How To Calculate The Power Factor Of Load Bank？

Friends who are often exposed to dummy load testing know that AC circuits contain terms such as resistance, capacitance or resistance, inductance (active power and reactive power). So in order for us to calculate the total power consumed, we need to know the phase difference between the sinusoidal waveforms of the voltage and current.

In AC circuits, the voltage and current waveforms are sine waves, so their amplitudes change over time. Since we know that power is voltage times current (P = V*I) , maximum power occurs when the two voltage and current waveforms are aligned with each other. That is, their peaks and zero-crossings occur at the same time. When this happens, the two waveforms are said to be "in phase".

By defining the total impedance of the circuit, the three main elements in an AC circuit that can affect the relationship between the voltage and current waveforms and their phase difference are resistors, capacitors, and inductors.

The impedance (Z) of an AC circuit is equivalent to the resistance calculated in a DC circuit, and the impedance is measured in ohms. For AC circuits, impedance is usually defined as the ratio of the voltage phasor to the current phasor produced by the circuit elements. A phasor is a straight line drawn in such a way that the magnitude of the voltage or current is represented by its length, and its phase difference relative to other phasor lines is represented by its angular position relative to the other phasor lines.

AC circuits contain resistance and reactance that combine to provide a total impedance (Z) that limits the flow of current around the circuit . But the impedance of an AC circuit is not equal to the algebraic sum of the ohmic values of resistance and reactance, because pure resistance and pure reactance are 90o  out of phase with each other . But we can use this 90o  phase difference as the sides of a right triangle, called an impedance triangle, where impedance is the hypotenuse determined by the Pythagorean theorem.

This geometric relationship between resistance, reactance, and impedance can be visually represented by using the impedance triangle as shown.

Note that impedance is the vector sum of resistance and reactance, and it has not only a magnitude (Z) , but also a phase angle ( Φ ) , which represents the phase difference between resistance and reactance. Also note that as the frequency changes, the triangle changes shape due to the change in reactance (X) . Of course, the resistance (R) will always remain the same.

We can take this idea a step further by transforming the impedance triangle into a power triangle that represents the three elements of power in an AC circuit. Ohm's Law tells us that in a DC circuit, the power (P) in watts is equal to the square of the current (I  2  ) times the resistance (R) . So we can multiply the three sides of the impedance triangle above by I  2 to get the corresponding power triangle as:

Active Power P = I  2  R Watts, (W)

Reactive power Q = I  2  X reactive volt-ampere, (VAr)

Apparent power S = I  2  Z volt- ampere, (VA)

real power in an AC circuit

Active Power (P) , also known as Active Power or Active Power, performs "real work" in a circuit. Real power (in watts) defines the power dissipated by the resistive portion of the circuit. Then the actual power (P) in the AC circuit is the same as the power P in the DC circuit . So just like a DC circuit, it is always calculated as I  2  *R , where R is the total resistive component of the circuit. 

Since the resistance does not create any phasor difference (phase shift) between the voltage and current waveforms, all useful power is transferred directly to the resistance and converted to heat, light and work. Then the power dissipated by the resistor is the real power, basically the average power of the circuit.

To find the corresponding active power value, the voltage and current rms values are multiplied by the cosine of the phase angle.

Active Power P = I  2  R = V * I * cos ( Φ ) Watts, (W)

But since they have no phase difference between the voltage and current in a resistive circuit, the phase shift between the two waveforms will be zero (0) . Then:

Actual power (P) is in watts, voltage (V) is in rms volts, and current (I) is in rms amps.

The actual power is then the I2   *R resistive element measured in watts , which is what you read on your utility meter in watts (W) , kilowatts (kW), and megawatts (MW) . Note that the real power P is always positive.

Reactive power in AC circuits

Reactive power (Q) , (sometimes called reactive power) is the power dissipated in an AC circuit that does no useful work but has a large effect on the phase shift between the voltage and current waveforms. Reactive power is related to the reactance created by inductors and capacitors, which can counteract the effects of active power. There is no reactive power in a DC circuit. 

Unlike active power (P) , which does all the work , reactive power (Q) takes power away from the circuit due to the creation and reduction of induced magnetic fields and capacitive electrostatic fields, making it harder to supply active power directly to a circuit or load .

The power stored by an inductor in its magnetic field attempts to control the current flow, while the power stored by the capacitor's electrostatic field attempts to control the voltage. The result is that the capacitor "produces" reactive power and the inductor "consumes" reactive power. This means that they both consume power and return power to the source, so they don't consume any real power.

To find the reactive power, the voltage and current rms values are multiplied by the sine of the phase angle.

Reactive power Q = I  2  X = V*I*sin( Φ ) reactive volt-ampere, (VAr's)

Since there is a 90o  phase difference between the voltage and current waveforms in pure reactance (inductive or capacitive), multiplying V*I by sin( Φ ) yields a vertical component that is 90  out of phase with each reactance oOther , so:

where reactive power (Q) is in reactive volt-amperes, voltage (V) is in rms volts, and current (I) is in rms amperes.

Then reactive power represents the product of volts and amperes, 90 degrees out of phase with each other , but in general, there can be any phase angle Φ between voltage and current.

Therefore, reactive power is an I  2  X reactive element, and its units are volt-ampere reactive (VAr) , kilovolt-ampere reactive (kVAr), and megavolt-ampere reactive (MVAr) .

Apparent Power in AC Circuits

We have seen above that the active power is dissipated by the resistance and the reactive power is supplied to the reactance. Therefore, the current and voltage waveforms are not in phase due to the difference between the circuit resistive and reactive components.

Then there is a mathematical relationship between active power ( P ) and reactive power ( Q ), called complex power. The product of the rms voltage V applied to an AC circuit and the rms current I flowing into that circuit is called the "volt-ampere product" ( VA ), symbol S , and its magnitude is often referred to as apparent power.

This complex power is not equal to the algebraic sum of the active and reactive powers added together, but rather the vector sum of P and Q given in volt- amperes (VA) . It is a complex power represented by a power triangle. The rms value of the volt-ampere product is often referred to as apparent power, because "obviously" this is the total power dissipated by the circuit, even though the actual power doing work is much less.

Since apparent power consists of two components, resistive power is in-phase power or active power in watts, and reactive power is out-of-phase power in volt-amperes, we can show the vector sum of these two power components in terms of power triangles form appears. A power triangle has four parts: P , Q , S and θ.

The three elements that make up a power source in an AC circuit can be represented graphically by the three sides of a right triangle, roughly the same as the impedance triangle above. As shown, the horizontal (adjacent) side of the power triangle represents the circuit active power ( P ), the vertical (opposite) side represents the circuit reactive power ( Q ), and the hypotenuse represents the apparent power produced ( S ).

P is the I  2  * R or real power to perform the work, in watts, W

Q is I  2  *X or reactive power in volt-amperes reactive, VAr

S is I2  *  Z or apparent power in VA, VA

Φ is the phase angle in degrees. The larger the phase angle, the greater the reactive power

Cos( Φ ) = P/S = W/VA = power factor, pf

Sin( Φ ) = Q/S = VAr/VA

Tan( Φ ) = Q/P = VAr/W

Power factor is calculated as the ratio of real power to apparent power, since this ratio is equal to cos( Φ ) .

The power factor cos( Φ ) is an important part of the AC circuit, and it can also be expressed by circuit impedance or circuit power. Power factor is defined as the ratio of real power (P) to apparent power (S) , usually expressed as a decimal value such as 0.95 , or as a percentage: 95% .

The power factor defines the phase angle between the current and voltage waveforms, where I and V are the magnitudes of the rms values of the current and voltage . Note that it doesn't matter whether the phase angle is the difference between current and voltage or whether the phase angle is the difference between voltage and current. The mathematical relationship is as follows:

We said earlier that in a purely resistive circuit, the current and voltage waveforms are in phase with each other, so when the phase difference is zero (0  o  ) , the actual power dissipated is the same as the apparent power. So the power factor is:

Power factor, pf = cos 0  o    = 1.0

That is, the watts consumed are the same as the volt-amps consumed, resulting in a power factor of 1.0 or 100% . In this case it is called unity power factor.

We also said above that in a pure reactive circuit, the current and voltage waveforms are 90o  out of phase with each other . Since the phase difference is ninety degrees (90  o  ) , the power factor will be:

Power factor, pf = cos 90  o    = 0

That is, the wattage consumed is zero, but there is still voltage and current supplying the reactive load. Obviously, reducing the reactive VAr component of the power triangle will result in a decrease in θ, thereby increasing the power factor to 1 , ie unity. It is also desirable to have a high power factor, as this makes the most efficient use of the circuit carrying current to the load.

We can then write the relationship between active power, apparent power, and circuit power factor as:

An inductive circuit whose current "lags" the voltage (ELI) is said to have a lagging power factor, while a capacitive circuit whose current "leads" the voltage (ICE) is said to have a leading power factor.

A wire wound coil with an inductance of 180mH and a resistance of 35Ω was connected to a 100V 50Hz power supply. Calculate: a) the impedance of the coil, b) the current, c) the power factor, and d) the apparent power dissipated.

Also draw the resulting power triangle for the coil above.

Data given: R = 35 Ω, L = 180mH , V = 100V and ƒ = 50Hz .

At a power factor of 0.5263 or 52.63% , the coil requires 150 VA of power to produce 79 watts of useful work. In other words, at 52.63% power factor, the coil needs 89% more current to do the same job, which is a lot of wasted current.

Adding a power factor correction capacitor ( 32.3uF in this case ) across the coil to increase the power factor above 0.95 or 95% will greatly reduce the reactive power consumed by the coil as these capacitors act as reactive current generation machine, thereby reducing the total amount of current consumed.

Power Triangle and Power Factor Summary

We have seen here that the three elements of electrical power in an AC circuit, namely active power , reactive power and apparent power , can be represented by the three sides of a triangle called the power triangle . Since these three elements are represented by a "right triangle", their relationship can be defined as: S  2   = P  2   + Q  2 , where: P is the active power in watts (W ) and Q is the active power in watts (W) Reactive power in volt-ampere reactive (VAr) , S is the apparent power in volt- ampere (VA) .

We also saw that in an AC circuit, the quantity cos( Φ ) is called the power factor. The power factor of an AC circuit is defined as the ratio of the active power (W) consumed by the circuit to the apparent power (VA) consumed by the same circuit . So this gives us: Power Factor = Real Power / Apparent Power, or pf = W/VA .

Then the cosine of the resulting angle between the current and the voltage is the power factor. Usually power factor is expressed as a percentage, such as 95% , but it can also be expressed as a decimal value, such as 0.95 .

When the power factor is equal to 1.0 (units) or 100% , i.e. when the actual power dissipated is equal to the apparent power of the circuit, the phase angle between the current and the voltage is 0  o , because: cos  -1  (1.0) = 0  o . When the power factor is zero (0) , the phase angle between the current and voltage will be 90 degrees because: cos  -1 ( 0  ) = 90 degrees . In this case, the actual power dissipated by the AC circuit is zero, regardless of the circuit current.

In a real AC circuit, the power factor can be between 0 and 1.0 , depending on the passive components in the connected load. For resistive loads or circuits (the most common case), the power factor will "lag". In a capacitive-resistive circuit, the power factor will "lead". AC circuits can then be defined as having unity, lagging or leading power factor.

A poor power factor with a value close to zero (0) will dissipate wasted power and thus reduce the efficiency of the circuit, while a circuit or load with a power factor close to one (1.0) or unity (100%) will be more efficient. This is because a circuit or load with a low power factor requires more current than the same circuit or load with a power factor close to 1.0 (units).