CHAPTER THREE instrumentation. Let's consider a pure inductive load in an AC circuit. Because instantaneous power delivered to inductor is the product of the instantaneous voltage drop across it inductor time z. Instantaneous current. In other words, p is equal to i times e to the power equals zero whenever the instantaneous current or voltage is zero. Whenever the instantaneous current and voltage are both positive, above the line here, the power is positive. And as with the resistor for example, the power is also positive when the instantaneous current and voltage are both negative.
However, because the current and voltage waves are 90 degrees out of phase, there are times When one is positive while the other is negative, resulting in an equally frequent occurrence of negative instantaneous power. Notice on these curves that when ever the voltage is zero, the power is zero even if the current is at a maximum, when the current is zero even though the voltage is at a maximum in this case maximum positive the power is also zero. As here when the current is zero, even though the voltage is at a maximum negative the power is still zero. And here when the current is at a maximum, the voltage is at zero. So the output voltage or output power or power dissipated or power dissipation is zero also. Now let's consider a pure capacity.
Load in an AC circuit. As you might have guessed, the same unusual power wave that we saw with the simple inductor circuit is present in this simple capacitor circuit. As with a simple inductor, there is a 90 degree phase shift between the voltage and current but this time the current leads the voltage and results in a power wave that alternates between positive and negative. This means that a capacitor does not dissipate power. As it reacts against the change in voltage it merely absorbs and releases power. Alternatively, notice that when the current is at a maximum, the voltage is at zero, so the power output is zero.
When the current is at zero and the voltage of the maximum negative as here, then the power is still zero, as it is here when the current is zero and the voltage is at a maximum positive value. Power is still zero. And lastly here the current is at a maximum, but the voltage is at zero. So the power is zero also. Clearly it can be seen that at times power is negative, but what does negative power mean? It means that in the case of the inductor, after having built up a magnetic field while the current is flowing into the inductor, it is now releasing power back into the circuit, while a positive power means that it is absorbing power from the circuit as it rebuilds the magnetic field.
Since the positive and negative power cycles are equal in magnitude and duration over time, the inductor releases just as much power back into the circuit as it absorbs it over a span of a complete cycle. What this means in practical sense is That the reactance of an inductor dissipates a net energy of zero. quite unlike resistance of a resistor which dissipates energy in the form of heat, mind you, this is for a perfect inductor only, which has no resistance even in the wires. This is the same thing that happens in a capacitor circuit on the power flow in and out is due to the buildup of an electric static field while the current was flowing into the capacitor, then releasing power back to the circuit during the negative power cycle. Let's take a closer look at an AC sinusoidal current and voltage over time that are out of phase.
The instantaneous power, which we will call p subscript I plotted here in blue is calculated by multiplying the instantaneous voltage, the subscripts plotted in green times the instantaneous current i subscript I plotted in red. The instantaneous voltage is mathematically given by VM sine omega t, where VM is the peak or the maximum voltage and omega is the angular velocity of the phasor where omega t is a particular angle at a particular time, the instantaneous current is given by I am sine omega t minus phi where I am is the peak or maximum current and phi is the displacement angle from the voltage phasor. So, the instantaneous power is mathematically given by the product of these two which is VM, I am sine omega t times sine omega t minus phi, which is as I said the instantaneous voltage times instantaneous current notice The power in is sometimes positive and it is sometimes negative.
It is positive when both V and I are either both positive or both negative and it is negative when one is positive and the other is negative. The average power is a constant over time. And it is given by this formula, which is VM times I am over two times cosine of phi. This is derived through a trigonometric manipulation which will follow and, but the result is more important here than the actual proof but I will go through that in the following slides. The average power is an important value in the world of commerce as it is this term that is equated to the consumption of energy It is the metered quantity that is used for billing of power. Okay, I'm very quickly going to go through the proof, I'm not going to spend a lot of time in the details, you can go back and and figure out the details yourself.
But I'm going to start with the formula that we have for calculating instantaneous power, which is VM, I am sine omega t times sine omega t minus phi. Now, what you see here in yellow is an identity, a trigonometric identity that we'll use to reduce this equation to something that we will we want to be able to work with and that is average power. So we're going to use this identity to reduce the last part of that term sine omega t minus phi, and that's going to give us this resultant formula. However, we still need to use a couple of identities. To reduce this, this equation even further, so we want to reduce sine squared omega t, and we'll use this identity to reduce sine squared omega t. And we'll use this identity to reduce the product of sine omega t cosine omega t. And that will give us this equation here, which is the instantaneous power is made up of two terms, V subscript m i subscript M over two times cosine phi minus v m, I am over two cosine the quantity two times omega t plus phi.
Now, we're looking for average power. So, the first term remains the VMI M over two cosine phi will remain However, this term because it is a omega t is in evolved in it over time coasts to omega t doesn't matter if it's phase shifted or not, will average out to zero. So, this term on the average over time goes to zero leaving us with just this term or this equation for the average power consumption p average is equal to v m m or two cosine of phi. We now know that the average power is given by this equation here, which can be rewritten by putting VM all over root two and I am all over root two, because two can be split into root two times root two. This actually means that if we remember that the RMS value for voltage is VM over root two and the RMS value for current i is i am all over root two.
So, we can say that the average power can be found by calculating the RMS voltage by the rms current times the phase angle between the two. And if we plot the phasers they would look something like this. p average is as find as a phaser itself, p average which is given by the RMS times high RMS times cosine of the angle between the two of them. And it represents real power in walks the actual work done by an electric current or actual energy consumed by the load to create for example he liked or Motion on a rectangular coordinate plane it is a phasor plotted along the horizontal or real axis. electrical systems normally have inductors and capacitors which are referred to as reactive components. Ideas ideal reactive components do not dissipate any energy but they draw currents and create voltage drops, which makes the impression that they are actually dissipating energy when they're not.
This is called reactive power. Its average value over a complete AC cycle is zero because the phase shift between the voltage and current it doesn't contribute to the net transfer of energy but circulates back and forth between the source and the load and places a heavier load on the utility reactive power is measured in volt amps reactive or bars and is calculated by the RMS voltage time CRMs current times the angle between them the summing of real power watts and reactive power bars and that is the the phaser Some are vector some produces a third phaser which we call apparent power as VA which gives us a better indication of the load on utility and is calculated simply by multiplying the RMS voltage times the rms current. This becomes more significant in industrial loads. The angle between the real power and the apparent power shown here is phi is called the power factor angle.
Cosine of this angle is known as the power factor and it is a number that is less than one, it goes from zero to one. It is important to note that phi is the same angle that exists between the voltage and the current. We have already derived how to calculate the average power from the RMS voltage and current, we now can define reactive power which is designated q average, which is measured in volt amperes reactive or bars is calculated by multiplying the RMS voltage times the rms current times the sine of the angle between them. And apparent power, which is the vector summing of the real power watts and the reactive power bars which is given by the RMS times I RMS and measured in VA or volt amps. Here are some examples plotted in real time, the voltage is in blue, the current is in red and the power is in green.
The voltage leads the current by 90 degrees giving a power factor of zero. The average power of course is zero and this type of load would be an inductor. In this case, the voltage is still blue and the Current is red and the instantaneous power is given by green, the power factor is still zero, this time the voltage leads the current by 90 degrees. And this is an example of a pure capacitive load. In this example, the voltage leads the current by 60 degrees, giving us a power factor of point five. The average power given by the M I am over two cosine theta would work out to one times one over two times point five, which works out 2.25 as indicated on the graph In this example the voltage and the current are in phase, giving us a power factor of one.
Again, the same formula for the average power is given by the M i m over two times coasts cosine or the power factor which is one, which gives us one times one over two times one, which is equal to point five. Now let's have a look at some of the instruments that measure power and energy. The electric dynamometer type meter has two fixed coils that are used to produce a magnetic field. The fixed coils are connected in series and a position coaxial that is in line with a space between them. These fix coils are connected in series with the load therefore the load current will flow through these coils which are called the current coil. It also has two moveable coils that are suspended in the field produced by the two fixed coils.
The two movable coils are also positioned coaxial and are connected in series. These movable coils are connected across the load and therefore measure the potential drop across the load these coils are called the potential coil. The main shaft on which the movable coils are mounted is restrained by spiral springs that restore the pointer to zero when no current is flowing through. The design of these components are such that the movement will measure the average power which is equal to the volts RMS times the I RMS to the load and the at the coast times the cosine of the angle between them. In an electronic or digital version of this electric dynamometer, two springs and coils are replaced by transducers, one for each the current and the voltage and each have an analog to digital converter, which feed into a digital processing unit, which drives a digital readout.
And that digital readout will make it will read out in the average power, which again measures the RMS voltage times the rms current times the cosine of the angle between them. The primary advantage of this these meters is that they can be used to measure alternating as well as direct current. Remember that AC RMS is equivalent to DC This slide shows the dynamometer watt meter connected in a circuit measuring the power delivered to a load. The current through the fixed coil is proportional to the load current and produces a magnetic field that reacts with the magnetic field of the potential coils that creates a torque that counteracts with the restraining springs of the meter movement that results in a reading of the average power delivered to the load. In an electronic version of this, the springs and coils of course are replaced by transducers and analog to do converters that feed into a digital processing unit that drives a digital readout.
Regardless of the type of meter, all watt meters have basically three connections current to the load is made to flow through terminal number one or into terminal number one and out terminal number two the load the voltage across the load is connected to terminals one number one and number three. Sometimes what meters have a fourth terminal, terminal number one and terminal number two are connected to the current element of the watt meter, while current or terminal number three and terminal number four connected to the voltage coil of the element of the watt meter. terminals one and four are then jumpered together. The two terminals the watt meter are marked indicate the polarity of the elements that will give of deflection of the meter for power flow into the load. terminals number one and terminals number four are usually jumpered together internally hence, terminal number four is not always showing and not always available on a watt meter.
Power is equal to the rate at which electrical energy is transferred to a circuit load. And that is given by the formula p is equal to v times AI and power is measured in watts. Energy is the amount of electrical energy transferred to the circuit load over time and energy is equal to power times time in watt hours. And is given by the formula e is equal to v times i times T and watt hours, which is the area underneath the curve of the power graph on a power versus time chart. The energy meter, measures power and records or register it registers it over time providing and measurement of energy. Just about every house has one.
Shown here is the older mechanical type kilowatt hour meter which is now being replaced by the more modern solid state, digital or electronic type meter. The older type watt hour meter has approximately eight major components the cover Which is basically a protection against the environment from moisture, rain, dust, rodents or anything else and it's usually glass and and it screws on to the base of the meter. The register is the recording device of the watt hour meter it keeps an accurate record of the total amount of energy that has passed through it over a period of time. The register is connected to the electrical components of the meter through a series of gears that are turned by the rotation of the disk. It is designed to show revolutions in kilowatt hours or kWh. The rotor is a part is that part of the meter that rotates as power is being used.
It consists of a disc mounted on a shaft. The disc is a round flat piece of aluminum that acts as a conductor. There are markings on the upper surface of the disk that are used for calibrating the meter. The shaft is held vertically in place in the meter by bearings. Depending on the vintage of the meter, the bearings may be of jewels or now a little bit more modern type, even though it's an old school meter, would more commonly they would be held in position by a cushion of magnetic bearings. The returning magnets are used to accomplish exactly what their name is standing for it retards or slows the operation of the meter specifically, they don't slowly entire operation, but only that of the rotating disc.
The magnets regulate the speed of the disk and prevent it from coasting or accelerating greater than that provided by the driving torque. Which of course is equivalent to the power consumption. The electromagnets the potential coils and current coils to potential coil. The main purpose of the potential coil is to convert voltage through the meter into a magnetic field proportional to the circuit voltage. The coil is extremely inductive and consists of many turns of insulated wire while not a high dielectric spool mounted on laminated silicon steel core. laminated steel concentrates the flux field as well as reducing both circulating currents and flexes in the laminates themselves.
The current coil and core The primary purpose of the current coil and core is to convert the circuit current into metal magnetic field proportional to the load current, the coil is connected in series with the load. Using a small number of large wire turns to carry the load current it produces a flux field in phase with the current. This requires the coils to have a large cross section, low inductance and low resistance. laminated silicon steel cores are used to concentrate the fluxes and reduce the losses. The coils are coated with an epoxy resin to insulate them electrically and mechanically from the iron core. When we talk about meter elements, this is what's meant a current coil and a potential coil make up one element of a meter.
A current coil by itself is considered a half element. And a potential coil by itself is considered a half element. About more of this later as we look at standard type kilowatt hour meters and their connections to the systems, the elements of the kilowatt hour meter are designed such that the rotor, disk and shaft rotate at a speed that is proportional to the power drawn by the load. The frame is where all of the aforementioned components are mounted. And the base is where the frame is mounted and the cover is screwed to the exterior which then provides the connections to the line and the load. The speed of the disk or its angular velocity is proportional to the power being consumed by the metered load.
The disk angular velocity or speed is proportional to the RMS voltage vector in the potential coils times the rms current factor in the current coils, times the cosine of the angle between them, which is equivalent to the power represented by those factors. A single phase watt hour meter is essentially an induction motor whose speed is directly proportional to the voltage applied to it and the amount of current flowing through it. The phase displacement of the current as well as the magnitude of the current is automatically taken into account by the meter. In other words, the power factor influences the speed and the moving element disk rotates with a speed proportional to real power. The register is simply a means of registering the revolutions and by proper gearing is arranged to read directly in kilowatt hours. In some cases, the meter reading must be multiple by a factor called a registered constant or a meter multiplier to obtain the total and accurate kilowatt hours.
The aluminum disk acts as a squirrel cage rotor torque being produced as a result of eddy currents induced in it by the potential and current coils. Normally, there is very little friction present in the meters, and if no additional retarding force other than friction were placed in the meter, the rotating element would travel at a relatively high speed. The necessary retarding action is provided by a magnetic brake consisting of a permanent magnet operating on the aluminum disc. A detent or a ratchet is sometimes attached to meters to prevent rotation in the reverse direction when it is desired not to read SR reverse power flow. This usually consists of a collar having notches or pins, which is placed on the disc shaft and appall attached to some fixed part of the meter, which engages with the notches or pins upon reverse rotation, but slides easily over them in the forward direction.
The kh of kilowatt hour meter is called the one hour constant, and equals the number of watt hours per one turn of the disk of a electromechanical meter. The kh wat our constant of a meter is indicated on the face of the meter as shown here. Also, the manufacturer places a marker on the disk that makes it easy to see and count the disk rotations. Stated another way The meter constant is the number of watt hours Wyland on the register for one revolution of the desk. The amount of energy represented by one revolution of the disk is denoted by the symbol kh which is given in units of watt hours per revolution. The value that is very common is 7.2.
Using the value of k h one can determine the power consumption at any given time by timing the disk rotation with that stopwatch and the power is equal to 303,600 times the K h all over T, where T is in seconds, and p is in power in watts. For example, if k h is equal to 7.2 as above, and one revolution one revolution took place in 14.4 seconds, the power is 1800 watts or 1.8 kilowatts. Here are some values that are common to the General Electric Westinghouse sang them sang gamma and Dunkin type meters. Most meters can record up to 10,000 kilowatt hours. This is accomplished through a series of four dials that read from right to left in significance. Each significant dial is geared to the previous significant dial in a ratio of one to 10.
There are some five dial meters that are read in a similar fashion. There is usually a test style that is geared to Returning 10 times faster than the least significant dial making making it easier to detect meter movement and fractions of the least significant dial. Reading the dials can be tricky as dial rotations alternate from clockwise to counterclockwise to clockwise and counterclockwise etc. As you progress upscale. Here are some examples. When determining the total consumption from dial readings, it must be done from right to left.
If the pointer is between two numbers, the lower number is the correct reading. For example, this register is reading 1799. The reading of this register would be one 5961 and the reading of this would be 50497. Up to this point we have only described the functionality of mechanical watt hour meters. However today the majority of meters are electronic or digital type meters, which vary in functionality but basically, they replace the current and potential coils with transducers, one for each the current and the potential and they feed in To add noise or analog to digital converters, again one for each the current and the voltage and that feeds into a digital processing unit that has several functions and it depends on the manufacturer and the when the watt meter was produced just what those functions are. However, one of which is a register, that takes the reading and it is an EEPROM.
That's electrically erasable programmable read only memory, which is a type of non volatile memory that is used to store the data and saves it in the case the power goes off. It doesn't lose the reading on the kilowatt hour meter, just as the old fashioned dial type never lost the register reading as well. Digital meters use a multi segmented indicator instead of dials to provide us with a reading to manually display the energy consumption as well as other factors such as demand. It will also as it cycles through the various readings, check all the segments of the digital display wants to make sure they are all they are all working. And for test purposes, the manufacturers have provided an electronic disk emulator as you'll see in the next slide. Much the same as the case of each of the mechanical meter, the electronic meters will also have a case of each and sometimes their designated case of AI because sometimes these meters put a pulse output and it's called a case of AI which is has the same relative meaning as a case of age.
Further auxiliary units can be added such as wireless and remote readings for signal and signaling devices. Electronic metering today has become much more computerized and this functionality is paramount to a medium to large size utility. This is a picture actually it's going to be a video of the kilowatt hour meter that was placed on my house by hydro one and I just wanted to show you what the display looks like and how the disk emulator works. It works much the same as a rotating disc. The three dots are repetitive and you can time the repetitions and then use the case of each of the meter to determine the load. And as you can see the case of each is 7.2 on this meeting And it is printed on the faceplate of the meter.
Energy meters regardless of their makeup, whether they're mechanical or electronic, are considered to be made up of meter elements. And one element is made up of voltage half element, which is either potential coils in the old school meters or voltage transducers plus an A to D converter in a new electronic meters. And the other half of the element is the current element, which is made up of a current coil or a current transducer and A to D converter. Together they make up a single element of a meter that can be used to measure and record power and energy consumption. by a single phase load and from now on when we start to consider these energy meters, we're going to disregard the fact that they are either electronic or mechanical. And we're only going to be talking about measuring devices with elements.
This is the end of chapter three.