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Power factor

The power factor of an AC electric power system is defined as the ratio of the real power flowing to the load to the apparent power in the circuit, and is a dimensionless number between 0 and 1 (frequently expressed as a percentage, e.g. 0.5 pf = 50% pf). Real power is the capacity of the circuit for performing work in a particular time. Apparent power is the product of the current and voltage of the circuit. Due to energy stored in the load and returned to the source, or due to a non-linear load that distorts the wave shape of the current drawn from the source, the apparent power will be greater than the real power.

In an electric power system, a load with a low power factor draws more current than a load with a high power factor for the same amount of useful power transferred. The higher currents increase the energy lost in the distribution system, and require larger wires and other equipment. Because of the costs of larger equipment and wasted energy, electrical utilities will usually charge a higher cost to industrial or commercial customers where there is a low power factor.

Linear loads with low power factor (such as induction motors) can be corrected with a passive network of capacitors or inductors. Non-linear loads, such as rectifiers, distort the current drawn from the system. In such cases, active or passive power factor correction may be used to counteract the distortion and raise the power factor. The devices for correction of the power factor may be at a central substation, spread out over a distribution system, or built into power-consuming equipment.

Power factor in linear circuits

In a purely resistive AC circuit, voltage and current waveforms are in step (or in phase), changing polarity at the same instant in each cycle. All the power entering the loads is consumed. Where reactive loads are present, such as with capacitors or inductors, energy storage in the loads result in a time difference between the current and voltage waveforms. During each cycle of the AC voltage, extra energy, in addition to any energy consumed in the load, is temporarily stored in the load in electric or magnetic fields, and then returned to the power grid a fraction of a second later in the cycle. The "ebb and flow" of this nonproductive power increases the current in the line. Thus, a circuit with a low power factor will use higher currents to transfer a given quantity of real power than a circuit with a high power factor. A linear load does not change the shape of the waveform of the current, but may change the relative timing (phase) between voltage and current.

Circuits containing purely resistive heating elements (filament lamps, strip heaters, cooking stoves, etc.) have a power factor of 1.0. Circuits containing inductive or capacitive elements (electric motors, solenoid valves, lamp ballasts, and others ) often have a power factor below 1.0.

Definition and calculation

AC power flow has the three components: real power (also known as active power) (P), measured in watts (W); apparent power (S), measured in volt-amperes (VA); and reactive power (Q), measured in reactive volt-amperes (var).

The power factor is defined as:


In the case of a perfectly sinusoidal waveform, P, Q and S can be expressed as vectors that form a vector triangle such that:

S^2\,\! = {P^2\,\!} + {Q^2\,\!}.

If \varphi is the phase angle between the current and voltage, then the power factor is equal to the cosine of the angle, \left|\cos\varphi\right|, and:

|P| = |S| \left|\cos\varphi\right|.

Since the units are consistent, the power factor is by definition a dimensionless number between 0 and 1. When power factor is equal to 0, the energy flow is entirely reactive, and stored energy in the load returns to the source on each cycle. When the power factor is 1, all the energy supplied by the source is consumed by the load. Power factors are usually stated as "leading" or "lagging" to show the sign of the phase angle.

If a purely resistive load is connected to a power supply, current and voltage will change polarity in step, the power factor will be unity (1), and the electrical energy flows in a single direction across the network in each cycle. Inductive loads such as transformers and motors (any type of wound coil) consume reactive power with current waveform lagging the voltage. Capacitive loads such as capacitor banks or buried cable generate reactive power with current phase leading the voltage. Both types of loads will absorb energy during part of the AC cycle, which is stored in the device's magnetic or electric field, only to return this energy back to the source during the rest of the cycle.

For example, to get 1 kW of real power, if the power factor is unity, 1 kVA of apparent power needs to be transferred (1 kW Ã· 1 = 1 kVA). At low values of power factor, more apparent power needs to be transferred to get the same real power. To get 1 kW of real power at 0.2 power factor, 5 kVA of apparent power needs to be transferred (1 kW Ã· 0.2 = 5 kVA). This apparent power must be produced and transmitted to the load in the conventional fashion, and is subject to the usual distributed losses in the production and transmission processes.

Electrical loads consuming alternating current power consume both real power and reactive power. The vector sum of real and reactive power is the apparent power. The presence of reactive power causes the real power to be less than the apparent power, and so, the electric load has a power factor of less than 1.

Power factor correction of linear loads

It is often desirable to adjust the power factor of a system to near 1.0. This power factor correction (PFC) is achieved by switching in or out banks of inductors or capacitors. For example the inductive effect of motor loads may be offset by locally connected capacitors. When reactive elements supply or absorb reactive

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Question:I am doing a project for university. He has told us to use a t-test. He has also told us that if we use another statistical test instead then we could gain extra marks. Obviously, I want to get the best grade possible because this is for my degree. I want to impress my supervisor by using a different statistical test. I will briefly give details of what I am analysing. Please can someone say whether I should use ANOVA, or to stick with the t-test... I am analysing 3 sets of data 1st data set (A) - 29 values 2nd data set (B) - 30 values 3rd data set (C) - 37 values Our task: To compare the mean and standard deviation A and B using the paried t-test A and C using the non-paired t-test B and C using the non-paired t-test I randomly came across the ANOVA test. It seems relevant as it is meant to be used to compare 3 means of different data. It looks useful and I think Excel does it automatically aswell. Do you think I should use it, or stick with the t-test function in excel? If it **is** suitable, then I am not sure about a few other things: Q1 what are factors in ANOVA tests? Q2 in the ANOVA test which I will perform, should it be 1- or 2-tailed? **Please can anyone answer this question for me** Im really stuck... the library is closed and there are no good Internet sites :-( thankyou for your kind help :) Ive heard of that programme before and i know my university definitely has it. I just havent used it because we never needed to for my course. I think my university sells SPSS CDs to students for a lower price. I might be totally wrong though Does anyone know whether I should perform a one- or two-tailed test for anova? thanks again I know about SPSS... :) Im just not sure if the ANOVA test is applicable, and if it is then do I need to do a 1 or 2-tailed test?

Answers:SPSS and SAS packages have tests to compare means The tests are called Tukey, SNK, LSD http://www.webpages.uidaho.edu/~brian/stat401ch9_02.pdf For you SPSS would be easier but the program is not free.

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Answers:I dont know about b., but for a lobsters need saltwater i think and a lake is fresh... and c is the opposite i think (they need freshwater and they were put in salt). Maybe the one for b is that there isnt enought room to grow.

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Answers:I'm not a biologist or anything but my guess would be that factors would include overpopulation, lack of space, chemicals put into ponds, supply of food, and quality of food. Anything that would effect the pond and its habitants would effect everything else.

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