Resistance, Reactance and Impedance

hello today we're looking at it electrical resistance reactance and impedance this video follows on from the video on alternating current AC which I did in the a level physics revision series this video is itself not part of the a-level revision series because as far as I know there is no a level subject which covers impedance but please tell me if I'm wrong and I will put it in the a level playlist but as far as I'm aware this material is more appropriate for students at university so although we'll be looking at the AC effects the alternating current effects on impedance let's first just remind ourselves what we mean by direct current and the resistance associated with direct current direct current for example comes from a battery and if you were to put a voltmeter across a battery and measure the voltage this is the voltage over a period of time you would expect that as long as the battery has a useful life that the voltage would remain the same unchanged if it's a 9-volt battery then the voltage will be a direct nine volt no change if you then put that battery here's the battery across a resistance R then a current will flow that current is determined this battery has voltage V that current and the voltage are related by Ohm's law V equals I R and so if the voltage is constant and of course the resistance is constant then that means the current is constant so if you were to measure the current would say an ammeter what you would find is this is the current this is time as you measured the current going through this circuit for the useful life of the battery the current also will be constant and so you've got in DC a constant voltage constant current going through constant resistance now let's take this circuit here and take the battery out and instead replace it by an alternating current supply so we have our alternating current supply going through our resistance R and the supply will have a voltage and there will be a current flowing through the circuit and Ohm's law will still apply V equals I R but what you have to remember is that when you've got an alternating current as we saw in the video that I've made on alternating current the voltage is constantly changing the resistance of course remains the same so if the voltage is constantly changing the current is constantly changing and the mathematical representation of that is that V equals V naught sine Omega T that's V at any time T and the current at any time T is equal to I naught sine Omega T so if we plot these on the graph first we'll plot the voltage that will have a sine wave shape so at the voltage at any time the maximum is V zero that's where this comes from a sine term always goes between plus one and minus one so if you multiply that by V naught it means that the total voltage will vary between plus V naught and minus V naught in the sine wave and the current will look like something like this that's the current obviously the size of the current the maximum of the current will be determined by the size of the resistance and you can also say that V at any particular time T is equal to I at any particular time T times R so anyway you choose if we choose a time say there this is time of course then V at time T is equal to I at time T times the resistance and you'll notice that the voltage and the current are what is called in phase that is to say when the voltage is at its maximum the current is at its maximum when the voltage is zero the current is zero when the voltage is at its negative maximum the current is at its negative maximum it's in phase and I want to introduce you to a concept called a phasor diagram a phasor diagram purports to draw the resistance and the phase angle so if I draw a little one up here just to show you if I were to draw something like that then the length of the vector as it were this length would represent the magnitude of the resistance and this angle here Phi would represent the extent to which the current was displaced from the voltage in other words instead of the current being exactly in phase the current might actually start here and go up like this as it happens that doesn't happen with resistance so the phase angle is zero there is no difference in phase between the current and the voltage so on our phasor diagram you would simply draw the resistance like this the length of the line is equal to the value of the resistance and the angle which is zero tells you that the current and the voltage are completely in phase now the reason I do that and that means that the so-called phase angle is zero there is no difference between where the voltages and where the current is the reason I do that is because we're now going to discover two electrical devices where that is not the case these two devices are capacitors and inductors let's first think about a capacitor a capacitor is really just two metal plates opposite one another with a small gap between them and if we think of it firstly in DC terms so we're going to put it across a battery then what happens the important thing is that although a current flows at least initially no current ever flows ever ever ever flows between the two plates never assume a current flows indeed if there is a spark that jumps across then it messes up the capacitor the whole point of a capacitor is that no current flows between the two plates that's crucially important and indeed for a proper capacitor there's usually a piece of insulating material between the two to make sure that no current can flow so what happens well be as it were the conventional current flows from plus two miners but what is actually doing in it what is what is actually happening in electricity is that electrons are flowing and they fly flow from minus to plus so the electrons will flow until they get to this plate and then they will congregate on the plate because they cannot jump across the gap that's the crucial point and if you get a congregation of electrons negatively charged electrons on this plate those electrons will repent repel the electrons on this plate and push them the electrons will now flow in this direction back to the positive terminal leaving a residue of positively charged particles on this plate in other words energy has been stored you could if you wanted to disconnect this capacitor which is now fully charged so it's negative on one side positive on the other and if you were to connect back to say a bulb as flash bulb then the capacitor will discharge you the flash bulb and it will flash and that's the principle on which many flashlights in cameras work you initially charge a capacitor with the battery in the camera and then when you flash the flashlight the capacitor suddenly discharges through the bulb producing a very high intensity but short-lived light and what we say is that the capacitance of this capacitor is equal to the charge it is capable of storing on one of its plates divided by the voltage so you can see that as the voltage increases so the charge that the capacitor is capable of storing will also increase now let's think about the other device the inductor an inductor is basically just a coil of wire sometimes called a solenoid so the current if we connect this now to a DC device the current will flow through the coil but what do we know about a current flowing through a wire here's just an ordinary wire and it has a current I flowing through it we know that a wire that's got a current flowing through it will generate a magnetic field that magnetic field will be around the wire so if you think about the coil at the top of the coil you've got a whole load of wires let's say the current is going in to the paper and at the bottom of the coil you've got a whole load of wires coming out of the paper so what we're doing is we're looking at the inductor and seeing the coil go round so at the top the call goes in at the bottom the coil comes out and the current is going in the direction of my pain so when the current goes in it will create a magnetic field around each of the wires in a clockwise direction and when it comes out it will create a magnet a magnetic field in the anti-clockwise direction so you'll notice that inside the solenoid at the top the field is traveling in that direction because it's going around like this so when it gets to the bottom the field is traveling in this direction at the top the field is also traveling because at the top the magnetic field goes down clockwise at the bottom the magnetic field goes round anti-clockwise which means that at the top of it the field is going in this direction consequently within the centre of the solenoid you've got a magnetic field going from one side to the other so that could be north that side and South that side you create a magnetic field within the solenoid and the flux associated with that field Phi is L the size of the inductance times the current flowing through the wire L is the measure of a conductor of an inductor and is measured in henries C is the measure of a capacitor and is measured in farad's now let's go back to our capacitor and this time we're going to put it in a circuit which has an alternating current source so there's the capacitor it's exactly the same diagram out here except that instead of the battery we now have an AC supply and the current of course because it's an AC supply is going to constantly change direction so electrons are going to form on this plate then they're going to form on this plate then this plate then this plate constantly changing and the question is is there a V equals I R equivalent for the capacitor and the answer is yes there is what you find is that V is proportional to I the voltage is proportional to the current flowing through the flight through the wire and and the as it were the resistive effect which we actually call the reactance is given by XC and by experiment it is found that XC is equal to 1 / Omega C where Omega equals two pi f and f is the frequency of the alternating current the number of cycles per second and see is just the capacitance of the capacitor so you find that there's a resist resistive effect which we call the reactance which is equal to one divided by Omega C and what that means of course is that as the frequency increases remember Omega is two pi times the frequency as the frequency increases the reactance will decrease so as the frequency goes up the reactance will go down so unlike a resistor which doesn't care what the frequency of the AC is when it comes to a capacitor the actual resistive effect of the capacitor depends on the frequency of the alternating current now why should that be why should that be well let's think about it what's happening is that the electrons are flowing onto the plate and they are going to have the effect of repelling electrons from this plate leaving it net positively charged but of course they also do something else not only do they repel the electrons from this plate but they also impede any further electrons that are coming in this direction wanting to join them on the plate as the charge builds up then light charges repel consequently the charges that are already on this plate will repel the charges that are trying to come and join them on the plate and ultimately they will stop the charges from flowing altogether but it takes a bit of time for that to happen so what is happening is as the charge builds up on the plate so the impedance of further charge being added to that plate grows with time but of course it doesn't get much time to do this because then the phase of the alternating current changes and now the electrons are moving in this direction so everything changes sign there are now electrons on this plate there is a net positive charge on this plate and the electrons coming in this direction are impeded and the longer the time is the greater will be the impedance and now you can see why the resistive effect Falls as the frequency increases because as the frequency increases the amount of time that you've got for electrons to build up on the plate before the electrons start going the other way is reduced higher frequency means less time for each cycle so consequently you've got no time or very little time for the electrons to build up on this plate and then begin to impede the flow of electrons before the electrons are now coming the other way so the higher the frequency the less impact there will be because there will be fewer electrons building up and therefore a lesser effect of repelling the electrons that are still coming along whereas if the frequency is so long that for example there are several seconds before the current changes then the electrons will fully build up on this plate to the extent that it is even possible that they will impede completely any flow or any further flow of electrons and so the current will completely stop so the higher the frequency basically the less time there is for electrons to build up on this plate and impede the other electrons that would like to try and join them before the current changes direction but there is something else that is noted by experiment and that is that if you plot the voltage across the capacitor and remember it's an alternating voltage so it's going to have a sine wave effect V equals V naught sine Omega T and then you measure the current which is flowing in the sir it you find that the current put some points there so and you can see what I'm doing the current looks something like this it is no longer in phase with the voltage in fact the current starts first and then the voltage comes second so the voltage lags behind the current this is the voltage this is the current and the voltage is lagging behind the current does this the voltage starts as it were later and in fact the angle is one quarter of a full wavelength and one since a full wavelength is two pi radians a quarter of that will be pi over two so in fact the voltage is minus PI over two behind as it were the current it's a quarter of a wavelength behind the current and what you can also see from this is therefore that the voltage at time T most emphatically is not equal to the current at time T times the reactance that is no longer true because when the voltage is it at its maximum the current is zero and when the current is at its maximum the voltage is zero so this certainly voltage no longer equals the current times the reactance for any given time what we can also do is to plot our phasor diagram you remember the phasor diagram for resistance for resistance we plotted the resistance along here because the angle was zero in other words there was no phase angle between the voltage and the current they were completely in phase but now the phase angle is minus PI over two which is of course minus ninety degrees so if we're plotting the reactance the size of the reactance sxc is plotted as the length of that line and the angle is 90 degrees in fact it's minus 90 degrees which is why I've plotted it downwards so our phasor diagram for capacitors shows the modulus and if you like the size of the reactants the resistive effect of the capacitor is represented by the length of the line and the phase angle is represented as minus 90 degrees minus PI over 2 on the phasor diagram incidentally although this formula is not true because you cannot take the voltage at a particular time what will be true is that the root means square of the voltage which we dealt with in the earlier AC video is equal to the root mean square of the current times XC so in others what we're saying is taken over the entire cycle of the sine wave the voltage root mean square is equal to the current root mean square times the reactants caused by the capacitance now let's take the inductor and put that into an AC circuit so here's the coil or solenoid and here's the AC supply and what we want to know again is is there an Ohm's law type of effect is there a V equals IR kind of equivalence as far as this is concerned by experiment you find that yes there is V is equal to I times what's called the reactance the resistive effect of the inductor but the reactance XL and on this occasion XL is equal to Omega times L where once again Omega is 2 pi times the frequency so whereas the reactant sorry the reactance of a capacitor was 1 over Omega C for an inductor the reactance is Omega L and what that means is that of course as Omega increases in other words as the frequency increases the reactants will increase so XL will increase as the frequency increases with the capacity remember it was the other way around as the frequency increased the reactance decreased but here with the inductance if the frequency increases so does the reactance now why should that be well let's just think for a moment what's happening I showed you earlier that the generated in an inductance Phi equals I L the current multiplied by the inductance value in this case of course there is going to be a changing flux because there's going to be a change in current and so the rate of change of flux is going to be L which is constant times the rate of change of current and we know that EMF is equal to minus the rate of change of flux which of course is going to be minus L the a di by DT so what you've got in this coil in this solenoid is a change in current which is generating a changing electric field indeed the Elector and sorry a changing magnetic field indeed the magnetic field is going to be going from north to south one way and then as the current changes direction it will go from north to south the other way so the magnetic field is going to be pointing in that direction then in that direction then in that direction then in that direction a constantly changing magnetic field and what happens if you have here is the inductor here is the current which for the time being is flowing in this direction of course it's going to change direction and because the frequency will mean that it will change its polarity but for the time being let's just think about the current flowing in this direction through the inductance and generating a magnetic field a changing magnetic field what does a changing magnetic field do in the presence of a wire answer it creates an induced current which direction does that current flow answer lenses or tells you it flows in the opposite direction so as to oppose the change that caused it that's what lenses law tells us so although there is a current flowing in this wire and it's a changing current the changing current produces a changing magnetic field and the changing magnetic field produces an induced current that flows in the other direction and thus the current the main current is impeded by the induced current of course very quickly the actual current will be flowing in the opposite direction and then the induced current will oppose that now why therefore should it be that as the frequency increases the reactance the resistive effect of the inductor should also increase well think the higher the frequency the more rapidly the magnetic field is changing which means that the rate of change of flux is greater so if you have a greater rate of change of flux you'll have a greater EMF and that greater EMF will produce a greater induced current so in other words as frequency increases that means that the rate of change of flux increases because it's changing more frequently so that goes up that means that the EMF increases because EMF is equal to minus T Phi by DT and that means that I induced increases and that means that the resistive effect increases so as the frequency increases the resistive effect of the inductor increases and that's why the to go up hand-in-hand now there is something else that we can note by experiment and that is that if we look at the voltage and the current going through the inductor here is the voltage again sine wave from the alternating current supply what happens to the current on this occasion well this time the current lags behind the voltage so in the case of the capacitor the voltage was ahead sorry was behind the current this time with the inductance the voltage is ahead of the current this is the voltage this is the current so the voltage starts out first and then the current follows behind and again the phase angle is PI over 2 but this time it's plus PI over 2 because the voltage is ahead of the current and you'll notice again that you cannot say that the voltage at any particular time is equal to the current at any time times R you cannot say that that is not true because when the voltage is at a maximum the current is zero when the current is at a maximum the voltage is zero but you can say again that the taken over the entire cycle V RMS root mean square is equal to I RMS times R which of course is the reactants x-l and now if we draw our phasor diagram remember for resistance it went out like this the phase angle was zero for capacitance it went down like this the phase angle was minus PI over two and you can probably guess that for an inductor the size of the inductance is measured by the length of the line and the angle is 90 degrees or PI over 2 so just to recap on the phasor diagrams if you're drawing a resistance it will be R in this direction with a phase angle of 0 if you're drawing the capacitance then the reactance value is the length of that line and the angle is 90 degrees and if you're drawing an inductance then the reactance effect is the length of that line with of plus PI over two and so the question comes what happens if you put a resistance a capacitor and an inductor in the same circuit well let's consider if you simply have three different resistances in a circuit so here is resistor r1 r2 and r3 all in series we know that the total resistance of that circuit will be r1 plus r2 plus r3 and that will be measured of course in ohms so is it true that if you have a resistance and an inductor and a capacitor all in series we know that the resistive value of the resistance is just R we know that the resistive value called the reactance of the inductor will be Omega L and we know that the resistive effect the reactance of the capacitor will be 1 over Omega C is it true that the total resistance is equal to R + Omega L plus 1 over Omega C and the answer is no that is wrong and the reason that that is wrong is that you have taken no account of the fact that there is a phase angle associated with the capacitor and with the inductor let's for simplicity just consider a circuit in which we have an inductance and a resistance we'll come to a capacitance in a moment capacitor in a moment but for the moment let's just keep two elements in our circuit you would have to draw your phase phasor diagram which would have the resistance here and the reactance of the so this would be length R for the resistance and your reactance would be in the vertical direction incidentally you can show dimensionally that the reactance or the inductor and indeed the reactance for the capacitor are in units of ohms so they have a resistance dimensionality and the way you calculate the net what's called impedance of these two devices is to apply the rules of pythagoras pythagoras says that the length of this diagonal line is the impedance effect of the two devices and I think you'll see relatively straightforwardly that Zed is equal to the square root of R squared + x-l squared it's just plain Pythagoras you can also see that the phase angle which in this case is Phi which is neither 0 nor pi over 2 that the tangent of Phi is equal to XL / R so the phase angle the angle by which the voltage and the current will be out of phase is determined by the relative sizes of these two devices you'll see that because it's positive that means that the voltage will lead the current now in order to ensure that we don't ever mix up the reactants and the resistance what we do is we draw the reactants on the y-axis which is often called the imaginary axis for those who don't know the eye is the square root of minus 1 since the square root of minus 1 doesn't exist it's called an imaginary term so we tend to draw an imaginary axis and a real axis and we put the resistance on the real axis and we put the reactance on the imaginary axis and this means that we can say in vector terms that the impedance Z is equal to R plus I XL that way we've kept the two things entirely separate because you can never mix real numbers and imaginary number it's just a mathematical device there's nothing imaginary about the reactants it's just a mathematical device to keep the two entirely separate and now we can do a little bit more maths I think you will see that from this triangle here R is Z cosine Phi right because R over Z is cosine Phi so R is e 2 equal to z times cosine Phi similarly you also say that XL is equal to Z sine Phi because x over Z is the sine of Phi so X is 8 times the sine of Phi and that means that Z is equal to R which is the value of Z cos Phi plus I times XL which is Z sine Phi and that means that Z is equal to the modulus have said that's what these terms actually are the length of the value of the impedance times the modulus of Z into cosine Phi plus I sine Phi because I've taken the Z's out both sides so the total value of the impedance is equal to the modulus of the impedance the value of its length times cosine Phi plus I sine Phi but mathematically this term is e to the I Phi so you get that the total value of the impedance is equal to the modulus of the impedance that is the as it were the absolute value of the impedance times e to the I Phi because this term is mathematically equivalent to that term so what are we going to do if we have all three components in a circuit a resistance and inductance and a capacitance all in the same circuit well we plot our phasor diagram which is going to have three elements on it firstly it's going to have an inductance XL then it's going to have a capacitance XC and then it's going to have a resistance R and how are we going to construct our diagram in order to do our Pythagoras well it's quite simple the net effect of the two reactances which are both in the same plane they're both in on the scale this is the imaginary scale this is the real scale you can simply subtract one from the oven the net effect of this will be this distance here which is simply XL minus XC it will depend which is the larger obviously if XL is greater than XC then there will be a small surplus in this direction if XC is greater than X L there will be a small net effect in this direction but it's the net effect of XL minus XC which is what you construct and then the total value of Z will equal the square root of R squared plus the net reactance squared which is simply XL minus XC all squared and the angle the the phase angle which here is Phi will be given such that tan Phi is equal to XL minus XC divided by R because this value here is XL minus XC and this line here is off and the angle Phi will be the extent to which the voltage lags or leads the current in this case it's a slight positive angle and so consequently if we draw the voltage then the voltage will lead the current by this angle whatever that turns out to be so the current will be a little behind the voltage by the angle that we have calculated now although resistances and reactances have the same dimensionality they can all be measured in ohms there is a very important difference between the two so let's now take our circuit it's a an alternating current circuit in which we have a resistance an inductor and a capacitor now the important thing about the difference is this that a resistance will dissipate power as current flows through it and the power loss in a resistor is given by the voltage times the current or you can sometimes using Ohm's law you can say that that equals I squared R we've dealt with this in previous videos on electricity so there is a power loss in resistance but there is no power loss in a perfect inductor or a perfect capacitor I say perfect because of course in practice there will be some element of resistance in both but in theory at any rate and in terms of exams you usually think in terms of the theory there is no resistance associated with these only a reactance consequently what happens with these two devices there is no power loss in reactance what you get is energy storage and dissipation the capacitor charges up and collect and as it charges up if it collects and stores energy and then as it discharges it releases that energy the inductance is building up a changing magnetic field which is a buildup of energy and then a dissipation of that energy so there's no power loss in these two devices only in the resistance if we measure the voltage drop across these devices firstly put our voltmeter across the resistance and get a voltage reading of V R which is the potential drop across and they measure the total voltage drop across the two reactances the inductor and the capacitor and that will give us the voltage drop the X let's say the total voltage of the AC supply is V you might be tempted to think that the total voltage is equal to the voltage dropped across the resistor plus the voltage dropped across the reactants and again you would be wrong because you would have forgotten the fact that we've got phasor diagrams to have to deal with so once again we have to draw our phasor diagram we will have VX along the imaginary axis and VR along the real axis and so again you have to do your Pythagoras and you will find that the net voltage which is the root mean square voltage is given by the square root of this term squared V R squared plus this term squared VX squared so you don't just add them together you have to do the Pythagoras now what is the power loss in this circuit well we've already said the only place you lose power at least in theory is in the resistance so consequently the power loss is going to be V R times the current flowing through the circuit so the power loss and we're going to do the average power loss through the resistance R because that's the only place you lose power is going to be the root mean square of the current times the voltage across R but if you look at this diagram here you'll see that V R over V RMS is equal to the cosine of the angle Phi V R a VMs harness is the cosine of the angle Phi and that means that VR is equal to V RMS time's the cosine of the angle Phi so we can substitute this VR I should have used the same R this R VR for here and that means that we get that the power loss across the resistance is equal to I RMS to this term here times VR which is V RMS times the cosine of the angle Phi so if you know the root means square of the current and the root mean square of the voltage multiply that by the cosine of the phase angle and that gives you the power loss in the resistance and how do you find Phi well if you know the voltage drop across the reactants is and the voltage drop across the resistor then you will readily see that VX / vr is the tangent of Phi VX divided by V R is equal to the tangent of the angle Phi so if you know VX and V R divide one by the other and that's the tangent of the phase angle now finally before I do some worked examples because people have been asking me to do some examples but before I do that I just want to look at what's called resonance and you get resonance in LRC circuits that's where all three of the components apply and the usefulness of this just to tell you where we're going is this is precisely how you tune in your radio and your television so what you've got is what's called an LRC circuit so we're going to have an alternating current supply going through a resistor an inductor and a capacitor and usually we make that a variable capacitor and the way you make a capacitor variable is you've got two plates I'm using my hands to represent the two plates of the capacitor now I turn it h on and as you turn the dial on your radio what you're effectively doing is turning one plate against the other and therefore the degree of overlap the amount of area or that is common to both reduces changes and that of course changes the capacitance so the way you change a capacitor is you just take you swivel one of the plates so that the amount of area that is common to both reduces or increases as you turn it and that therefore becomes a variable capacitor so let's just remind ourselves that the total impedance of this circuit is going to be the square root of the resistance squared plus the capacitance squared and in that meat case we've got that the total impedance is the square root of R squared plus XL minus XC all squared and that is the square root of R squared plus well the capacitance of elfs of XL as we've said is Omega L minus the capacitance XC is 1 over Omega C all squared now think about this term here this term squared will always be positive because any term squared will be positive it matters not whether Omega L e is bigger than or smaller that 1 over Omega C it doesn't matter whether this term is positive or negative when you square it it will come out positive so Z will always be the square root of R squared plus a positive term that means that Z the impedance will be at its minimum when this term is zero and that means if the impedance is at its minimum the current will be at its maximum because V equals IR B equals I said when Z is small I will be large so when this term is zero you get the minimum impedance maximum current so what does it mean for Omega L to equal 1 over Omega C because if that's the case then this term will be zero well multiply this out and you get that Omega squared LC equals 1 which means that Omega is equal to one over the square root of LC which means since Omega is two pi F it means that the resonant frequency the frequency at which this term equals zero and therefore the impedance is minimized and therefore the current is maximized that frequency is going to be one over the square root of two pi so it 1 over 2 pi times the square root of LC and if you draw a picture showing the current against the varying frequency what you will find is that it will suddenly shoot up when you get to the resonant frequency because that's the point at which the impedance is minimized so the current is maximized and this is typically what you're doing your radio is picking up all sorts of frequencies but only at the frequency that is as it were maximized where the current is maximized because you have achieved a minimum Zed by making this term equal to zero it is the current for that frequency which is maximized and that's why you can hear that station on your radio all the other frequencies are down at minimal levels so you don't get to hear them but if you change the capacitor then you will change the point at which the frequency at which this term becomes zero and so if you change the capacitor the resonant frequency changes to hear a second resonant frequency so this may be one of your favorite charile channels and that may be your second favorite channel and by varying the capacitor you can change the frequency at which the signal will be maximized that's how essentially radio and television tuners work now I said I'd do a couple of examples and which I dread doing because I'm terrified I shall make a mistakes if I do you'll tell me but let's take the situation where we have an alternating current and a circuit which is simply a resistance and capacitance and the values are that the resistance is 100 ohms the capacitor is 10 microfarads the V RMS is 250 volts which means it's kind of broadly the united kingdom power of supply that comes from the wall the frequency in the united kingdom is 50 hertz and what we want to know is what is the root mean square current that is flowing in the circuit well we remind ourselves that V RMS is equal to I RMS times Z that is essentially the Ohm's law for impedance we also know that the impedance has a value of the square root of R squared plus the reactance squared reactance of course is the capacitor in this case and we know that XC is 1 over 2 pi F times C we know F as 50 Hertz we know C that's 10 times 10 to the minus 6 fan out remember you have to convert the micro farad into 10 to the minus 6 if you put that all in to a calculator you should get that this is the equivalent of 318 OHS remember reactance has the same dimensions as resistance ohms and then Zed using this formula here will equal the square root of R squared R remember 100 ohms so R squared is 10,000 plus 318 squared which will be 101 3 2 1 approximately and if you take the square root of that you should get 333 owns so the net impedance of the two devices is 333 ohms which means that the root means square of the current which is what we want is V RMS divided by Z this is of course affinity Ohm's law I equals V over Z and that is equal to 250 which is the root mean square voltage divided by 333 which is the net impedance and that comes to approximately naught point seven five amps which is what we were asked for but let's just go a little bit further and think about the voltage traveling through the across the resistance well the voltage across the resistance will simply be the root mean square current times the value of the resistance well that's nor point seven five is the current we just calculated that times a hundred ohms because that's the value of the resistance and that's seventy five volts so if you measure the voltage across the resistance that's what you'd measure the voltage across the capacitor is going to be i rms times the reactance of the capacitor which is again Noor point seven five because that's what we calculated times 318 because that remember was the value of the reactance of the capacitor and that comes to two hundred and thirty eight point five volts and you might be tempted to add the two together and get three hundred and thirteen point five volts and say that can't be right because the voltage is 250 volts and of course you've made the mistake of adding together these two voltages you can't do that remember because they're on different axes ones on the imaginary axis ones on the real axis so you have to plot the phasor diagram and that means that V is going to be equal to the square root of the R squared plus VX squared which is the square root of the r-squared 75 squared plus the C squared which is two hundred and thirty eight point five all squared and if you take the square root of that you should get two hundred and fifty volts and now let's look at an LRC circuit we've got a resistance and inductance and a capacitor all in the same circuit I can tell you that the resistance is 25 ohms the capacitance is 50 micro farad's the inductance is 200 milli henries the RMS voltage again is 250 volts and the frequency is 50 Hertz what you'd get from the UK mains broadly and what we want to know firstly is the total impedance of the circuit secondly we want to know what the I RMS the root mean square current is and thirdly we want to know what is the phase angle what is the either the lagging or leading of the voltage by the current well we remind ourselves that the impedance is always going to be the square root of R squared plus the net effect of the reactance squared which in this case will be R squared plus XL minus XC squared and that's going to be the square root of R squared plus 2 pi F L because Omega L is the reactance of the inductor minus 1 over 2 pi F C I the ie 1 over Omega C all squared and if you substitute in here arc remember was 25 so the 25 squared is 625 plus if you work this out you get 62 if you work this out you get sick see three all squared so 62 minus 63 you're almost at resonance this is all no zero and that will come out to the square root of 626 which is broadly 25 ohms so the net impedance of the entire circuit is 25 ohms we know that V RMS is equal to I RMS times Z and of course we know that V RMS is 250 volts we've just calculated Z so obviously I RMS is going to be V RMS 250 divided by Z which is 25 that means that I RMS is equal to 10 amps which is a high current and the tangent of the angle the phase angle is given as you know by x over R and the net effect of X is 60 to minus 63 which is minus 1 over r well the resistance was 25 ohms and so you've got an at an angle which is a negative angle which means that the will lead the voltage or the voltage will lag behind the current it has a capacitive effect and Phi is minus 2 point 3 degrees so it's a very small angle so the current will barely lag will barely lead the voltage at all but only just the the current will be just ahead of the voltage in this particular circuit and just to show you what I mean by resonance remember in that circuit we had a total current of 10 amps suppose that the frequency instead of being 50 Hertz which is what I said it was suppose it were a hundred Earth's then you would have that Z is equal to the square root of R squared plus two pi F L minus one over two pi FC whole squared but this time you would use F is a hundred instead of F is fifty and so you would get that the total impedance would be R squared which is 625 Plus this term with F as 100 will now come out at 125 and this term will come out at 31 so now you get 125 minus 31 squared and that will give you a total impedance if you calculated of 97 ohms and now we can still say that V RMS is equal to I RMS times Z the total value of the impedance and that means that I RMS is V which is of course still 250 divided by Z which is 97 so I RMS is going to be equal to 250 divided by 97 which is about 2.5 amps so when the frequency was 50 Hertz as in the previous page the current was 10 amps when the frequency goes to a hundred Hertz the current Falls to 2.5 amps it's moved quite a long way from resonance 2.5 amps of course is still a quarter of 10 amps so if you were tuning and that you get quite a lot of interference the skill with LRC circuit is to adjust the values of L R and C such that at resonance you get a very high peak of current but everywhere else you get hardly any current at all

Channel: DrPhysicsA

Views: 327,110

Rating: 4.9322457 out of 5

Keywords: Resistance, Reactance, Impedance, Capacitor, Inductor, Electricity, Resonance, LRC circuit, AC, DC, voltage, current, frequency, phasor, Ohms Law

Id: FEERuJlwBxE

Length: 59min 37sec (3577 seconds)

Published: Mon Apr 15 2013