AC Basics Capacitors in AC Circuits

today I'm going to talk about capacitors with alternating current before I do I want to review how capacitors act in DC particularly how they act in charging and discharging cycles and you might want to review the video I made about that which I've linked below before we watch this because I'm going to just give a quick overview of that and it's important to understand how capacitors charge and discharge to understand why they do what they do in alternating current so let's get started I'm going to use the demonstration circuit I used to demonstrate how capacitors charge and discharge I start with a battery a switch a resistor our capacitor and back to our battery and I have another switch here just to facilitate showing that the capacitor is completely discharged when I start the charging cycle so I'm going to start out with one volt 1 ohm and one farad because first of all the easy numbers to work with and they show a particular parameter of what we need to know about capacitors charging and discharging so what I'm going to do is throw that switch short the capacitor will have a resistor in series but this will cause the capacitor to completely discharge if I leave that for a few seconds now I'm going to open that switch and now I can start the charge cycle so I close the switch and what happens well current surges over to the capacitor but because the capacitor is two conductors separated by an insulator that current cannot go through the capacitor but it spreads out on to the plate that it reaches now of course what's really happening is we have electrons flowing the opposite direction these electrons pile up on this plate over here and they drive electrons off the opposite plate but in this class as is done in the electronics Industry and most of Academia we talk about conventional current where we pretend that the current actually flows the opposite direction because it's easier to analyze circuits if we imagine our current going from positive to negative high pressure to low pressure that makes more sense than trying to figure out what goes from more negative to less negative that gets confusing so we're going to stick with our conventional current so the current flows into the capacitor that drives current off the other side now of course it's not the same current but we see current go in we see a current come out it looks like the current's going right through the capacitor and if we put a voltmeter here we will see that that capacitor is zero volts now this is the way it is at the moment we throw that switch so what does it look like we have current flowing through the capacitor no voltage across it that capacitor looks like a short circuit so it might as well be just that a short circuit so at that moment in time we have nothing but our one volt battery and our one ohm resistor and of course we then have one amp of current flowing so let's draw a quick graph over here of what's happening and what's going to happen so at this particular moment in time I've just flipped the switch so our time is zero here's time on the horizontal axis of our graph and our voltage and current are going to be over here I'll draw that in later so at the moment we throw the switch our voltage is down to zero and our current is at its maximum and of course in this case one volt one ohm we get one amp of current so I will put our one amp there but after time that capacitor starts to build up electricity and it acts sort of like a compressed air tank so a compressed air tank you put a little air in you get some pressure you put more air in you get more pressure you put more air in you get more pressure and you can keep doing that until one of two things happen either you just can't push anymore or the tank explodes a capacitor works the same way except of course our pressure is voltage so we put a little electricity in we get a little voltage but more in we get more voltage and we keep adding electricity we keep getting more and more voltage until either we can't push anymore or the insulation breaks down and the capacitor fails and it could actually even explode so as that charges it starts to push back so we start getting some voltage across it after one second in this case we have one farad and one ohm and so we multiply those together to tell us how long it will take to get to 63.2 percent of our battery voltage which of course is going to be 0.632 volts so one farad one ohm one times one equals one that tells us one second to reach the magic number of 63.2 percent of that voltage so there is after one second it's going to be a little curvier than that it's kind of important to know that so it starts to charge up it's starting to slow down because as it charges it pushes back more and more and the charging slows down but after one second we're going to get the 0.632 volts and the current is going to drop down from one amp to 0.368 amps so we start out at first we have our maximum current and zero volts But as time goes by in the case of these two resistors that voltage is going to fully charge after about five seconds so this is going to charge slower and slower and slower and it's going to finally flatten out and stop charging when the capacitor reaches the source voltage so it's going to reach one volt it's going to stop charging and our current has dropped down to zero so what happened when we first flipped the switch we have our maximum current and zero volts so when we have our maximum current we have zero volts but then it climbs up and we have our maximum voltage and now we have zero amps so when we have maximum amps we have zero volts when we have maximum volts we have zero amps now I'm going to redraw this to facilitate something I need to show here actually I'm going to take this out of the way because I need a little more room to draw here so I'm going to give ourselves a little more time and we let that capacitor charge and it charges up and then it just stays there stays there at one volt in that case or whatever the battery voltage is so if that was a 10 volt battery this would be 10 volts if it was 100 volt battery it would be 100 volts so it charges up to the battery voltage and then just stays there and then the current starts at maximum drops down to zero and just stays there so once the capacitor charges it just stays charged unless we discharge it so now I'm going to flip those switches I'm going to open the charging switch and close the discharging switch and this is going to start discharging so what's going to happen let's do it right about there when I hit the discharge it's going to discharge following a curve that looks pretty much like the charge curve inverted and in one second or one time constant it's going to get down to 63.2 percent of the starting voltage the current is going to do something kind of funny because let me redraw part of the circuit here or the whole thing actually well I'll just there resistor and capacitor and I'm going to be discharging so I'll have the switch closed and that switch open so when I was charging current was flowing this way through the capacitor but when I discharge it's going to flow the opposite way and so when it's flowing one way that's positive current flowing the other way it's negative current it's rather arbitrary but if I say negative current that's all I mean positive current is Flowing One Way negative current is Flowing the other way and we'll graph that as a negative current so when I start the discharge I open the switch close that switch it starts discharging at first this has one volt so it looks like a one volt battery and I have one volt and one ohm so I get one amp of current there's my one amp but that current is going to drop back down and eventually it's going to hit zero so we're back to where we started with zero volts and zero amps okay now let's let this stay charged for a while just to point out something I'm going to redraw this with some more space so I can demonstrate something so let's go ahead and let that charge oh we'll charge it all the way and good current is going to go down to zero and then I'm going to flip those switches and let it discharge so this is going to discharge that goes down to a negative current and comes back up then I'm going to flip the switches again it's going to charge so this is going to go charge again the current is going to jump back up and then go back down and so I'm spending more time with a high current I have high current here High current there high current there and it's going to do the same thing when I start the discharge here so let's flip the switches even faster at a higher rate at a higher frequency and see what happens so let's erase that to there so right at this moment here I flip the switch and let it discharge that discharges that jumps down to negative and I'm not going to let it just charge all the way now I'm going to start the charge again and then I'm going to hit the that's going to jump up to a high current and then drop back down then I'm going to start discharging it then I'm going to flip the switch again and we're going to keep seeing this cycle so if I flip the switches faster I'm spending even more time in a high current state so that's now let's flip them really fast so I start to charge and then I discharge start to charge and then I discharge start to charge and then I discharge so I'm flipping those switches really fast and of course the current is going to jump up to here come down a little bit then it's going to jump down to there come down a little bit jump up to here come down a little bit jump down to there come down a little bit so we have the current look what's happening we're spending a lot of time with a lot of current so the faster I flip those switches or the higher frequency that I flip those switches and change them from charge to discharge the more time I spend conducting current so I get more current through the capacitor so at a low frequency I don't get a lot of current but at a high frequency I get a lot of current so let's look at another device and which is just a resistor and so if I get a lot of current that means I have a low resistance but if I get a small amount of current I have a high resistance so low resistance High current High Resistance low current in the case of the capacitor high frequency High current low frequency low current so that capacitor as my frequency goes up as I charge and discharge it more and more often and let it go through less of the cycle it conducts more current so at higher frequencies it looks like a small resistor or a low value resistor and at low frequencies it looks like a higher value resistor so the capacitor changes its opposition to current flow with the frequency the higher the frequency the less opposition the current flow the lower the frequency the more opposition to current flow we have a name for that it's called capacitive reactants it's represented by the letter X and a sub c so X means reactants and c means with a capacitor and that's measured in ohms so it's just like resistance it opposes current flow and it's measured in ohms but it changes with frequency higher frequency fewer ohms lower frequency higher ohms so that's the way the capacitor reacts to alternating current let's start taking a closer look at that so let's put our source voltage on a graph this graph shows one cycle of the source voltage remember that a sine wave is related to a circle so we assign 360 Degrees to our sine wave each vertical line represents 10 degrees on this graph so we have 360 going from left to right with this complete cycle each horizontal line represents 20 volts so let's put zero volts in the middle so the top is positive 100 volts and the bottom is negative 100 volts our 50 volt RMS Source voltage Peaks at approximately 70 volts to calculate the capacitor voltage in real time would take some complicated calculus there is a simpler way to calculate the voltage but I don't want to tackle that right now so let's just put the voltage here and analyze what it's doing we'll join the show in progress right here where the capacitor voltage is rising and just crossing zero volts just as in DC the current is flowing through the resistor and into the capacitor the capacitor is charging but as it charges The Source voltage is increasing so the capacitor voltage is not leveling off like it does with DC right here it's almost following a straight line however here the source voltage Peaks and starts to drop as long as the source voltage is higher than the capacitor voltage the capacitor will continue to charge but it starts to charge slower so our curve begins to heal over finally right here the capacitor voltage and Source voltage meet after this point the source voltage is lower than the capacitor voltage so the capacitor begins to discharge this is a key point in the cycle the capacitor voltage will always cross the source voltage at the capacitor's peak voltage if we reduce the resistance or the capacitance the capacitor will charge faster and reach a higher Peak voltage but the capacitor voltage will still cross the source voltage at its peak it has to after this point the source voltage is lower than the capacitor voltage and the capacitor will begin to discharge here we decreased the resistance or the capacitance and the capacitor charges faster the capacitor charges to a higher voltage and crosses the source voltage earlier but it still crosses at the capacitor Peak voltage here we increased the resistance or capacitance so the capacitor charges more slowly and takes a little more time to charge and the capacitor voltage crosses the source voltage later but still at its peak in the DC circuit the capacitor charges until it equals the battery voltage the maximum voltage across the capacitor equals the battery voltage with an AC Source the capacitor of voltage reaches its peak point at the time when it equals the source voltage in AC if you freeze time the maximum capacitor voltage equals the source voltage at that time just as in DC the voltage across the capacitor is a sine wave however that sine wave is delayed compared to the source voltage this delay time depends on the resistor capacitor combination remember that we measure the points in a cycle of a wave in degrees a complete cycle being 360 Degrees the delay in the peak of the capacitor voltage is a certain number of degrees after the peak of the source voltage let's go back to our circuit with our 30 Ohm resistor and our 100 microfarad capacitor at a frequency of 60 hertz we see that the capacitor voltage lags behind the source voltage by approximately 49 degrees remember that number we will run into it later in the DC circuit when the capacitor finished charging we have the maximum voltage across it and zero current the capacitor was acting exactly what it is it is two conductors separated by an insulator so it's an open circuit so at that time we had our maximum voltage and zero current will it do the same thing in an AC circuit let's take a look the current in the circuit is determined by the value of this resistor and the voltage difference across it in the DC circuit where we had a battery which was one volt in that case when we started charging the capacitor we had one volt here and zero volts there for a difference of one volt that was a one ohm resistor so we had one amp of current but as the capacitor charged this voltage Rose and eventually it reached one volt also and so if we have one volt here and one volt there there's no voltage difference therefore there's only zero volts across the resistor remember that there are voltage difference is what matters so there's no voltage difference that means zero volts and therefore there was Zero amps of current flowing through the capacitor in the case of the alternating current we have voltages that are constantly changing so this voltage is going up and down and this capacitor is charging and discharging reversing polarity and then charging that way then discharging so these two voltages are going up and down but they're not synchronized one is going up and down at one time the other one at another time they're not synchronized and they're not opposite each other it's more like a little game here where one's coming and going and so the voltage across this resistor is constantly changing in the AC circuit The Source voltage and the capacitor voltage are constantly rising and falling these voltages are not synchronized so the voltage across the resistor is continually changing this means that the current is constantly changing the timing of this change is related to The Source voltage and the capacitor voltage let's look at these voltages at different points in time and see what the current is at those times we'll start here where the capacitor voltage is 0 volts that's 50 volts on this side and zero volts on this side for a total voltage of 50 volts and 30 ohms for the resistor so we are going to get a current of 1.66 amps at that moment in time through the whole circuit through the resistor and therefore through the whole circuit and this current is flowing through the resistor from left to right we'll use a different graduation of our scale for current otherwise it will be hardly visible let's make it one amp per vertical division now let's put a dot right here at 1.66 amps to represent the current at this point in time now let's look here where the source voltage Peaks The Source voltage is at 70 volts and the capacitor voltage is at about 28 volts that's 70 volts on this side and 28 volts on that side for a total difference of 42 volts and that now gives us a current of 1.4 amps now let's look here where the capacitor voltage Peaks here the source voltage is about 44 volts and the capacitor voltage is also 44 volts that puts a difference of zero volts across the resistor so the current is 0 amps right here let's do one more right here the source voltage is about 11 volts and the capacitor voltage is about 37 volts the difference is now 26 volts but now the voltage on the right is higher than the voltage on the left so we go positive to negative here so the current is flowing in the opposite direction which will plot as a negative current now we have a current of minus 0.866 amps right here now let's put the whole current plot on the graph the current lines up such that when the current Peaks the capacitor voltage is at zero volts when the capacitor voltage Peaks the current is 0 amps no matter where we look if the capacitor voltage is at its maximum positive or negative the current is zero likewise the capacitor voltage is zero wherever the current is at its maximum positive or negative this is true regardless of the frequency or the values of the resistor or the capacitor this means that the current will always Peak 90 degrees ahead of the capacitor voltage where does that put the current in relation to the source voltage let's figure that out the capacitor voltage always Peaks as it crosses the source voltage sometime after the source voltage Peaks the current always Peaks 90 degrees before the capacitor voltage Peaks therefore the alignment of the current to the source voltage will be the lag angle of the capacitor voltage minus 90 degrees in our circuit the capacitor consider voltage lags behind the source voltage by 49 degrees the current is 90 degrees ahead of the capacitor voltage that puts the current approximately 41 degrees ahead of the source voltage in our circuit the current also has an RMS value of 1.25 amps all of this finally shows us the relationship between the source voltage and the current let's relate that to a simple DC circuit let's say you have a 50 volt battery and a 40 Ohm resistor you will have 1.25 amps of current Ohm's law except there's one small problem if we put a 40 Ohm resistor across our 50 volt RMS Supply the peak current will happen at the same time as the peak voltage however in our circuit the current Peaks approximately 41 degrees before the source voltage Peaks this is because there is another source voltage the capacitor wreaking havoc with the circuit we showed that the current Peaks right where it should nevertheless in this RC circuit the current Peaks before the source voltage by 41 degrees it's the nature of the Beast so there we have it in this particular circuit the combination of the resistor and capacitor in series gives us approximately 40 ohms of opposition to current flow also the current leads the source voltage by 41 degrees is there an easier way to tackle this yes it doesn't explain why everything happens the way it does it also doesn't demonstrate that an AC circuit acts exactly like a DC circuit at any moment in time but it does calculate the opposition to current flow the impedance in ohms without any complex map just some simple algebra we can also calculate the timing difference or phase angle between the source voltage and the current using a simple trigonometric function found on any scientific calculator so let's look at those calculations so here's our circuit 50 volts RMS 60 hertz 30 ohm resistor 100 microfarad capacitor we just have to go through a few steps to find out the same information that we found out by simply plotting out all of the voltages and seeing how they acted at certain moments in time to get the current and its phase relationship to our source voltage so we'll start by finding out how much opposition to current flow this capacitor has and that's called the capacitive reactants so we have our capacitance and we have our frequency and knowing those we can calculate how much opposition to current flow this capacitor has at that frequency the formula is x sub c the capacitive reactants or the reactance of a capacitor is equal to 1 over 2 Pi FC so 2 pi is 6.28 f is the frequency and C is the capacitance in farads now we have to do the reciprocal because remember as our frequency goes up our capacitive reactance goes down so as our frequency goes up this looks like a smaller and smaller resistor as our frequency goes down we get more capacitive reactants so this looks like a bigger and bigger resistor so frequency goes up capacitive reactance goes down and vice versa so there are the reciprocals of each other so to calculate the capacitive reactants we need to take the reciprocal of these terms so what we're going to do is take 1 over 6.28 times 60 times .0001 so to do this on a typical calculator you hit 6.28 times 60 times .0001 equals and I get a number that's kind of useless at this point but now I take the reciprocal so I hit the reciprocal button and I get 26.53 so at 60 hertz this 100 microfarad capacitor has a capacitive reactance of 26.53 ohms so basically it acts like a 26.53 ohm resistor at this frequency of course it's not a resistor it's reacting to the frequency it's not resisting it because as we showed the voltage across this capacitor is not happening at the same time as the current so we have some things going on here that we have to take into account as we calculate the total impedance of the circuit so we started out with that we have 26.53 ohms there and we have 30 ohms there now can we just simply add these together like two resistors in series no we can't because remember the voltage across here is going to be 90 degrees out of phase with the current so I'm going to draw a little graph over here I'm going to draw a line about that long and see that's our resistance so I make a horizontal line for the resistance and that's going to be 30 units long I can actually do this on graph paper and calculate this up by drawing a graph and just measuring the units but we're going to do this by math now to represent the capacitor I need to make a line that's at a right angle to my resistance remember the voltage and the current are 90 degrees out of phase so that means I need to draw a 90 degree angle here and that's going to be 26.53 units 26.53 ohms and my impedance is going to be the distance between these two points and there's a really easy way to figure that out it's called the Pythagorean theorem a little bit of simple geometry so the Pythagorean theorem says that the square of this distance is equal to the sum of these two squares in other words if I take 30 and square it add that to 26.53 and square that so I add those two together I Square it Square it add them together and then take the square root I will have the length of the hypotenuse of that triangle so let's pull out my trusty calculator and do that so it's going to be 30 hit my square button I'll store that in memory now I take 26.53 26.53 and I square that now I add that to what I just had in memory plus recall hit the equal sign now I have a useless number but now I take the square root of that number and I got 40.04 so the length of this line is going to be 40 and that represents the impedance of 40 ohms so I could do this on graph paper I find exactly the same thing but I used the Pythagorean theorem to figure that out so we already learned that this combination of resistor and capacitor is going to act sort of like a 40 Ohm resistor except the problem is the current through the circuit is not going to be happening at the same time as our source voltage it's going to lead that Source voltage in this case by 41 degrees we figured that out when we plotted out all those voltages so how do we calculate that angle it's actually going to be this angle right there so the angle of the current to our source voltage is going to be the angle of this triangle how do we calculate that well it takes a little trigonometric function we're going to talk about that in a completely different video I'll have some math review but just for now we just have to know that there's a little button on here that does this function and the formula is get this out of the way our phase angle represented by the Greek letter Theta is equal to the arc tangent which is the reciprocal of the tangent which is a trigonometric function I'll talk about those later but what I can say quickly is that your sine and your cosine in your tangent are the ratios of these different sides so the ratio of these two sides is the cosine the ratio of these two sides is the sine and the ratio of these two sides is the tangent and the arc tangent is simply the reciprocal of the tangent so my angle is the arc tangent of my capacitive reactants divided by my resistance so I take this side divide it by that side take the arc tangent or the reciprocal of the tangent and that will give me this angle so what are the numbers well that's going to be equals to the arc tangent of 26.53 over R fairly simple to do it's a three-step process so 26.53 divided by 30 equals that number is useless but I take the arc tangent so where's my arctangent button right there and I got 41.48 degrees and so remember when we did all those gyrations and figured out how all those sine waves lined up with each other just because of how that circuit works if we freeze time and see that the circuit acts like a DC circuit if we freeze time we found that our current was leading the source voltage by 41 degrees approximately well there we go 41.48 degrees so I did all those gyrations to show you why these things line up the way they do I wanted to point out that a AC circuit acts exactly like a DC circuit if you freeze time so all that happens under alternating current is the circuit is reacting to the changes in voltage but at any moment in time it just acts like a DC circuit and we could take that circuit analyze it like a DC circuit at different times and see how those voltages all lined up and we got this angle by lining those things up we don't have to go through that we understand why that happens and now we know why this all works so the steps to find out my impedance how much opposition to current flow and of course I take this 40 ohms divide that into 50 volts I get 1.25 amps which is my RMS current and we know that the current has a phase angle of 41.48 degrees once again how we did that was we calculated our capacitive reactants with this formula capacitive reactants equals the reciprocal of 2 pi times the frequency times the capacitance once we had that we could use the Pythagorean theorem to see our resistance here our reactants there they make a right angle and the Pythagorean theorem tells us the length of that line there or we could have drawn it on graph paper but we could do the calculation and then this formula the arc tangent of our reactants over our resistance gives us the phase angle or the angle that the current leads the source voltage in this particular circuit and just finish up here how do we notate those things well once we calculate our impedance and our phase angle that would be expressed in this case we had an impedance of 40 ohms with a phase angle of approximately 41 degrees and that's how we would write that out our impedance and our phase angle there's another way to write this out that can be useful this is called polar notation because we're looking at the angle so let's draw our triangle again so there's our triangle and we're looking at this angle and the length of that line the other way to express this is called rectangular notation where I simply talk about the length of this line and the length of that line and that would be expressed as well this is 30 ohms and this is 26.53 ohms so rectangular notation I would say it's 30 ohms minus J 26.53 so polar and rectangular notation now I use minus J because it's a capacitor now this is a complex number if you're familiar with complex numbers you would say hey isn't that a lowercase i Well normally it is but in electronics a lowercase i represents AC current and So to avoid confusion in electronics we use J but in most mathematics they would use an I so capacitive reactants we use minus J and when we get to inductors in AC circuits we will have a plus J there because they go different directions as a matter of fact I have this backwards I really should have that go down so for capacitance I should have the resistor horizontal and the capacitor goes down for a minus J whatever and there's our hypotenuse it still works but for an inductor I would be a plus J for my inductive reactants and using this rectangular notation if I have both inductors and and capacitors I can actually add those together working with this so sometimes this is useful but typically we express an impedance as a number of ohms and a phase angle representing the length of the hypotenuse of our triangle and the angle of between our resistance and our impedance and so that is how our resistor and capacitor interact with alternating current if you found this video useful and informative please give me a thumbs up down below it really helps the channel and subscribe because that not only 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Channel: Vocademy - Electronics Technology

Views: 10,066

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Id: gl067OyGeSI

Length: 36min 7sec (2167 seconds)

Published: Wed May 24 2023