AC Power Analysis: Basics

Instantaneous Power

The formula for power is: P=V⋅I, Power= Voltage*Current. We call that, instantaneous power. Even when V and I are changing in time, P=V⋅I applies for each instant. It matters not if the V or I changes are sinusoidal, what their frequency is, or even if they are aperiodic. P=V⋅I, is something you can use always. There are no exceptions that I can think of.

You don’t need Ohm’s Law to use V⋅I, because V refers to the voltage at one-point relative to ground, and I refers to the current flowing past that point. But suppose you have a resistor R, and you want to calculate the power dissipated by it. Using Ohm’s Law, the voltage drop across R is V=I⋅R. In this case, V is the voltage drop between two points in a circuit. The power dissipated is (V)⋅I=(I⋅R)⋅I=I²⋅R. If you have a capacitance C instead of R, then you need to use the differential equation I=C dV/dt. If you have an inductance L, then you need to use the differential equation V=L dI/dt.

Sinusoidal Power

Let’s jump directly to sinusoidal waveforms such as we have in AC circuits. Let V and I be sinusoids of the form cos(ωt+ϕ). What happens to the instantaneous power? Well, P=V⋅I still applies, but now P varies with time, and how it varies depends on the phase angle of V relative to the phase angle of I. If the relative phase is zero, we say that voltage and current are “in phase”. That is depicted below.

Note that the frequency of V⋅I is twice the frequency of V or I, and that the value of V⋅I is zero twice in the cycle, and that the average of V⋅I is not zero.

Suppose V and I are not in phase? In this picture, I (dotted line) is shifted by an angle 90 degrees relative to V (blue). V⋅I is shown as red-green shaded areas; red means power flowing right (+) and green means minus power flowing left (-). Note that the areas of red and green are equal in this case, so they cancel. Power flows just as much to the left as to the right, so the net energy over the entire cycle is zero. What should we call this peculiar state? Referring to the entire cycle, we call it pure imaginary power (also called reactive power, also called VARs). Don’t be fooled, even in this case instantaneous V⋅I remains real.

Below, I is shifted only 45 degrees relative to V. We see that V⋅I is predominantly red, but green for part of the cycle.

We can now generalize to all possible phase shifts. We discuss averages for the entire cycle.

• Phase 0 is pure real power, phase 180 is pure real power flowing in the opposite direction.
• Phase 90 is pure imaginary power, phase 270 is pure imaginary power flowing in the opposite direction.
• Any other phase is a linear combination of nonzero real power and nonzero imaginary power.

Complex Power

In 1893, Charles Proteus Steinmetz published a paper that explained the great advantages of complex AC analysis. The other electrical geniuses of the day (including Nikola Tesla) were all using tedious integral calculus and expressions using definite integrals of cos⁡(ωt+ϕ), or the Euler form e^−jωt+ϕ. Steinmetz left them in his dust because he recognized the combination of fortuitous luck with coincidences of mathematics that were relatively obscure at the time. Namely:

• Manufacturers were already making AC generators in 1893 that generated sinusoidal voltages.
• The sinusoid is the only mathematical function that has the property that differentiation and integration return a function of the same form but shifted 90 degrees.

Steinmetz found that by restricting his equations to an integer number of whole cycles, and by replacing real quantities by complex ones, we obtain quasi-static equations that are hugely simplified relative to integral calculus. Simpler how? 

• DC P=V⋅I becomes AC S̅=V̅⋅I̅, where S̅ is complex power, usually written as S̅=P+jQ where P is real power and Q is imaginary power. (Actually, it should be S̅=V̅⋅I̅* but, I’m ignoring the sign of Q.)
• DC Ohm’s Law V=I⋅R becomes AC V̅=I̅⋅Z̅ where Z̅ is the complex impedance. Z̅ includes resistance, inductance, and capacitance.
• Differential equation terms in DC (like C dV/dt and L dI/dt ) become algebraic in AC.
• Series, parallel, Kirchhoff’s Laws, mesh analysis, matrix analysis: essentially all the tools and methods of DC circuit analysis become directly applicable to AC if we just use complex and whole cycles.

Next, think once again of the pictures from above with the red-green areas depicting V*I. Instead of time-varying instantaneous V*I, we will focus on just the whole cycle averages, P (as measured by an AC Watt meter) and Q (as measured by an AC VARs meter). P and Q will be constant in time, but they will vary as we change the phase shift. The meter readings versus ϕ are shown in the table.

Maglev Suspension Systems

Shanghai Transrapid

In previous article, we discussed the Maglev technology and principle on which it is working. Here, we will discuss about the different technologies or suspension system used to levitate the train, although the basic concept remains same. Even though Maglev looks like new, futuristic mode of transportation, concept was first proposed about a century ago by Hermann Kemper. First commercial Maglev developed by German company Transrapid International was tested in Shanghai, China, in 2002. Shanghai Transrapid is functional since December 2003, with average speed of 430 Km/h.

Several other countries have started working on Maglev projects. Enough research is going on to improve the technology to levitate the train. Engineers are testing there prototypes of train for commercial use.

Electromagnetic Suspension System (EMS):

Electromagnetic Suspension  (EMS) System

In Germany, the company named, Transrapid International had developed Electromagnetic suspension (EMS) System. In this system, the bottom of the train wraps around a steel guideway. Electromagnets attached to train's undercarriage are directed up towards the guideway, which levitates the train about 1 centimetre above the guideway. The system keeps the train levitated even when it is not moving. The system is provided with emergency battery backup in case of power failure. Other guidance magnets embedded in the train's body keep it stable during the travel. The company also has train in commercial use implementing EMS, known as Transrapid and can reach the speed of 500 Km/h with people onboard.

Electrodynamic Suspension System (EDS):

Electrodynamic Suspension  (EDS) System

US & Japanese engineers are working on a competing version of Maglev trains SCMAGLEV, that uses Electrodynamic Suspension (EDS) System. The key difference is, Japanese trains uses supercooled, superconducting electromagnets, This kind of electromagnets can conduct electricity even after the power supply is shut off. The trains using this technology can levitate nearly 10 centimetres above the guideway. But, the train using EDS, must roll on the rubber tires until it reaches a lift off speed about 100 Km/h. Also, wheels have advantages during power failure of system to slow down the train.

EDS saves the energy by cooling coils, however, also the cryogenic system use to cool the coils can be expensive. Also, passengers with pacemakers would have to be shielded from magnetic fields generated by the superconducting electromagnets.

Inductrack:

Inductrack

Inductrack is a newer type of EDS that used permanent room temperature magnets to produce the magnetic field instead of super-cooled superconducting electromagnets. Inductrack also uses auxiliary wheels to accelerate until it begins to levitate. The permanent magnets made from neodymium- iron boron, which generates intense magnetic field are used. Magnets are arranged in a Halbach array so that, the intensity of magnetic field concentrates above the array to create enough levitating force.

Introduction to MagLev - Magnetic Levitation for Transportation

Maglev Train

In last discussion, we discussed about the electromagnetic induction and application of induced eddy current in induction cooktop. Here, we will discuss about Electromagnets application in transportation field. Electromagnets are temporary magnets produced due to flow of electric current through conductor coil. It also has poles which can attract or repel depending upon the polarity. Repulsion force of the electromagnet used for levitation of objects. This electromagnetic levitation is about to bring the revolution in the transportation.

Maglev train is new way of transportation which uses magnetic levitation to transport the vehicle without touching ground. Magnetic levitation is used to create both lift and propulsion. Maglev moves more smoothly and quietly. Due to absence of friction power consumption for propulsion is relatively less and thus it holds some of high speed records for rail transportation.

Maglev Propulsion:

Maglev Guideway

Maglev is short for Magnetic Levitation. Unlike conventional train maglev do not have engine to pull cart along the track. Maglev floats over guideway using basic principle of magnetic repulsion and same used to propell the train. The magnetised coil running along the track, called guideway, repels the large magnets on the train's undercarriage, allowing the train to levitate 1 to 10 centimeters above the guideway. Once the train is levitated, power is supplied to coils within guideway walls to create a unique pattern of magnetic field that pull and push the train along the guideway. The electric current supplied to the coils in the guideway walls is alternating to change polarity of magnetized coils. This change in polarity causes magnetic field in front of train to pull the vehicle forward, while the magnetic field behind the train push the train. The propulsion is same as linear motor.

Linear Motor Propulsion for Maglev

Lack of friction and trains aerodynamic design allows the train, to reach ground speed of more than 500 km/h. In comparison, Boeing 777 can reach top speed of 905 Km/h. Developers say that Maglev will eventually link cities that are up to 1000 miles apart.

Germany and Japan both are developing maglev train technology. Although, based on similar concept, both have distinct differences in technology used to create lift and thrust. We will discuss more about it in next section.

How The Maglev Works

Eddy Current, One that blows Transformers, Cooks Food in Kitchen

Eddy currents or Foucault currents are circular currents induced within the solid block of conductor due to time-varying magnetic field around it. Eddy currents take time to build up and can persists for long time (Actually, it is analogous to turbulence in the water or swirls known as eddies). It is undesirable effect commonly observed in the electric transformer, causes heating of the core. Overheating of the transformer may lead to failure of the transformer.

Eddy Current in Unlaminated & Laminated Core

According to Faraday’s law of induction, the high frequency moving/ varying magnetic field possess the ability to induce current in the conductor in its field. The coil wound around the core of the transformer generates the electromagnetic field which passes through core. The electrons in the core experience the Lorentz force. Electrons start to move in direction such that the magnetic field produced due to their motion will oppose the magnetic field which cause them to move. According to Lenz’s law this induced current moves circular loop (swirls) within the solid core and keep generating heat in the core. Sometimes transformer fails due to excessive heat generated. That’s why the core of the transformer is made of laminated plates stacked together instead of a solid metal core or poorly conductive material (ferrite core). Core made of laminated plates stacked together minimises the eddy currents.

Eddy Currents Formation

Eddy currents are not always undesirable. Eddy current used for electromagnetic braking of locomotives. Since there is no contact with brake shoe and drums no mechanical wear. Eddy currents are used for non- destructive testing of materials. The phenomenon is fundamental of some of sensors (Proximity sensor).The heating effect is used for electric heating of ores or metals to melt them in foundry. Induction heating is innovative and clever way to cook the food.

Induction Cook-top Working Principle

The induction cooktop uses same phenomenon which causes transformer core to heat. Induction cooktop has electric coil placed below, which produces magnetic field. The cookware to be placed on top must be made up of conductive or ferromagnetic material and with flat bottom so as to make more contact with magnetic field. The current in the coil induces the small eddy currents of about one volt in the bottom of the cookware causes to heat the bottom of pot. These currents come equipped with their own magnetic fields result is a lot of swirling, churning, pushing and pulling of the molecules within the material in cookware.

Other than commercial and industrial use eddy currents has some attractive experiments to demonstrate like levitation and repulsion of metallic objects. When magnet is allowed to fall through copper tube, the speed of a magnet considerably reduces. This is because eddy currents are produced in the copper tube due to motion of magnetic field associated with magnet. The electromagnetic force produces by these eddy currents in the copper tube opposes the magnet field of magnet, hence the motion of magnet. Experiment explains the electromagnetic braking fundamentals.