CHAPTER THREE and inductors opposition to change in current translates to an opposition to alternating current in general, which is by definition sinusoidal and always changing in instantaneous magnitude and direction. This opposition to alternating current is similar to resistance, but different in that it is always resulting in a 90 degrees phase shift between the current and the voltage and it dissipates zero power on average. Because of the differences it has a different name reactance reactance to AC is expressed in ohms, just like resistances except that it is mathematically symbolized by the letter X instead of R to be specific reactance associated with an N Doctor is usually symbolized by the capital letter X with a letter L as a subscript like this. We have already seen the relationship between the voltage drop across an inductor and the current through it. That is the voltage will lead to current by 90 degrees.

The inductor of a given size will impede AC current similar to the resistance of a resistor, that resistor resistance is defined by ohms law to be proportional to the voltage across it divided by the current through it. In phasor terms, it is defined by V divided by AI and is in phase with the current. The reactance of an inductor is also defined by ohms law to be proportional to the voltage across it, divided by the current through it, but in phaser term When divided when dividing a V by AI, it is calculated to be 90 degrees leading the current. This phasor is sometimes written using the J operator as j times x C. So we have concluded that the reactance is a phaser itself, and it is at 90 degrees to the current that's going through it. And it's described as having a magnitude of x L. And what is that magnitude?

Well, let's have a look at that. Now. Since the inductor drops voltage in proportion to the rate of current change, they will drop more voltage for faster changing currents and less voltage for slower changing currents. What this means is the reactance in ohms for any inductor is directly proportional to the frequency of the alternating current. Also the reactance is directly proportional to the inductance of the inductor. The exact formula for determining reactance is as follows.

X l is equal to two pi f L, where f is equal to frequency in hertz or cycles per second, pi is the infamous 3.14159 and L is inductance in Henry's. Let's consider a series RL circuit with a supply voltage of EA T. The resistor will offer resistance to the AC current while the inductor will offer resistance in the form of reactance to the AC current. This combined opposition will be a combination of resistance and reactance in order to explain Press this opposition succinctly, we need a more comprehensive term for opposition to current other than resistance and reactance alone. This term is called impedance, its symbol is Zed, and is also expressed in units of ohms. Just like resistance and reactance. As we have already seen resistors and inductors react quite differently to voltages and current and in order to express the impedance in terms of how they react in combination, we will have to examine it just a little bit further.

And that impedance Zed T is the vector sum of X r plus x L, the impedance of the resistor plus the impedance of the reactance. If we take a closer look at This series resistor inductor circuit, we know that the current through each element of the series circuit is the same, because that's a characteristic of a series circuit. So the current in each element of the circuit is equal to AI. In other words, the current through the resistor is equal to the current through the inductor is equal to I have the circuit and those currents are vectors or phasers, but I'll call them vectors for now. And we just learned from the previous slides that the voltage drop across the resistor is in phase with a current in the circuit and that the voltage drop across the inductor is 90 degrees out of phase with the current In other words, the voltage leads the current by 90 degrees, giving us this orientation of the current and voltage vectors of the circuit.

We know from current shops voltage laws, that the voltage drops around the circuit have to add to zero. Or in other words, the voltage drops across the resistor and the inductor have to equal the voltage of the source. And in that case, this is what the vectors would look like are the phasers, the phaser er when added to the phaser el has to equal the phaser et of the source you can see from the phasor diagram that the current is at zero degrees. In other words, we're using the current as our reference phasor. In which case, the voltage across the resistor will be in phase with that current than it's shown in the diagram as such, and the voltage across the inductor will be 90 degrees to the current, we can rewrite our purchase Voltage Law equation using polar coordinates this time, and we'll write in the angles of the phasers and the angle of the voltage phasor.

For the whole circuit, in other words, the angle of E t the source voltage will be the sum of er plus e L, which is er at zero degrees plus the phaser e l at 90 degrees, giving us the result of et at five degrees as I've indicated in the diagram. I'm going to move that equation down here for the moment, and I'm going to divide each element of the equation by the current. Now I can do that as long as I do it to both sides of the equation. So I've divided the ET at five degrees by one zero degrees, and I've divided the voltage er at zero degrees with the current is zero degrees, and I've divided l at 90 degrees by the current at zero degrees. Now if you look at the equation, you can see that the resistance voltage er at zero degrees all over I at zero degrees, is nothing more than the reactance of the resistor or it's just the resistance of the resistance.

Because that's ohms love voltage over current will equally resistance. I've indicated it as a reactance here, but it's really at a resistance x subscript R. And the last component of that equation, e l l o l at 90 degrees all over one at zero degrees is the reactance of the inductor. We already saw that a couple of slides back, and if we add those two together, we will get Zed t, which is the total impedance of the circuit, as long as we add up the phasor results of the components. Now Zed T will equal at at five degrees all over one zero degrees, which would give us the magnitude Zed He, which is x r plus x l at five degrees, and that angle is the exact angle that Zed T is that with reference to our current zero, or the voltage across the resistor, which is in phase with the current.

And if you look up in the diagram just above it, you can see it's the same angle that the voltage angle is at for the circuit. We're going to replace the inductor with a capacitor this time and it's still a series circuit. So the current through each element of the circuit is the same I'm going to designate that current through the capacitor as I see the voltage drop across the capacitor as E c, and the time domain graph which relates the voltage to the current will look like this. You will remember that the current through a capacitor is a reaction against the change in voltage across it. Therefore, the instantaneous current is zero whenever the instantaneous voltage is at a peak zero change or a level slope on the voltage sine wave and the instantaneous current is at a peak whenever the instantaneous voltage is at a maximum change that is the points of the steepest slope on the voltage wave.

That is where it crosses the zero line. This results in a voltage wave that is 90 degrees out of phase with the current wave. Looking at the graph, the current wave seems to have a head start on the voltage wave. You might say that the current leads the voltage or the voltage lags behind the current by 90 degrees. The phasers would look like this, the capacitor voltage lags the circuit current by 90 degrees. Or you might say the current leads the capacitor voltage by 90 degrees.

I'm also going to show the voltage drop across the resistor, which is of course in phase with the current. When dealing with AC, as it's applied to a capacitor circuit, it's sometimes confusing to try to remember whether the current lags the voltage or the voltage lags the current. Especially if you're also dealing with inductive circuits. It's easy to remember that there is always a 90 degree phase shift because that happens with both capacitors and inductors. But it's the fact of whether the voltage leads or the current leads. And what I tried to remember is going back to first print Because there are acronyms out there that you can memorize.

But that's one more thing that you've got to remember. But you've already got the first principles in your head. So what I would like to do is replace the AC generator with a battery. As you can see, I've replaced the AC generator with a battery, which is essentially a DC generator. And it will try to hold the voltage of the circuit at a constant value, and indeed it will it'll and the current will flow once I've closed the switch, and the current will be limited by the passive elements in the circuit. In other words, the resistor in the capacitor.

However, there's zero current flowing at time equals zero. So when I close the switch, the first thing that happens is the voltage will start to rise on the capacitor as it starts to charge but it'll start off at zero, but the capacitor because it's not sure charged looks like a short circuit to the current as soon as the switch is closed, so the current immediately rises to its maximum level which is limited only by the resistor. So the what the graph shows you here is that the current will lead the voltage. And you can take that into an AC circuit and say that there's a 90 degree phase shift, however, the voltage will lag the current. Now, let's replace the capacitor with an inductor and keep everything else the same. Once the switch is closed in this DC circuit, the voltage will rise immediately across the inductor.

However, the current is impeded by the fact that it takes some time to build the magnetic field. So you can see from the graph that as soon as the switch is closed, the voltage instantaneously raises raises to its a maximum value. But the current takes some time to rise to its maximum value until the magnetic field is built. So you might say, in this DC circuit, and you can very well see it actually, that the current will lag the voltage. Now, if you extrapolate this to a, an AC circuit, then you can say the same thing. And that is, we know there's a 90 degree phase shift between the current and the voltage, but just as in this DC circuit, the voltage will lead the current and because it's 90 degrees, the voltage will lead the current by 90 degrees.

So the two takeaway here is if the phasers are rotating in a counterclockwise direction, then the voltage across the capacitor is 90 degrees lagging the current in the circuit. Let's look at a capacitor in an AC circuit considering only the voltage and current in the capacitor itself. Because instantaneous power delivered to the capacitor is the product of the instantaneous voltage drop and the instantaneous current p is equal to EC times IC. The power equals zero whenever the instantaneous current or the voltage is zero. Whenever the instantaneous current and voltage are both positive, that is above the zero line, the power is positive as with the resistor example, The power is also positive when the instantaneous current and voltage are both negative that is below the zero line. That is two negatives multiplied by each other result in a positive.

However, because the current and voltage waves are 90 degrees out of phase, there are times when one is positive and the other is negative, resulting in equally frequent occurrences of negative instantaneous power. You will notice that there is on the average as many positive power flows as negative power flows resulting in a net average power flow into and out of the inductor of zero. So, what exactly is going on with the capacitor during this positive and negative flow of energy during the positive flow of Energy the current is building an electrostatic field in the capacitor. This requires energy which is then stored in the electrostatic field. As long as both the current through the capacitor and the voltage drop across the capacitor are positive, this will happen. Once the current and voltage have opposite values one positive one negative, the power flow will be negative.

This is when the electrostatic field around in the capacitor starts to collapse and the stored energy will dissipate back into the circuit. During the next positive flow of energy, the electrostatic field is building in the opposite direction. In the capacitor, this requires energy again which is stored in the reverse electric electrostatic field. As long as both the current through the capacitor and the voltage drop across the capacitor are negative. This will happen Once the current and the voltage have opposite values again one positive one negative, the power flow will be negative. This is when the electrostatic field around the capacitor again starts to collapse and the stored energy will again dissipate back into the circuit.

This process will repeat itself, at the same rate as the frequency of the power flow. And the net average power flow is zero over time. The current flow in a capacitor is caused by the changing voltage which is sinusoidal. And defined in opposition to alternating current in general. This voltage which is sinusoidal and by definition always changing in instantaneous magnitude and direction, for any given magnitude of AC voltage at a given frequency, a capacitor of a given time size will impede a certain magnitude of AC current. Just as the current through a resistor is a function of the voltage across the resistor and the resistance offered by the resistor.

The AC current through a capacitor is a function of the AC voltage across it and the reactance offered by the capacitor. This opposition to alternating current is similar to resistance but different in that it is always resulting in a 90 degree phase shift between the current and the voltage and it also dissipates zero power. We've seen this 90 degree phase shift and and have seen a way of remembering whether it's leading and lagging already. As with inductors, the reactance of a capacitor is expressed in ohms and is symbolized also by the letter X, but this time we use a subscript C to denote the fact that this reactance is capacitive reactance. We have already seen the relationship between the voltage drop across a capacitor and the current through it. That is, the voltage will lag the current by 90 degrees.

As I have said a capacitor of a given size will impede AC current similar to the resistance of a resistor, that resistor resistance is defined by ohms law to be proportional to the voltage across it divided by the current through it. In phaser terms, the resistor is defined by V over I and it's in phase with the current. However, the reactance of a capacitor is also defined by ohms law to be proportional to the voltage across it divided by the current through But in phaser terms when defined by V over I, it is calculated to be minus 90 degrees lagging the current. This phaser is sometimes written using the J operator as minus j times XC. Since capacitors conduct current in proportion to the rate of voltage change, they will pass more current for faster changing voltages and less current for slower changing voltages. What this means is that reactance in ohms for any capacitor is inversely proportional to the frequency of the alternating current and notably also inversely proportional to the capacitance of the capacitor.

The actual formula being XC of the magnitude of x e is equal to one divided by to pi f c, where f is the frequency in hertz or cycles per second, pi is 3.14159 and C is the capacitance of the capacitor in ferrets. Let's have a closer look at this series RC circuit, the resistor will offer resistive reactance to the AC current well the capacitor will offer resistance in the form of capacitive reactance to the AC current. Because the resistors reactance is in phase with the current and the capacitors reactance is at a minus 90 degrees. That combined effect of the two components will be an opposition to current equal to the complex sum of the two numbers. Zed expressed in units of ohms and this collective summation is known as impedance. Because this is a series circuit, the current in all of the elements of this circuit are equal.

And we have just learned that the voltage drop across a resistor in a circuit is in phase with the current and the voltage drop across a capacitor is 90 degrees lagging and the impedance of the circuit is made up of the vector sum of the impedance of the resistor and the reactance. In the case of the voltages kerkhof voltage law also tells us that the sum of the voltages in the series circuit have to add to zero. In other words, the sum of the voltage drop on the resistor plus the voltage drop on the capacitor is equal to the source voltage which is e t. I'm going to rewrite the equations in polar notation, I'm going to let the current be our reference phasor. In other words, the current I'm setting at one at zero degrees and all other phasers will be referenced to our current phaser.

So the voltage equation, written in polar notation is at at five degrees. Now phi will be a product or actually a sum of the voltage in the resistor and the voltage drop across the capacitor. And we don't know what that is yet until we know the magnitudes of era ec. We do know the angle, but we don't know the magnitudes. So we'll just set the EP at five degrees, which will be a variable and we can calculate that at a later time. That's going to equal er at zero degrees plus EC at minus 90 degrees.

I am now going to divide each of those voltage elements of the equation by the same thing and I'm going to divide them each element by the current in the circuit I at zero degrees. Now, as long as I do that to both sides of the equal sign in the equation, that's legitimate as far as as far as algebra is concerned. So we have this new formula here which is at at five degrees over the current er at zero degrees over the current and EC at minus 90 degrees all over the current and of course the current is one, zero degrees. Now if we have a closer look at that equation, you can see that the element of the equation that's er at zero degrees all over i o at zero degrees, because of old Law is actually the resistance in the circuit, which is the voltage over the current and they're both at zero degrees.

So that gives me a phasor which is equivalent to the reactance of the resistor and it is at zero degrees because both the numerator and denominator are at zero degrees and we can plot that as I have done on the right. The reactance element of that voltage equation is EC at minus 90 degrees all over the current one is zero degrees. And that because of ohms law is the reactance of the capacitor and that is a phaser at minus 90 degrees which was which I will plot to the right as you can see, now the sum of those two phasers are indeed Zed T. Which is he at phi degrees all over one at zero degrees and that is the impedance of the circuit. And that angle of that impedance is five degrees. One thing you will notice is that the angle phi, the contained angle of Zed t with the horizontal is the same angle as et with the current, which is not surprising because the triangle formed by x r XC and Zed T is proportional to the triangle formed by er EC and et.

Because in the equation they're divided by the same thing, so The angles should all be the same. It's just the magnitudes that are different. So the triangle formed by x r XC and Zed T is definitely and should be proportional to the voltage triangle. This slide summarizes what we have gone over in regard to resistors, inductors and capacitors in an AC circuit. They are react ances themselves or ultimately, our phasers. The summarization shows their magnitude and their direction, as well as what the plot of the phaser might look like.

This ends the chapter