Chapter Five power flow in AC circuits. Let's consider for a moment a pure inductive load in an AC circuit. Because instantaneous power is the product of the instantaneous voltage and the instantaneous current, that is P equals i times E, the power equals zero whenever the instantaneous current or voltage is zero. Whenever the instantaneous current and voltage are both positive, above the line, the power is positive. As with resistors the power is also positive when the instantaneous current and voltages are both negative below the line. 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 equally frequent occurrences of negative instantaneous power.

Notice on the graph here that the power goes from zero to a maximum back through zero up to a positive maximum, back to zero down to a negative maximum as the current and the voltages swing through their positive and negative crests. You'll notice that when the current slope is zero at the crest of its wave, the voltage is just crossing the zero line so voltage is zero. Hence, power is zero. And indeed, that's what the green line shows here. Again, where the current slope is is equal to a maximum. In other words, its, its slope is at a maximum slope, where is it where it's crossing zero, voltage is a maximum, but because current is zero at this point, so is the power of zero.

And zero point for the power is where the current slope may be a maximum and the voltage may be a maximum, but the current slope or the current is zero, so the power is zero. And where the current slope is zero, and the voltage is zero, then of course we have zero power. Now let's consider a pure capacitive load in an AC circuit. As you might have guessed, the same Unusual power wave that we saw with a simple inductor circuit is present in a simple capacitor circuit as well. As with a simple inductor circuit, the 90 degree phase shift between the voltage and the current results in a power wave that alternates equally between positive and negative. This means the capacitor does not dissipate power, as it reacts against the changes in voltage, it merely absorbs and releases power.

Alternatively, and a couple of highlights to look at on this graph, when the current slope is zero, when the current is maximum, voltage at this point happens to be zero so the power would be zero. At this point, the current slope is a maximum, but it is also the current is equal to zero. Even though the voltage is maximum, the power input will be zero. And at this point the current slope is a maximum, but it is zero and even though the voltage is at a maximum, the power is zero and at this point, current slope is equal to zero, and the voltage is equal to zero, so the power would be zero at this point as well. Clearly it can be seen that at times the power is negative, but what does negative power mean? It means that the inductor after having built up a magnetic field while the current was flowing into the inductor is now releasing power back into the circuit while a positive power means that it is absorbing power from the circuit as it builds its 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 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, while unlike resistant, the result of a resistor, which dissipates energy in the form of heat, mind you, this is for a perfect inductor only, which has no wire resistance at all. This is the same thing that happens with a capacitive circuit only the power flow in and out is due to a build up of electric static field while the current was flowing into the capacitor, then releasing power back into the circuit during the negative power cycle. For a moment, let's talk about instantaneous power. What you see in front of you are or is two, basically sinusoidal waves one representing the current, which is in red and one representing the voltage which is in black.

Now the voltage starts at zero goes up to a maximum comes back through zero goes to a minus maximum, and so on and so on. And it's a nice uniform sinusoidal wave, the current depending on the impedance of the circuit, of course, will either be shifted left or right, a certain amount. So, we will we would either say that the current is going to be leading or lagging the voltage, but it's represented by this wave shape on the on a time curve or on a time graph. Now if if we indeed want to calculate What the instantaneous power at any one moment is, what we would do is take a point on the graph at a particular time, and we would say the instantaneous power is given by the voltage at that point in the current. Let's have a look at that. We'll take a slice of time right here.

And we will say that the power at that particular time is being dissipated is given by the voltage, the instantaneous voltage, and the incident times the instantaneous current. And that can either be a positive or a negative number, or it could be zero depending as we seen in our examples in the previous slides, were on the graph that you're going to actually take your instantaneous power. So that's not a big deal. I mean, if you want to get instantaneous power, however, in power circuitry and power systems instantaneous power isn't always a thing that we're after what we are usually looking for is average power. In other words, what's the average power consumption of a hot water heater? What's the average power consumption of an industrial motor.

So the average power is a thing that becomes significant. It's a thing that we would like to deal with, on a very simple basis. So let's take a current in a voltage and apply it to a circuit. And that's this pursuit. The circuit is made up of some reactive material, and it's made up of some resistant material. So indeed, the current could be represented by the red sinusoidal curve there and the voltage could be represented by the green a sinusoidal curve there, and you can see that the voltage is following the current by a lag of a certain amount.

And the power as we've seen in our previous slides is given by at least the instantaneous value is given by the instantaneous voltage times the instantaneous current and at times it can be positive and at times it can be negative. The voltage sinusoidal wave can be described in mathematical terms or trigonometric terms or algebraic terms whenever you want to call it the voltage can be the maximum voltage where BM is the maximum voltage times sine omega t, where t is sometime in along the time curve on The bottom in seconds. And a term omega is actually related to the system frequency. We call it angular velocity, but I don't want to get hung up on terminology is just that it's related to the frequency of the system. And it's actually a term that is in degrees per second. So if we multiply it by t seconds, we're left with a an angle.

So omega t, changes as we move along the T line, generating an angle that goes sinusoidal from positive maximum to negative maximum and crosses zero. So we can describe the voltage in terms of the maximum point of that voltage times sine omega t. The current can also be described the same way as a maximum current times sine omega t. However, it is a head of the voltage by a certain phase. shift or a phase angle, and we'll call that phi for now. So it's a phi is a set angle. So we have to include that in our formula so that we can now place the current curve out somewhere along the T line with respect to the voltage. If we multiply the left hand terms together, P, that would give us the instantaneous power for the circuit.

And that right hand side would become the voltage maximum times the current maximum times sine omega t times sine omega t minus phi. Now we're getting a little bit complicated in formulas and I'm not that interested in the getting to the final results as I am with the final results. What I want to do is end up with a formula that will calculate how much power really Power is consumed by my circuit. And the amount of real power that is consumed by the circuit is shown in purple. In this diagram here, everything above the positive line for power is power consumed. All the yellow stuff below the zero line is what we call reactive power.

It's where the reactive components are absorbing the power into their either fields of some sort, whether it's a magnetic field or it's a electrostatic field in the case of a capacitor regardless, it pulls the the the power in, but then it releases it back to the circuit. So we have a net effect of zero power or dealing with reactive, we are mostly in interested in positive power consumption. This power consumption is we call it the average power consumption of an AC circuit and it would be the power that would be supplying, say a resistive load like baseboard heaters, that could be a hot water tank, or it could be a motor that is has no reactive, or it's just we're just want to calculate the amount of power that the motor is delivering mechanically in the form of mechanical power or some of the heat losses.

Regardless, we are interested in in the real power consumption. So what we're interested in is, what is the average of that purple section. That's there, we need a formula that would say, Okay, this is the power the average power that we're consuming for that particular load. Now, the way we get that p average is To take that term that we've just developed there be instantaneous power, and we want to manipulate it so that we kind of level off the peaks and fill in the valleys. And somebody has done that hard work for us. I'm not going to go through it, the average power through a trigonometric and integration workings mathematically, you can get that p average, which is real power consumption with this formula.

In other words, the P i formula boils down to p average, equals the voltage maximum times the current maximum, all over two times cosine of the angle between the current and the voltage. So we have a formula for calculating real power consumption in a circuit, which is we designated as p average. It's the average Have the purple humps in that diagram that we have there. And that is actually a measure of the real power consumed by the circuit, whether that's heat light mechanical energy or whatever. It's given by the formula p average is equal to the i m over two cosine phi. And if we were to draw a right angle triangle whose contained angle is phi, then we would have a hotpot noose for now would be VM I am over to because VM I am over to coast CLI is what are the averages so the hypothesis would be given by that term VM I am over to the quadrature or the perpendicular side of the triangle to average just happens to be what we call Q and Q average is given trigonometric Lee speaking by that triangle, VM, I am over two sine phi.

Now it just so happens that this Q average happens to be the average reactive power that's in yellow humps on our diagram. And that is the power that's pulled out of the system but then pumped back into the system with a net exchange of zero real power. The p average. We've already started discussing it in terms of real power and it is real power and it is represented in watts and designated by the letter p. q average is what we call reactive power. And it's usually designated by Q and its measured in bars. The term VM I am all over two is what we call a parent power, or and it's designated as S. And it's measured in the VA, sometimes kV, a or MVA in bigger circuits, but it is apparent power and it's just a product of the voltage and the current.

Now from our previous slide, we have what we call the power triangle made up of real power with a quadrature of reactive power. That gives us a measure of our what we call apparent power. The Fly angle is known as the power factor angle. And cosine of that angle cosine phi is known as the power factor. So we have a way of calculating what the real power, real average power consumed in a circuit is. It's v m i m over two cosine phi, or phi is the angle between the voltage and the current.

And VM and Iam are the maximum values of the sinusoidal waves that we're using to describe the voltage and the current, we can take VMI M over two and split it out into two terms and still not change the meaning of the equation, we could say that the p average is equal to VM over root two times i m over root two times cosine theta. But you remember that the RMS value of voltage is equal to VM over root two and the RMS value for i is i am over two. So the average power consumption real power can assumption in this circuit is if you measure the voltage in RMS and the current in RMS terms and multiply it by the phase angle the cosine of the phase angle between the two of them, you come up with p average. Similarly, the Q average can be calculated using the same v rms times I RMS, this time, sine of the angle between them.

And the apparent power VA can be you calculated using just v rms times the I RMS. Let's look at what is sometimes referred to as an electric pendulum. We're going to look at applying a DC voltage. And as soon as we have a look at that we'll swing over to how it reacts with AC but it helps to look at how parallel LC circuit works in terms have applied DC voltages or DC voltage. capacitors store energy in the form of electric fields and electrically manifest that stored energy. As a book potential static voltage inductors store energy in the form of a magnetic field and electrically manifests that stored energy as kinetic motion of electrons, current.

Capacitors and inductors are flip sides of the same reactive coin storing and releasing energy and complimentary modes. When these two types of reactive components are directly connected together, they're complimentary tendencies tend to store energy and will produce an unusual result. If we assume that both components are subject to an application of voltage, say from a battery as you see here, the capacitor will very quickly charge and the inductor will oppose the change in current leaving the the capacitor in a charge state and the inductor building its magnetic field and building the current flow through it. When the supply voltage is removed, by opening the switch, the capacitor will begin to discharge, its voltage decreasing. Meanwhile, the inductor will begin to build up a charge in the form of a magnetic field and the current increases in the circuit. The inductor still charging will keep electrons flowing in the circuit until the capacitor has been completely just discharged, leaving zero voltage across it.

And I've indicated the voltage E on the capacitor in blue and the current through the inductor by the red dotted line. So At this point in time, the capacitor is fully discharged, but there is current flowing maximum maximum Li through the inductor. The inductor then will maintain the current flow even with no voltage applied. In fact, it will generate a voltage like a battery in order to keep current in the same direction. The capacitor being the recipient of the current will begin to accumulate a charge in the opposite polarity as before. When the inductor is finally depleted of its energy reserves and the electrons come to a halt, the capacitor will have reached its full voltage charge in the opposite polarity as compared to what it was when it was started.

The capacitor As before, we'll begin to discharge through the inductor causing an increase in current in the opposite direction as before, and a decrease in the voltage as it depletes its own energy reserves. Eventually the capacitor will discharge to zero volts leaving the inductor fully charged and full current through. The inductor desiring to maintain the current in the same direction will act like a source again generating a voltage like a battery to continue the flow. In doing so the capacitor will begin to charge up and occurred will decrease in magnitude. In time, eventually the capacitor will become fully charged. Again, as the inductor expand expends, all its energy reserves trying to maintain the current, the voltage will once again be at its positive peak and the current and zero this Complete one full cycle of the energy exchange between the capacitor and the inductor.

This oscillation will continue with steadily decreasing amplitude due to power loss from say straight resistance in the circuit until the process stops. If there is no resistance in the circuit, which is impossible, but it's, it can be hypothesized, this oscillation will continue to happen over and over again. Overall the behavior is akin to that of a pendulum as a pendulum mass swings back and forth, there is a transformation of energy taking place from kinetic motion to potential motion or hype. In a similar fashion to the way energy is transferred from the capacitor inductor circuit, back and forth. In the alternating forms of current kinetic motion of electrons and voltage potential electric energy. At the peak height of each swing of the pendulum the mass briefly stops and switches directions.

It is at this point that potential energy the height is at its maximum and the kinetic energy motion is at zero. As the mass swings back the other way, it passes quickly through a point where the string is pointed straight down. At this point potential energy height is at zero, and the kinetic energy motion is at a maximum, like this circuit. A pendulums back and forth oscillations will continue with steadily dampening amplitude, the result of air friction or resistance, dissipating the energy Also like the circuit, the pendulums, position and velocity measurements trace two sine waves 90 degrees out of phase over time. In physics, this kind of natural sine wave oscillation for a mechanical system is called harmonic motion. The same underlying principle governs both the oscillations of the capacitor inductor circuit and the action of the pendulum.

It is an interesting property of any pendulum that its periodic time is governed by the length of the string holding the mass and not the weight of the mass itself. That is why a pendulum will keep swinging at the same frequency as the oscillations decrease in amplitude. The oscillation rate is dependent on the amount of energy stored in it. Same is true for the capacitor inductor circuit the rate of oscillation is strictly dependent on the size of the capacitor inductor, not on the amount of the voltage or the current at each respective peak in the waves. The ability for such a circuit to store energy in the form of oscillating voltage and current has earned it the name a tank circuit. Its property of maintaining a single natural frequency regardless of how much or how little how little energy is actually being stored in it gives it a special significance in electric circuit design.

However, this tendency to oscillate or resonate at a particular frequency is not limited to circuits, exclusively designed for that purpose. In fact, nearly any AC circuit with a combination of capacitance and inductance, commonly called an LC circuit will tend to manifest Usually effects when the AC power source frequency approaches that natural frequency. This is true regardless of the circuits intended purpose. If a power supply frequency for a circuit exactly matches the natural frequency of the circuits of the LC combination, the circuit is said to be in a state of resonance. The unusual effects will reach maximum in this condition for resonance. For this reason, we need to be able to predict what the resonant frequency will be for various combinations of L and C. And be aware of what the effects of resonance are.

A condition of resonance will be experienced in a tank circuit, when the reactance is of the capacitor and inductor are equal to each other. That is to say when two pi f l is equal to one over two pi f c. Because inductive reactance increases with increasing frequency and capacitive reactance decreases with increasing frequency, there will only be one frequency where these to react and says we'll be equal. To find that resonant frequency, we simply solve the equation for F. And we find the natural frequency for this tank circuit is one over two pi times the square root of the inductance times the capacitance. So let's look at a practical example. In the above circuit, let's say we have a 10 micro farad capacitor and 100 milli Henry inductor, since we know the equation for determining the reactance of each of the given frequencies, and we're looking for that point where the two reactants are equal to each other, we can set the to react and supportiveness equal to each other and solve for the frequency algebraically LC is equal to 10 times 10 to the negative six times 100 times 10 to the minus three, which gives gives us 0.000001.

The square root of that is point 0012 pi times that is point 00628318. And one over two pi the square root of that is 159.4. Five Five. So, there we have it the formula to tell us the resonant frequency of a tank circuit given the values of inductance. In Henry's and the capacitance in ferrets plugging it in to the values, we arrive at the resonant frequency have 159.155 hertz what happens at resonant frequency is quite interesting with capacitive and inductive reactance is equal to each other the total impedance increases to infinity meaning that the tank circuit draws no current from the AC power source. We can calculate the individual impedances of the capacitor which works out to mathematically using the resonant frequency the reactants to the capacitor is 100 ohms and the reactance of the inductor is also 100 ohms using the resonant frequency that we calculated.

Now, the parallel impedance, the total parallel impedance of the two impedances are the two reactance is there is given by the formula one over one over the set L plus one over the Zed C. And if you work those through, you will end up with a final number of one over zero. What this means in practical terms is that the total impedance of the tank circuit is infinite. Hence be behaving like an open circuit So let's hook up our, our two react instance now in series and a similar effect will happen is they will start to resonate when they're hooked up in series. And the resonant frequency if we set it at 159.155 hertz, we find that the impedance of the inductor is plus j 100 ohms and the impedance of the of the capacitor is minus j 100 ohms. The total series impedance would be given by Zed l plus C, which would be given by plus 100 ohms plus j 100 ohms plus minus j 100 ohms which would equal zero which essentially means that we have a short circuit And this is characteristic of a resonant frequency circuit.

And we have to be careful when we're connecting these circuits up of the frequencies that we're dealing with. This ends chapter five