Reactive Power

Reactive Power can best be described as the quantity of “unused” power that is developed by reactive components, such as inductors or capacitors in an AC circuit or system. In a DC circuit, the product of “volts x amps” gives the power consumed in watts by the circuit.

  

However, while this formula is also true for purely resistive AC circuits, the situation is slightly more complex in an AC circuits containing reactive components as this volt-amp product can change with frequency.

In an AC circuit, the product of voltage and current is expressed as volt-amperes (VA) or kilo volt-amperes (kVA) and is known as Apparent power, symbol S. In a non-inductive purely resistive circuit such as heaters, irons, kettles and filament bulbs etc, their reactance is practically zero, so the impedance of the circuit is composed almost entirely of just resistance.

For an AC resistive circuit, the current and voltage are in-phase and the power at any instant can be found by multiplying the voltage by the current at that instant, and because of this “in-phase” relationship, the rms values can be used to find the equivalent DC power or heating effect.

However, if the circuit contains reactive components, the voltage and current waveforms will be “out-of-phase” by some amount determined by the circuits phase angle. If the phase angle between the voltage and the current is at its maximum of 90o, the volt-amp product will have equal positive and negative values.

In other words, the reactive circuit returns as much power to the supply as it consumes resulting in the average power consumed by the circuit being zero, as the same amount of energy keeps flowing alternately from source to the load and back from load to source.

Since we have a voltage and a current but no power dissipated, the expression of P = IV (rms) is no longer valid and it therefore follows that the volt-amp product in an AC circuit does not necessarily give the power consumed. Then in order to determine the “real power”, also called Active power, symbol P consumed by an AC circuit, we need to account for not only the volt-amp product but also the phase angle difference between the voltage and the current waveforms given by the equation: VI.cosΦ.

Then we can write the relationship between the apparent power and active or real power as:

 

 

Note that power factor (PF) is defined as the ratio between the active power in watts and the apparent power in volt-amperes and indicates how effectively electrical power is being used. In a non-inductive resistive AC circuit, the active power will be equal to the apparent power as the fraction of P/S becomes equal to one or unity. A circuits power factor can be expressed either as a decimal value or as a percentage.

But as well as the active and apparent powers in AC circuits, there is also another power component that is present whenever there is a phase angle. This component is called Reactive Power (sometimes referred to as imaginary power) and is expressed in a unit called “volt-amperes reactive”, (VAr), symbol Q and is given by the equation: VI.sinΦ.

Reactive power, or VAr, is not really power at all but represents the product of volts and amperes that are out-of-phase with each other. Reactive power is the portion of electricity that helps establish and sustain the electric and magnetic fields required by alternating current equipment. The amount of reactive power present in an AC circuit will depend upon the phase shift or phase angle between the voltage and the current and just like active power, reactive power is positive when it is “supplied” and negative when it is “consumed”.

Reactive power is used by most types of electrical equipment that uses a magnetic field, such as motors, generators and transformers. It is also required to supply the reactive losses on overhead power transmission lines.

The relationship of the three elements of power, active power, (watts) apparent power, (VA) and reactive power, (VAr) in an AC circuit can be represented by the three sides of right-angled triangle. This representation is called a Power Triangle as shown:

Power in an AC Circuit

 

From the above power triangle we can see that AC circuits supply or consume two kinds of power: active power and reactive power. Also, active power is never negative, whereas reactive power can be either positive or negative in value so it is always advantageous to reduce reactive power in order to improve system efficiency.

The main advantage of using AC electrical power distribution is that the supply voltage level can be changed using transformers, but transformers and induction motors of household appliances, air conditioners and industrial equipment all consume reactive power which takes up space on the transmission lines since larger conductors and transformers are required to handle the larger currents which you need to pay for.

Reactive Power Analogy with Beer

In many ways, reactive power can be thought of like the foam head on a pint or glass of beer. You pay the barman for a full glass of beer but only drink the actual liquid beer which is always less than a full glass.

This is because the head (or froth) of the beer takes up additional wasted space in the glass leaving less room for the real beer that you consume, and the same idea is true for reactive power.

But for many industrial power applications, reactive power is often useful for an electrical circuit to have. While the real or active power is the energy supplied to run a motor, heat a home, or illuminate an electric light bulb, reactive power provides the important function of regulating the voltage thereby helping to move power effectively through the utility grid and transmission lines to where it is required by the load.

While reducing reactive power to help improve the power factor and system efficiency is a good thing, one of the disadvantages of reactive power is that a sufficient quantity of it is required to control the voltage and overcome the losses in a transmission network. This is because if the electrical network voltage is not high enough, active power cannot be supplied. But having too much reactive power flowing around in the network can cause excess heating (I2R losses) and undesirable voltage drops and loss of power along the transmission lines.

Power Factor Correction of Reactive Power

One way to avoid reactive power charges, is to install power factor correction capacitors. Normally residential customers are charged only for the active power consumed in kilo-watt hours (kWhr) because nearly all residential and single phase power factor values are essentially the same due to power factor correction capacitors being built into most domestic appliances by the manufacturer.

Industrial customers, on the other hand, which use 3-phase supplies have widely different power factors, and for this reason, the electrical utility may have to take the power factors of these industrial customers into account paying a penalty if their power factor drops below a prescribed value because it costs the utility companies more to supply industrial customers since larger conductors, larger transformers, larger switchgear, etc, is required to handle the larger currents.

Generally, for a load with a power factor of less than 0.95 more reactive power is required. For a load with a power factor value higher than 0.95 is considered good as the power is being consumed more effectively, and a load with a power factor of 1.0 or unity is considered perfect and does not use any reactive power.

Then we have seen that “apparent power” is a combination of both “reactive power” and “active power”. Active or real power is a result of a circuit containing resistive components only, while reactive power results from a circuit containing either capacitive and inductive components. Almost all AC circuits will contain a combination of these R, L and C components.

Since reactive power takes away from the active power, it must be considered in an electrical system to ensure that the apparent power supplied is sufficient to supply the load. This is a critical aspect of understanding AC power sources because the power source must be capable of supplying the necessary volt-amp (VA) power for any given load.

Previous
Average Voltage Tutorial

Next
Harmonics

Other Tutorials in AC Circuits

• Passive Components in AC Circuits
• Harmonics
• Reactive Power
• Average Voltage Tutorial
• RMS Voltage Tutorial
• Parallel Resonance Circuit
• Series Resonance Circuit
• Parallel RLC Circuit Analysis
• Series RLC Circuit Analysis
• AC Capacitance and Capacitive Reactance
• AC Inductance and Inductive Reactance
• AC Resistance and Impedance

Harmonics

In an AC circuit, a resistance behaves in exactly the same way as it does in a DC circuit. That is, the current flowing through the resistance is proportional to the voltage across it.

  

This is because a resistor is a linear device and if the voltage applied to it is a sine wave, the current flowing through it is also a sine wave so the phase difference between the two sinusoids is zero.

Generally when dealing with alternating voltages and currents in electrical circuits it is assumed that they are pure and sinusoidal in shape with only one frequency value, called the “fundamental frequency” being present, but this is not always the case.

In an electrical or electronic device or circuit that has a voltage-current characteristic which is not linear, that is, the current flowing through it is not proportional to the applied voltage. The alternating waveforms associated with the device will be different to a greater or lesser extent to those of an ideal sinusoidal waveform. These types of waveforms are commonly referred to as non-sinusoidal or complex waveforms.

Complex waveforms are generated by common electrical devices such as iron-cored inductors, switching transformers, electronic ballasts in fluorescent lights and other such heavily inductive loads as well as the output voltage and current waveforms of AC alternators, generators and other such electrical machines. The result is that the current waveform may not be sinusoidal even though the voltage waveform is.

Also most electronic power supply switching circuits such as rectifiers, silicon controlled rectifier (SCR’s), power transistors, power converters and other such solid state switches which cut and chop the power supplies sinusoidal waveform to control motor power, or to convert the sinusoidal AC supply to DC. Theses switching circuits tend to draw current only at the peak values of the AC supply and since the switching current waveform is non-sinusoidal the resulting load current is said to contain Harmonics.

Non-sinusoidal complex waveforms are constructed by “adding” together a series of sine wave frequencies known as “Harmonics”. Harmonics is the generalised term used to describe the distortion of a sinusoidal waveform by waveforms of different frequencies.

Then whatever its shape, a complex waveform can be split up mathematically into its individual components called the fundamental frequency and a number of “harmonic frequencies”. But what do we mean by a “fundamental frequency”.

Fundamental Frequency

A Fundamental Waveform (or first harmonic) is the sinusoidal waveform that has the supply frequency. The fundamental is the lowest or base frequency, ƒ on which the complex waveform is built and as such the periodic time, Τ of the resulting complex waveform will be equal to the periodic time of the fundamental frequency.

Let’s consider the basic fundamental or 1st harmonic AC waveform as shown.

Where: Vmax is the peak value in volts and ƒ is the waveforms frequency in Hertz (Hz).

We can see that a sinusoidal waveform is an alternating voltage (or current), which varies as a sine function of angle, 2πƒ. The waveforms frequency, ƒ is determined by the number of cycles per second. In the United Kingdom this fundamental frequency is set at 50Hz while in the United States it is 60Hz.

Harmonics are voltages or currents that operate at a frequency that is an integer (whole-number) multiple of the fundamental frequency. So given a 50Hz fundamental waveform, this means a 2nd harmonic frequency would be 100Hz (2 x 50Hz), a 3rd harmonic would be 150Hz (3 x 50Hz), a 5th at 250Hz, a 7th at 350Hz and so on. Likewise, given a 60Hz fundamental waveform, the 2nd, 3rd, 4th and 5th harmonic frequencies would be at 120Hz, 180Hz, 240Hz and 300Hz respectively.

So in other words, we can say that “harmonics” are multiples of the fundamental frequency and can therefore be expressed as: 2ƒ, 3ƒ, 4ƒ, etc. as shown.

Complex Waveforms Due To Harmonics

Note that the red waveforms above, are the actual shapes of the waveforms as seen by a load due to the harmonic content being added to the fundamental frequency.

The fundamental waveform can also be called a 1st harmonics waveform. Therefore, a second harmonic has a frequency twice that of the fundamental, the third harmonic has a frequency three times the fundamental and a fourth harmonic has one four times the fundamental as shown in the left hand side column.

The right hand side column shows the complex wave shape generated as a result of the effect between the addition of the fundamental waveform and the harmonic waveforms at different harmonic frequencies. Note that the shape of the resulting complex waveform will depend not only on the number and amplitude of the harmonic frequencies present, but also on the phase relationship between the fundamental or base frequency and the individual harmonic frequencies.

We can see that a complex wave is made up of a fundamental waveform plus harmonics, each with its own peak value and phase angle. For example, if the fundamental frequency is given as; E = Vmax(2πƒt), the values of the harmonics will be given as:

For a second harmonic:

E2 = V2max(2×2πƒt) = V2max(4πƒt), = V2max(2ωt)

For a third harmonic:

E3 = V3max(3×2πƒt) = V3max(6πƒt), = V3max(3ωt)

For a fourth harmonic:

E4 = V4max(4×2πƒt) = V4max(8πƒt), = V4max(4ωt)

and so on.

Then the equation given for the value of a complex waveform will be:

Harmonics are generally classified by their name and frequency, for example, a 2ndharmonic of the fundamental frequency at 100 Hz, and also by their sequence. Harmonic sequence refers to the phasor rotation of the harmonic voltages and currents with respect to the fundamental waveform in a balanced, 3-phase 4-wire system.

A positive sequence harmonic ( 4th, 7th, 10th, …) would rotate in the same direction (forward) as the fundamental frequency. Where as a negative sequence harmonic ( 2nd, 5th, 8th, …) rotates in the opposite direction (reverse) of the fundamental frequency.

Generally, positive sequence harmonics are undesirable because they are responsible for overheating of conductors, power lines and transformers due to the addition of the waveforms.

Negative sequence harmonics on the other hand circulate between the phases creating additional problems with motors as the opposite phasor rotation weakens the rotating magnetic field require by motors, and especially induction motors, causing them to produce less mechanical torque.

Another set of special harmonics called “triplens” (multiple of three) have a zero rotational sequence. Triplens are multiples of the third harmonic ( 3rd, 6th, 9th, …), etc, hence their name, and are therefore displaced by zero degrees. Zero sequence harmonics circulate between the phase and neutral or ground.

Unlike the positive and negative sequence harmonic currents that cancel each other out, third order or triplen harmonics do not cancel out. Instead add up arithmetically in the common neutral wire which is subjected to currents from all three phases.

The result is that current amplitude in the neutral wire due to these triplen harmonics could be up to 3 times the amplitude of the phase current at the fundamental frequency causing it to become less efficient and overheat.

Then we can summarise the sequence effects as multiples of the fundamental frequency of 50Hz as:

Harmonic Sequencing

Name	Fund.	2nd	3rd	4th	5th	6th	7th	8th	9th
Frequency, Hz	50	100	150	200	250	300	350	400	450
Sequence	+	–	0	+	–	0	+	–	0

Note that the same harmonic sequence also applies to 60Hz fundamental waveforms.

Sequence	Rotation	Harmonic Effect
+	Forward	Excessive Heating Effect
–	Reverse	Motor Torque Problems
0	None	Adds Voltages and/or Currents in Neutral Wire causing Heating

Harmonics Summary

Harmonics have only been around in sufficient quantities over the last few decades since the introduction of electronic drives for motors, fans and pumps, power supply switching circuits such as rectifiers, power converters and thyristor power controllers as well as most non-linear electronic phase controlled loads and high frequency (energy saving) fluorescent lights. This is due mainly to the fact that the controlled current drawn by the load does not faithfully follow the sinusoidal supply waveforms as in the case of rectifiers or power semiconductor switching circuits.

Harmonics in the electrical power distribution system combine with the fundamental frequency (50Hz or 60Hz) supply to create distortion of the voltage and/or current waveforms. This distortion creates a complex waveform made up from a number of harmonic frequencies which can have an adverse effect on electrical equipment and power lines.

The amount of waveform distortion present giving a complex waveform its distinctive shape is directly related to the frequencies and magnitudes of the most dominant harmonic components whose harmonic frequency is multiples (whole integers) of the fundamental frequency. The most dominant harmonic components are the low order harmonics from 2nd to the 19th with the triplens being the worst.

Power in AC Circuits

In a direct current circuit, the power consumed is simply the product of the dc voltage times the DC current, VxI and is measured in watts. However, we can not calculate it in a similar manner for reactive AC circuits.

  

Electrical power is the “rate” at which energy is being consumed in a circuit and as such all electrical and electronic components and devices have a limit to the amount of electrical power that they can safely handle. For example, a 1/4 watt resistor or a 20 watt amplifier.

Electrical power can be time-varying either as a DC quantity or as an AC quantity. The amount of power in a circuit at any instant of time is called the instantaneous power and is given by the well-known relationship of P = VI. So one watt (which is the rate of expending energy at one joule per second) will be equal to the volt-ampere product of one volt times one ampere.

Then the power absorbed or supplied by a circuit element is the product of the voltage, V across the element, and the current, I flowing through it. So if we had a DC circuit with a resistance of “R” ohms, the power dissipated by the resistor in watts is given by any of the following generalised formulas:

Electrical Power

 

Where: V is the dc voltage, I is the dc current and R is the value of the resistance.

So power within an electrical circuit is only present when both the voltage and current are present, that is no open-circuit or closed-circuit conditions. Consider the following simple example of a standard resistive dc circuit:

DC Resistive Circuit

Electrical Power in an AC Circuit

In a DC circuit, the voltages and currents are generally constant, that is not varying with time as there is no sinusoidal waveform associated with the supply. However in an AC circuit, the instantaneous values of the voltage, current and therefore power are constantly changing being influenced by the supply. So we can not calculate the power in AC circuits in the same manner as we can in DC circuits, but we can still say that power (p) is equal to the voltage (v) times the amperes (i).

Another important point is that AC circuits contain reactance, so there is a power component as a result of the magnetic and/or electric fields created by the components. The result is that unlike a purely resistive component, this power is not only consumed but instead is stored and then returned back to the supply as the sinusoidal waveform goes through one complete periodic cycle.

Thus, the average power absorbed by a circuit is the sum of the power stored and the power returned over one complete cycle. So a circuits average power consumption will be the average of the instantaneous power over one full cycle with the instantaneous power, p defined as the multiplication of the instantaneous voltage, v by the instantaneous current, i. Note that as the sine function is periodic and continuous, the average power given over all time will be exactly the same as the average power given over a single cycle.

Let us assume that the waveforms of the voltage and current are both sinusoidal, so we recall that:

Sinusoidal Voltage Waveform

 

As the instantaneous power is the power at any instant of time, then:

 

Applying the trigonometric product-to-sum identity of:

 

and θ = θv – θi (the phase difference between the voltage and the current waveforms) into the above equation gives:

 

Where V and I are the root-mean-squared (rms) values of the sinusoidal waveforms, v  and i respectively, and θ is the phase difference between the two waveforms. Therefore we can express the instantaneous power as being:

Instantaneous AC Power Equation

 

This equation shows us that the instantaneous AC power has two different parts and is therefore the sum of these two terms. The second term is a time varying sinusoid whose frequency is equal to twice the angular frequency of the supply due to the 2ω part of the term. The first term however is a constant whose value depends only on the phase difference, θ between the voltage, (V) and the current, (I).

As the instantaneous power is constantly changing with the profile of the sinusoid over time, this makes it difficult to measure. It is therefore more convenient, and easier on the maths to use the average or mean value of the power. So over a fixed number of cycles, the average value of the instantaneous power of the sinusoid is given simply as:

 

where V and I are the sinusoids rms values, and θ (Theta) is the phase angle between the voltage and the current. The units of power are in watts (W).

The AC Power dissipated in a circuit can also be found from the impedance, Z of the circuit using the voltage, Vrms or the current, Irms flowing through the circuit as shown.

AC Power Example No1

The voltage and current values of a 50Hz sinusoidal supply are given as: vt = 240 sin(ωt +60o)Volts and it = 5 sin(ωt -10o)Amps respectively. Find the values of the instantaneous power and the average power absorbed by the circuit.

From above, the instantaneous power absorbed by the circuit is given as:

 

Applying the trigonometric identity rule from above gives:

 

The average power is then calculated as:

 

You may have noticed that the average power value of 205.2 watts is also the first term value of the instantaneous power p(t) as this first term constant value is the average or mean rate of energy change between the source and load.

AC Power in a Purely Resistive Circuit

We have seen thus far, that in a dc circuit, power is equal to the product of voltage and current and this relationship is also true for a purely resistive AC circuit. Resistors are electrical devices that consume energy and the power in a resistor is given by p = VI = I2R = V2/R. This power is always positive.

Consider the following purely resistive (that is infinite capacitance, C = ∞ and zero inductance, L = 0) circuit with a resistor connected to an AC supply, as shown.

Purely Resistive Circuit

 

When a pure resistor is connected to a sinusoidal voltage supply, the current flowing through the resistor will vary in proportion to the supply voltage, that is the voltage and current waveforms are “in-phase” with each other. Since the phase difference between the voltage waveform and the current waveform is 0o, the phase angle resulting in cos 0o will be equal to 1.

Then the electrical power consumed by the resistor is given by:

Electrical Power in a Pure Resistor

 

As the voltage and current waveforms are in-phase, that is both waveforms reach their peak values at the same time, and also pass through zero at the same time, the power equation above reduces down to just VxI. Therefore the the power at any instant can be found by multiplying together the two waveforms to give the volt-ampere product. This is called the “Real Power”, (P) measured in watts, (W), Kilowatt (kW), Megawatt (MW), etc.

AC Power Waveforms for a Pure Resistor

 

The diagram shows the voltage, current and corresponding power waveforms. As the voltage and current waveforms are both in-phase, during the positive half-cycle, when the voltage is positive, the current is also positive so the power is positive, as a positive times a positive equals a positive. During the negative half-cycle, the voltage in negative, so to is the current resulting in the power being positive, as a negative times a negative equals a positive.

Then in a purely resistive circuit, electrical power is consumed ALL the time that current is flowing through the resistor and is given as: P = VxI = I2R watts. Note that both V and I can be their rms values where: V = IxR and I = V/R.

AC Power in a Purely Inductive Circuit

In a purely inductive (that is infinite capacitance, C = ∞ and zero resistance, R = 0) circuit of L henries, the voltage and current waveforms are not in-phase. Whenever a changing voltage is applied to a purely inductive coil, a “back” emf is produced by the coil due to its self-inductance. This self-inductance opposes and limits any changes to the current flowing in the coil.

The effects of this back emf is that the current cannot increase immediately through the coil in-phase with the applied voltage causing the current waveform to reach its peak or maximum value some time after that of the voltage. The result is that in a purely inductive circuit, the current always “lags” (ELI) behind the voltage by 90o (ω/2) as shown.

Purely Inductive Circuit

 

The waveforms above shows us the instantaneous voltage and instantaneous current across a purely inductive coil as a function of time. Maximum current, Im occurs at one full quarter of a cycle (90o) after the maximum (peak) value of the voltage. Here the current is shown with its negative maximum value at the start of the voltage cycle and passes through zero increasing to its positive maximum value when the voltage waveform is at its maximum value at 90o.

Thus as the voltage and current waveforms are no longer rising and falling together, but instead a phase shift of 90o (ω/2) is introduced in the coil, then the voltage and current waveforms are “out-of-phase” with each other as the voltage leads the current by 90o. Since the phase difference between the voltage waveform and the current waveform is 90o, then the phase angle resulting in cos 90o = 0.

Therefore the electrical power consumed by a pure inductor, QL is given by:

Real Power in a Pure Inductor

 

Clearly then, a pure inductor does not consume or dissipate any real or true power, but as we have both voltage and current the use of cosθ in the expression: P = IVcosθ for a pure inductor is no longer valid. The product of the current and the voltage in this case is imaginary power, commonly called “Reactive Power”, (Q) measured in voltamperes reactive, (VAr), Kilo-voltamperes reactive (KVAr), etc.

Voltamperes reactive, VAr should not be confused with watts, W which is used for real power. VAr represents the product of the volts and amperes that are 90o out-of-phase with each other. To identify the reactive average power mathematically, the sine function is used. Then the equation for the average reactive power in an inductor becomes:

Reactive Power in a Pure Inductor

 

Like real power, P, reactive power, Q also depends on voltage and current, but also the phase angle between them. It is therefore the product of the applied voltage and the component part of the current which is 90o out-of-phase with the voltage as shown.

AC Power Waveforms for a Pure Inductor

 

In the positive half of the voltage waveform between the angle of 0o and 90o, the inductor current is negative while the supply voltage is positive. Therefore, the volts and ampere product gives a negative power as a negative times a positive equals a negative. Between 90o and 180o, both current and voltage waveforms are positive in value resulting in positive power. This positive power indicates that the coil is consuming electrical energy from the supply.

In the negative half of the voltage waveform between 180o and 270o, there is a negative voltage and positive current indicating a negative power. This negative power indicates that the coil is returning the stored electrical energy back to the supply. Between 270oand 360o, both the inductors current and the supply voltage are both negative resulting in a period of positive power.

Then during one full-cycle of the voltage waveform we have two identical positive and negative pulses of power whose average value is zero so no real power is used up since the power alternately flows to and from the source. This means then that the total power taken by a pure inductor over one full-cycle is zero, so an inductors reactive power does not perform any real work.

AC Power in a Purely Capacitive Circuit

A purely capacitive (that is zero inductance, L = 0 and infinite resistance, R = ∞) circuit of C Farads, has the property of delaying changes in the voltage across it. Capacitors store electrical energy in the form of an electric field within the dielectric so a pure capacitor does not dissipate any energy but instead stores it.

In a purely capacitive circuit the voltage cannot increase in-phase with the current as it needs to “charge-up” the capacitors plates first. This causes the voltage waveform to reach its peak or maximum value some time after that of the current. The result is that in a purely capacitive circuit, the current always “leads” (ICE) the voltage by 90o (ω/2) as shown.

Purely Capacitive Circuit

 

The waveform shows us the voltage and current across a pure capacitor as a function of time. Maximum current, Im occurs a one full quarter of a cycle (90o) before the maximum (peak) value of the voltage. Here the current is shown with its positive maximum value at the start of the voltage cycle and passes through zero, decreasing to its negative maximum value when the voltage waveform is at its maximum value at 90o. The opposite phase shift to the purely inductive circuit.

Thus for a purely capacitive circuit, the phase angle θ = -90o and the equation for the average reactive power in a capacitor becomes:

Reactive Power in a Pure Capacitor

 

Where -VIsinθ is a negative sine wave. Also the symbol for capacitive reactive power is QC with the same unit of measure, the voltampere reactive (VAR) as that of the inductor. Then we can see that just like a purely inductive circuit above, a pure capacitor does not consume or dissipate any real or true power, P.

AC Power Waveforms for a Pure Capacitor

 

In the positive half of the voltage waveform between the angle of 0o and 90o, both the current and voltage waveforms are positive in value resulting in positive power being consumed. Between 90o and 180o, the capacitor current is negative and the supply voltage is still positive. Therefore, the voltampere product gives a negative power as a negative times a positive equals a negative. This negative power indicates that the coil is returning stored electrical energy back to the supply.

In the negative half of the voltage waveform between 180o and 270o, both the capacitors current and the supply voltage are negative in value resulting in a period of positive power. This period of positive power indicates that the coil is consuming electrical energy from the supply. Between 270o and 360o, there is a negative voltage and positive current indicating once again a negative power.

Then during one full-cycle of the voltage waveform the same situation exists as for the purely inductive circuit in that we have two identical positive and negative pulses of power whose average value is zero. Thus the power delivered from the source to the capacitor is exactly equal to the power returned to the source by the capacitor so no real power is used up since the power alternately flows to and from the source. This means then that the total power taken by a pure capacitor over one full-cycle is zero, so the capacitors reactive power does not perform any real work.

Electrical Power Example No2

A solenoid coil with a resistance of 30 ohms and an inductance of 200mH is connected to a 230VAC, 50Hz supply. Calculate: (a) the solenoids impedance, (b) the current consumed by the solenoid, (c) the phase angle between the current and the applied voltage, and (d) the average power consumed by the solenoid.

Data given: R = 30Ω, L = 200mH, V = 230V and ƒ = 50Hz.

(a) Impedance (Z) of the solenoid coil:

 

(b) Current (I) consumed by the solenoid coil:

 

(c) The phase angle, θ:

 

(d) Average AC power consumed by the solenoid coil:

AC Electrical Power Summary

We have seen here that in AC circuits, the voltage and current flowing in a purely passive circuit are normally out-of-phase and, as a result, they can not be used to accomplish any real work. We have also seen that in a direct current (DC) circuit, electrical power is equal to the voltage times the current, or P = VxI, but we can not calculate it in the same manner as for AC circuits as we need to take into account any phase difference.

In a purely resistive circuit, the current and voltage are both in-phase and all the electrical power is consumed by the resistance, usually as heat. As a result, none of the electrical power is returned back to the source supply or circuit.

However, in a purely inductive or a purely capacitive circuit that contains reactance, (X) the current will lead or lag the voltage by exactly 90o (the phase angle) so power is both consumed and returned back to the source with the average power calculated over one full periodic cycle being equal to zero.

The electrical power consumed by a resistance, (R) is called the true or real power and is simply obtained by multiplying the rms voltage with the rms current. The power consumed by a reactance, (X) is called the reactive power and is obtained by multiplying the voltage, current, and the sine of the phase angle between them.

The symbol for phase angle is θ (Theta) and which represents the inefficiency of the AC circuit with regards to the total reactive impedance (Z) that opposes the flow of current in the circuit.