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Rabu, 11 April 2012

Reactive energy and power factor

The nature of reactive energy

Alternating current systems supply two forms of energy:
  • “Active” energy measured in kilowatt hours (kWh) which is converted into mechanical work, heat, light, etc
  • “Reactive” energy, which again takes two forms:
  - “Reactive” energy required by inductive circuits (transformers, motors, etc.),
  - “Reactive” energy supplied by capacitive circuits (cable capacitance, power  
     capacitors, etc)
All inductive (i.e. electromagnetic) machines and devices that operate on AC systems convert electrical energy from the power system generators into mechanical work and heat. This energy is measured by kWh meters, and is referred to as “active” or “wattful” energy. In order to perform this conversion, magnetic fields have to be established in the machines, and these fields are associated with another form of energy to be supplied from the power system, known as “reactive” or “wattless” energy.
The reason for this is that inductive circuit cyclically absorbs energy from the system (during the build-up of the magnetic fields) and re-injects that energy into the system (during the collapse of the magnetic fields) twice in every power-frequency cycle.
An exactly similar phenomenon occurs with shunt capacitive elements in a power system, such as cable capacitance or banks of power capacitors, etc. In this case, energy is stored electrostatically. The cyclic charging and discharging of capacitive circuit reacts on the generators of the system in the same manner as that described above for inductive circuit, but the current flow to and from capacitive circuit in exact phase opposition to that of the inductive circuit. This feature is the basis on which power factor correction schemes depend.
It should be noted that while this “wattless” current (more accurately, the “wattless” component of a load current) does not draw power from the system, it does cause power losses in transmission and distribution systems by heating the conductors.
In practical power systems, “wattless” components of load currents are invariably inductive, while the impedances of transmission and distribution systems are predominantly inductively reactive. The combination of inductive current passing through an inductive reactance produces the worst possible conditions of voltage drop (i.e. in direct phase opposition to the system voltage).
For these reasons (transmission power losses and voltage drop), the power-supply authorities reduce the amount of “wattless” (inductive) current as much as possible.
“Wattless” (capacitive) currents have the reverse effect on voltage levels and produce voltage-rises in power systems.
The power (kW) associated with “active” energy is usually represented by the letter P.
The reactive power (kvar) is represented by Q. Inductively-reactive power is conventionally positive (+ Q) while capacitively-reactive power is shown as a negative quantity (- Q).
The apparent power S (kVA) is a combination of P and Q (see Fig. L1).
Sub-clause "The Power Factor" shows the relationship between P, Q, and S


Fig. L1: An electric motor requires active power P and reactive power Q from the power system

Equipement and appliances requiring reactive energy

All AC equipement and appliances that include electromagnetic devices, or depend on magnetically-coupled windings, require some degree of reactive current to create magnetic flux.
The most common items in this class are transformers and reactors, motors and discharge lamps (with magnetic ballasts) (see Fig. L2).
The proportion of reactive power (kvar) with respect to active power (kW) when an item of equipement is fully loaded varies according to the item concerned being:
  • 65-75% for asynchronous motors
  • 5-10% for transformers


Fig. L2: Power consuming items that also require reactive energy

The power factor

The power factor is the ratio of kW to kVA. The closer the power factor approaches its maximum possible value of 1, the greater the benefit to consumer and supplier.
PF = P (kW) / S (kVA)
P = Active power
S = Apparent power
Definition of power factor
The power factor of a load, which may be a single power-consuming item, or a number of items (for example an entire installation), is given by the ratio of P/S i.e. kW divided by kVA at any given moment.
The value of a power factor will range from 0 to 1.
If currents and voltages are perfectly sinusoidal signals, power factor equals cos φ.
A power factor close to unity means that the reactive energy is small compared with the active energy, while a low value of power factor indicates the opposite condition.
Power vector diagram
  • Active power P (in kW)
  - Single phase (1 phase and neutral): P = V I cos φ
  - Single phase (phase to phase): P = U I cos φ
  - Three phase (3 wires or 3 wires + neutral): P = \sqrt 3 U I cos φ
  • Reactive power Q (in kvar)
- Single phase (1 phase and neutral): P = V I sin φ
- Single phase (phase to phase): Q = U I sin φ
- Three phase (3 wires or 3 wires + neutral): P = \sqrt 3 U I sin φ
  • Apparent power S (in kVA)
  - Single phase (1 phase and neutral): S = V I
  - Single phase (phase to phase): S = U I
  - Three phase (3 wires or 3 wires + neutral): P = \sqrt 3 U I
V = Voltage between phase and neutral
U = Voltage between phases
I = Line current
φ = Phase angle between vectors V and I.
  - For balanced and near-balanced loads on 4-wire systems
Current and voltage vectors, and derivation of the power diagram
The power “vector” diagram is a useful artifice, derived directly from the true rotating vector diagram of currents and voltage, as follows:
The power-system voltages are taken as the reference quantities, and one phase only is considered on the assumption of balanced 3-phase loading.
The reference phase voltage (V) is co-incident with the horizontal axis, and the current (I) of that phase will, for practically all power-system loads, lag the voltage by an angle φ.
The component of I which is in phase with V is the “wattful” component of I and is equal to I cos φ, while VI cos φ equals the active power (in kW) in the circuit, if V is expressed in kV.
The component of I which lags 90 degrees behind V is the wattless component of I and is equal to I sin φ, while VI sinφ equals the reactive power (in kvar) in the circuit, if V is expressed in kV.
If the vector I is multiplied by V, expressed in kV, then VI equals the apparent power (in kVA) for the circuit.
The simple formula is obtained: S2 = P2 + Q2
The above kW, kvar and kVA values per phase, when multiplied by 3, can therefore conveniently represent the relationships of kVA, kW, kvar and power factor for a total 3-phase load, as shown in Figure L3.


Fig. L3: Power diagram
An example of power calculations (see Fig. L4)

Type of circuitApparent power S (kVA)Active power P (kW)Reactive power Q (kvar)
Single-phase (phase and neutral)  S = VIP = VI cos φQ = VI sin φ
Single-phase (phase to phase)  S = UIP = UI cos φQ = UI sin φ
Example        5 kW of load10 kVA5 kW8.7 kvar
cos φ = 0.5
Three phase 3-wires or 3-wires + neutralS = \sqrt 3 UIP = \sqrt 3 UI cos φQ = \sqrt 3 UI sin φ
ExampleMotor Pn = 51 kW65 kVA56 kW33 kvar
cos φ= 0.86
ρ= 0.91 (motor efficiency)
Fig. L4: Example in the calculation of active and reactive power

Practical values of power factor

The calculations for the three-phase example above are as follows:
Pn = delivered shaft power = 51 kW
P = active power consumed

P=\frac {Pn}{\rho}=\frac{51}{0.91}=56\, kW
S = apparent power

S=\frac{P}{cos \phi}=\frac {56}{0.86}= 65\, kVA
So that, on referring to diagram Figure L5 or using a pocket calculator, the value of tan φcorresponding to a cos φ of 0.86 is found to be 0.59
Q = P tan φ = 56 x 0.59 = 33 kvar (see Figure L15).

Q=\sqrt{S^2 - P^2}=\sqrt{65^2 - 56^2}=33\, kvar

Average power factor values for the most commonly-used equipment and appliances (see Fig. L6)


Fig. L5: Calculation power diagram

Equipment and appliancescos φtan φ
  • Common induction motor
loaded at
  • Incandescent lamps
  • Fluorescent lamps (uncompensated)
  • Fluorescent lamps (compensated)
  • Discharge lamps
0.4 to 0.6
2.29 to 1.33
  • Ovens using resistance elements
  • Induction heating ovens (compensated)
  • Dielectric type heating ovens
  • Resistance-type soldering machines
  • Fixed 1-phase arc-welding set
  • Arc-welding motor-generating set
  • Arc-welding transformer-rectifier set
0.8 to 0.9
0.7 to 0.9
0.7 to 0.8
0.75 to 0.48
1.02 to 0.48
1.02 to 0.75
  • Arc furnace

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