + jQ (for inductive) or S = P – jQ (for capacitive).
Why Apparent Power is Important
Transformers, generators, and cables are usually rated in voltamperes (VA), not WATTS. The allowable output of these devices is usually limited to heat loss, which is determined by the voltage and current, almost independent of the power factor. The rating of a machine is defined as the maximum amount of apparent power that it is meant to supply without overheating.
Power Factor
A circuit in which the current lags the voltage (i.e., an inductive circuit) is said to have a lagging power factor. The opposite is true for a capacitive circuit (leading power factor).
26.4 Use of Complex Numbers for Determination of Power
Given voltage V = V ∠ α = V(cos α + jsin α) = a + jb and current I = I ∠θ = I(cosθ + jsinθ) = c + jd.
The phase ∠ between voltage and current is φ = α – θ.
Active Power P = VI cos( α – θ) = VI (cos α cosθ + sin α sinθ) = ac + bd.
Reactive Power Q = VI sin( α – θ) = VI (sin α cosθ – cos α sinθ) = bc – ad.
Summary: The expression for active power P = ac + bd and reactive power Q = bc – ad cannot be obtained by multiplying the complex voltage by the current. They can only be obtained by multiplying the voltage by the conjugate of the current.
26.5 Power Factor Improvement
A high power factor reduces the current flowing, which lowers power losses due to I2R and hence results in cheaper running costs. Suppliers of electricity introduce tariffs that reflect this (i.e., high power factor means lower bills).
Most residential and industrial loads are inductive, meaning they operate at a lagging power factor. In order to reduce the value of S (VA), a capacitor can be connected in parallel with the inductive load. The effect of the capacitor is to reduce the reactive power of the system without changing the active power.