Welcome back to another episode of We Are Being Transformed, a podcast where we explore the myriad of ways that people are transformed and in turn shape the culture around them, whether that be through ritual, lore, religion, etc. And today I just want to give a trigger warning. The content in this episode will be quite graphic at times. We discuss acts of sexual violence and human trafficking at moments. So if you or someone you know is a victim of sexual violence, please contact RAINN. You can contact them at 1-800-656-4673. Similarly, if you or someone you know is a victim of human or sex trafficking, please contact the National Human Trafficking hotline at 1-888-373-7888. And we will include the links in the description. Hello. So with that being said, I would like to welcome back to the show the one, the only Dr. Celine Lilly. How are you tonight? I'm doing well. Thanks Jason. How are you? I am fantastic. Honored to have you back on the show. Awesome. So in our last discussions, we really talked and got into the meat of the matter with Roman self-definition and the violence that entailed through mapping that masculine self-definition. We also, in another episode, we tackled how the Nag Hammadi texts like On the Origin of the World, Reality of the Rulers, and The Secret Book of John really tended to act as a sort of cultural critique. What these texts have in common, just academically how they're grouped together, they're usually considered not, sorry, cut, Neil. Okay, so three, two, one, okay, start here. So these texts, they're usually grouped together in academia as quote unquote, Scythian texts. They share a certain lore, certain aspects of the mythology, if you will. There's a story as Dylan Burns would say with all this that's going on. So I think what's interesting is that even though you have a very, how shall I say, a very complex cosmology here, cosmogony, it's not just like these patriarchal figures that are the figurehead of things. There are, for lack of a better term, Eve, Noria, Sophia are more or less, in a very real sense, savior figures. So I was wondering if you could expand upon that a little bit. Yeah, it's really interesting to see the ways, and especially just even thinking about, they get called Scythian literature. And like Scythian basically doesn't show, I mean, his name is mentioned once in reality of the rulers. And I'm trying to think of these mentioned at all on the origin of the world. Like Scythian is so minor there, he plays a little bit larger role in secret revelation of John. But in these two texts in particular, it's really, really Eve. And then obviously in reality of the rulers is the one where Noria shows up. And I think one of the things that I found really interesting. So A, so yes, just this fact that we have these female, these female figures that really form this conglomerate of these shifting names of, you know, the wisdom Sophia, Zoe life, Eve. The beast or the serpent who's also called the instructor is part of this kind of female androgynous, like it's an interesting thing to so much of the time. In these ancient texts, you get androgynous figures, but they tend towards the masculine. And here we get these androgynous figures who kind of tend towards the feminine. And particularly is named that in on the origin of the world. And so it's just a real, you know, in thinking about it, it excites me. There's so much that I could say about it. It's hard to kind of pick one track. But I guess maybe the two things that I'd say that are really interesting are in on the origin of the world in particular, when the rulers make their atom to try and seduce her.  And first, he's given a soul.  And Eve says to Eve says to him, you know, Adam arise stand. And the text actually says, and her word became a work. So much like the God at the beginning of Genesis, her words are actually efficacious. And she's able to save Adam from kind of from from this spiritless form that he has at this point. She's also the one technically who, you know, again, and what a melgoma of this, it's it's interesting. But he is also saved through the eating of the fruit through these this amalgam of figures, the tree, the serpent, the beast, the instructor, the the fleshy Eve. And also then to think about just these resonances with you know, with the Christ figure, if these indeed are Christian stories, which I think there's so much more we could say about that. But, you know, you have a penetrated, feminized savior figure in Jesus. And that was one of the things about crucifixion is that it was emasculating because it was penetrative. And here you have the same thing through sexual violence, and that, again, you know, these children end up becoming kind of the saviors of humanity, specifically with, with Noria, who we can say a lot more about. Yeah, and even just getting back to Eve for a second, I mean, it's not just this came out of thin air. I mean, if if you read the Septuagint and the Septuagint, which is the Greek version of the Old Testament, the Hebrew Bible, you'll notice that in the first three chapters, she's not named Eve. She's called Zoe. She's so this is not something that they're just kind of making up. It's something that they're they're gleaning from the text, and they're interpreting in their own way, which I, you know, I just find that fascinating. You know, I love that. But yeah, we could talk more about that, but maybe that's for another time. What I really wanted to ask you about was what can we kind of gather what little we can what can we gather about the communities who circulated these texts from these markers of these strong importance, female savior figures, like if anything at all? I mean, I think there, you know, I think there's so much speculation that we could there's so much that we could speculate. What can what can we know for sure? Who knows? But I think, you know, two, I was gonna say two, but there's probably there's probably three or more that I'll list. But some of the things that I think we can notice are that these are folks who are interested in they are interested in gender. There is a bunch of gender fluidity that's going on in these texts. Again, something, you know, in On the Origin of the World thinking about they have there's like this hymn to eve that mirrors thunder perfect mind.  And so noticing noticing these this interesting playfulness around this that happens there. Noticing the ways in which they they talk about gender and violence that it really upends this hierarchical model that we have in the in the Roman system. And so there is there are they're definitely interested in having what they would consider. And I don't I don't necessarily want to speculate the scope of this, but they are interested in justice. Despite the fact that many folks think that they're only interested in kind of the people who wrote these texts are only interested in evacuating this world, you know, the sparks get liberated back, you know, into the heavens. I think they're very interested in what happens in this world. And although I do think that they may when I look at the ends of some of the texts where there seems to not be there seems to not be a lot of hope around the possibility of things shifting at times where the only thing that they can imagine is kind of a fiery ending. And yet at the same time, there does seem to be a real community feel in a lot of these texts that it's not about individuals, but particularly with the relationship even between all of these female figures that that help one another with even Adam that there is something very, very important that it's saying about relationship and partnership and how people should be with one another and treat treat one another. And so these are communities that that at least care about striving for these things. Well said. Well said. Thank you for that answer. You were mentioning Thunder Perfect Mind and it kind of brought to mind. There's a section in Reality of the Ruler is a very graphic scene where after assaulting Eve, these rulers, these archons quote unquote defile Eve's voice. And it reminded me of Thunder Perfect Mind and there's because there's a section in there. And one of my favorite lines in there and the character or the figure goes, I am hearing adequate for everyone and speaking that cannot be repressed. And that just really kind of stood out to me. I was wondering what you thought the significance of these archons trying to silence Eve by defiling the voice, but they can't at the end do it. So I was wondering your thoughts on that. I do wonder and again, this is the place where I have to say, I'm a modern person in the modern world. And so this may be anachronistic, but I do wonder if some of it is really about silencing her that these types of violence are supposed to be unspeakable to protect the people who perpetrate them.  And this is again, not part of this imperial system. And I think it's one of the things that I find really compelling about the way that Karen King talks about this, that somehow, you know, Sufi is this fallen character and yet, and yet she's the one that's constantly calling out these male power dynamics. And so there's something really, there's something, there's something really fascinating happening with this where, where, and I think that moments like that really point to this place of parody, critique, how we think about satire in the contemporary world, that those places where there are disjunctures where it's like, this doesn't make logical sense. Well, when you start to put it in these other contexts and notice that it's trying to poke fun at something, all of a sudden there are these other, you know, a myriad of other possibilities for interpretation. Right, absolutely. It's kind of like, I mean, even like something like 20 years ago, like my son can still look at like a trailer for airplane and find that funny, but he's not going to get all the, all the subtle satire. And they're the same thing if you're reading Aristophanes, right? You're not going to get all of the parody that's in Lizzie Strato, right?  So yeah, a little said, final question for you. Why is it important, do you think, for readers today to know and discover figures like Noria, Eve, and Sophia? I think that they continue to intervene into the binary gender dynamics that we have inherited in so many ways. I think, you know, though in, you know, in Jewish tradition, God has no image, we have really managed to image God as a male, as a particular type of father figure that I actually do think also the early Christian texts subvert kind of what this father figure is supposed to be in certain ways. But again, this gets co-opted, gets co-opted by the empire. So I think there are pieces there that are so, that are just so, so very important about having this myriad of images that different people can relate to and see themselves in the story. The one thing I do think I just really want to say about Noria before we go is there have been, and I really understand these critiques, and it's something that I wrestle with, but this idea that as the rulers are threatening her, she cries out to the divine realm and O'Lelleth steps in and saves her. And there have been interpretations of this kind of saying, you know, once again, it's the male coming in to save the day for this female figure. And I just keep wondering, like, is there a way to think about this? And I do, I talk about this in my book as well, like, why is it that, why is the bravery of crying out for help and actually being met in that a bad thing? And I think this is one of the ways in which we, I myself do this too, tend to map our hyper individualistic society onto these constructs. And I just think that, you know, anybody, if there is a possibility of intervening on your own behalf in violence that's coming at you, by all means cry out for help to whoever is there.  And I think the thing that I also just want to point out about this is that, you know, it does say when she's born, the text says, so this is reality of the rulers, that she is the virgin who the rulers don't defile. So these are two things that go hand in hand, but it's so important to remember that being a virgin in the ancient world isn't just about sex, it's also about kind of being able to be autonomous. So if we think about like Artemis and what Daphne was actually trying to do, she didn't want to get married because she wanted to run around with Artemis or Diana and she wanted to hunt and she wanted to be carefree and she didn't want to be yoked to a man in a family. So because that was a really difficult position for some folks in the ancient world. So just to really think about this expansive, the expansive possibilities around someone like Noria, who might have been an example on more levels than is just obvious in the text. Well said, well said. Thank you for that. So Dr. Lilly, this has been another amazing discussion. I hope to have you back soon. But in the meantime, before we part, where can people find you? Thanks very much, Susan. It's just been really great to be here with you again. And people can find me at the University of Colorado Boulder, where they can find me at the Wester Institute. So westerinstitute.org. And the Wester Institute has some courses coming out, correct? So we will have we are starting an academy that will be that will begin in fall of 2023. And they also have ongoing opportunities to hear folks like John Dominic Crossen, other scholars. We have seminars that we run where the public can actually listen in on scholars debating a myriad of different issues in contemporary scholarship. So please check them out on a bunch of levels. It's a really interesting organization that keeps trying to push kind of into new areas of scholarship. Amazing, amazing. So yes, I encourage you all to check that out. Another forward public facing opportunity for you to learn more about the subject if you're interested. I know I like to take advantage of these things when I can. So you should as well. So until next time, Dr. Lilly, it's been a pleasure. Thank you very much. We will see you next time. Thanks.


Hello and welcome to NewsCreek. The competition commission of India has ordered a probe against search engine giant Google. This was after it found that Google misused its market dominance to basically discriminate against rivals. To talk more about this, we have with us Prabir Prakash sir. Hello Prabir. So there are not too many details about this case. This is based on a Reuters report which has seen the document. But there is already a case from the European commission from last year where Google was fined five billion dollars on very similar charge. And the competition commission sources have indicated that the case was quite similar. So could you just take us through what exactly that case was and what its implications are in India? See without getting the details of the case, the question that always comes up with respect to monopolies or what is in the US called antitrust is that if you have a market position in which you are a monopoly of something, can we use this to try and bundle things where you are not a monopoly and thereby also make that into your monopoly. Now this if we look back on say Microsoft which used to be the dominant player in the PC market. At that time it used to do with the internet explorer what's being charged against Google now that Chrome is being bundled in a similar fashion with Android. We'll come back to this issue. So what it means is you have to buy windows at that time PC platforms really Mac or Windows and the IBM PC compatible market was a much bigger one. So everybody who bought IBM PC compatible bought Microsoft Windows to run their machines and then a whole bunch of things came along with it. At the time there are other also shall we say essentially web browsers and internet explorer was a web browser. Other web browsers which are more dominant at the time had to give way to internet explorer mainly because of the Windows dominance. So that became the preeminent in that sense web browser. Android case is very similar that exactly the charge against internet explorer is the charges come out against Google. Microsoft had to debundle in that sense internet explorer that you could use other web browsers Firefox later group Chrome but at that time Firefox was a very popular web browser. So what Google has done is something very similar the Android system which is really based on Linux. So it was in fact a Linux kernel which was used. So this is open source product. So Google could not privatize the Android version could make my money out of that. But what Google did is it created dependencies on the Android version it was selling and said if you use preloaded Android then you have to also load Chrome as the web browser. You have also to put Google search as the basic search engine. So it was using it Android monopoly that it had and let's face it. The Android today is the dominant shall we say the smartphone operating system and which is the way the next play is of the internet. So this monopoly that Google has over the Android ecosystem is what gives it also the monopoly over the browser and other things. Now this is not the only play that Google has in fact the much bigger play is that it comes also with Google Play Store. So all the Android phones that you see in the world have Google Play Store and the Google Play Store forces or makes it very easy for the app developers to integrate with the Android ecosystem of Google but via the Apple Play Store sorry the Google Play Store and the Google Play Store is on the cloud that's a Google cloud services. Therefore the second dependence apart from the Android operating system itself and that dependency can be changed as what the European Commission is doing or what the competition commission in the MF was Google to do but the dependency on the Apple the dependency on the Google Play Store will still remain and that is the key problem that all Android ecosystems have and this essentially is also the problem by the way Huawei has at the moment and they are taken out of the Google Play Store that what happens to all the Google apps all the other apps that run seamlessly with the Android system and what will happen to that is the basic threat that Huawei phones will have outside China and that's something that is also the issue over here. So far the antitrust regulators both in Europe and India their solution has of course been to levy quite substantial fines and Google is also responded by giving users an option to download other browsers of course and also make the other browsers default browsers in their individual phones but like you said it does not really solve the problem of the Play Store which is actually where the more substantial issue is. You see this is the shall we say how monopoly is operated. This is not something that we are aware of. This is how all monopolies have behaved in the past if I have a monopoly in one sector how can I use it to have monopoly in other sectors. In this case it was really the operating system monopoly which paved the way by Microsoft earlier with the Windows operating system with Linux giving it competition and also the fact that Apple has also become a major player again in the PC market that is a weakening monopoly but still let's face it Windows or Microsoft one of the major revenue still of Microsoft is to sale of software. I think 60% of their revenue comes from Windows, Word and other services other software that they provide leaving that out for the time being if we come back to what Apple sorry Apple has done shall we say not Google but what Apple has done they have built their own ecosystem completely. Now anybody who wants to challenge Google today will have to consider how to build their own ecosystem completely new ecosystem otherwise Google's monopoly is going to remain not only in search engine not only on things like Google map not only on Gmail all of these things but the fact that Android is the de facto operating system today in India for instance 98% of phones apparently use Android. So the question that remains is do we have an alternative to the Android ecosystem not just the Android phone. You see the Android based operating system can be very easily changed with an alternative Linux kernel. Android itself actually was bought by Google at one point of time it was based on a Linux kernel and it cannot be made close source because it was always based on a open free and open source software. So that kernel is open that can be forked by anybody but the real problem is today the app based provisions because without the apps the phone is in some sense very very of limited use because smartphones use a lot of apps and at some of the apps are standard there you have the problem. So yes you can get shall we say the Google's Chrome you can use Firefox instead you can get other things out of the dependency that is there on the phone or which the phone manufacturers have had supposed to submit to Google if they want the Android as a pre shall we say pre's already downloaded stuff already when you order the phone it comes along with the phone. So if you do all of that even then that Android based dependency is really on the cloud. Now one must understand what happens about Google. Google is quite willing to pay the future five billion dollars in fines for having got Android into the preeminent position of the European market and say the Indian company the Indian arm or Google itself is an American company is fined by the Competition Commission say X number of dollars. Now having captured 98% of the Indian market Google might be quite willing to pay the fine as a capital investment shall we say in the monopoly it has created. So this fines they're quite happy to pay the fines in lieu of the monopoly they have got for five, eight, six years whatever the duration might be and that's a long game that Google is playing and many jurisdiction there will be no fines. So even the fines in some sense are expenditure which Google is willing to undertake in order to achieve the monopoly and would you see the possibility of say some a company like Huawei being able to actually generate its own ecosystem as an alternative because that is one of the options they are considering to build a Huawei Play Store which has its own series of apps. You know the challenge for the world is that if we don't want to succumb to companies like Google and so on internationally what are the options they will have and this is true for Chinese obviously, Russia little less so but definitely at any point of time any of the companies can be sanctioned. So these are those who already have strategic competition in the United States and you can be easily taken out of the US shall we say domains in by use of sanctions. So suddenly all the US products becomes out of bounds for it. So what do they do? So I think they have no alternative but to do what we India India called self reliance for a long time or shall we say technological sovereignty again which means you need to have options by which whether it is hardware or software you should be able to be completely self reliant. You may not be so but you have to aim to be so. So in case that you have an existential crisis like the Huawei has then you can say okay what is the gap now I need to cover in order to become independent and I think that is the challenge that to for instance Huawei has it's they have been looking at different things they have been looking at say companies which have now almost died and whether they have an app store which they can they can take over they're looking at Russians what is the Russian apps which are there. So they are on the work hunt for a quick way to create their play store equivalent and let's let's see there is there are two precedents to this so it's not that difficult to do either but big bucks. So one is Apple which has built its Apple store by which you can get the apps from Apple they integrate seamlessly with Apple because the hardware is all the same. So in fact the integration is even easier because the target hardware is not as varied and as shall be said different from each other as the Google ecosystem has and the second is Amazon Fire Stick for instance that also seems to have built a whole bunch of applications or apps which are independent of the Google Play Store and therefore these big companies are already thinking ahead that they cannot afford to be dependent on Google and this is the route that Huawei or any other company has to travel. It's clear that Huawei is being forced to travel this route they didn't want to they would like to compete in the European market and in the third world market with products which people don't see it as dissimilar which also have the familiar Google stuff on it because Google still controls a large part of the search engine, the Google Maps, the Gmail and also the Google Docs now etc. including the search engine which is where the contention has come out. So I think all of this we have to see how it goes but we are seeing a long-term battle against Google's monopolies or the monopolies of big American companies because it's clear that we are today entering into global trade wars and what a news click we have earlier says also Technological Wars or Tech Wars. So I think that's where it is. It's going to be a rocky road. I don't think the globalized United World which was the sort of the slogans it is all the slogans which was given is actually working at the moment but behind this is a really an attempt to reassert dominance of certain countries companies or through some companies certain countries asserting the dominance over rest of the world which is where the competition commissions are now waking up to the threats. The Europe is waking up to threats of American digital companies taking over their space and in fact the only ones today who are able to challenge it is because they have a huge market internal market is the Chinese companies to a lesser extent Russia because they have good technological strength but they really do not have as big obviously as bigger market as the Chinese have the population of Russia small Russian is not a popular language internationally so it doesn't have the numbers required to really create a large market base. So we have to see how Russia and China whether they will come together strategically even on the digital tech sector as it were. Thank you. That's all we have time for today. Keep watching news click.


We do a lot of attack missions here, but reconnaissance is a mission that we have to do. We work on the Avenger weapon system. We have missiles, we shoot down aircraft coming in. And from that we built an entire scenario around it that mimicked our real-life situation. Today we were told to go out and act as an enemy for the Apaches. We learned how difficult it is to identify vehicles in that forested, hilled terrain. Actually, I really wanted to be more in the medical field and now I really want to be in aviation.


Hi everyone, Chris here from IELTS Advantage with another mini course. So we haven't done one of these mini courses in a few weeks and we're very very popular and one that a lot of you have been Asking about is IELTS Vocabulary. So of all the things that you could improve Vocabulary is probably the one that it's going to give you the biggest bang for your buck By that I mean it it has a force multiplying effect and it's really going to help you in many many many different areas So this week we're going to do a three-day mini course showing you how to improve your vocabulary So first of all is vocabulary important. Is it worth your time spending? Improving your vocabulary. So let's look at writing and speaking first. So it's 25% of your total marks It's obviously very very important But for writing and speaking helps you express meaning if you can't express meaning and get the meaning across to the examiner Either through speaking or writing then you're going to be in all sorts of trouble and then the wider of vocabulary You have the better you're able to express meaning and express yourself and the better you are able to Discuss the different topics either in the speaking test or the writing test For listening and reading the wider your vocabulary the more you're going to be able to understand I'm sure that you've done Maybe reading and listening practice tests and you just have no idea what's happening because there are lots of words You don't understand and it also helps you locate the correct answer It's good to for skimming and scanning and it helps you decide the correct answer as well imagine you have you know a multiple choice question with three different possible answers with three similar meanings having a knowing that the meaning of those words and having a wide vocabulary is going to help you with that and Then being aware of synonyms and all of those things is going to really really help you as well So is vocabulary important it is extremely important and I'm not saying that you should just focus on vocabulary You need to focus on everything, but if you're not focusing on vocabulary and you're doing some of the the bad things That I'm going to show you in this video Then you're going to get into a lot of trouble and if you are learning vocabulary the wrong way It's actually going to lower your score So many of you are doing things that are not helping you at all They're actually lowering your score and that's what we're going to look at in today's video So if you need any extra help today You can go to our vocabulary page on our website IELTSadvantage.com If there's anything you're unsure of or you need some extra help and we have a lot of free resources on there You can check those out and my email is there as well If you need any extra help with any other areas of the IELTS test So is vocabulary important let me demonstrate with this sentence So if I said to you meet me at 11 p.m Next to the old cinema you should know exactly what I mean. There's no problem with meaning there But imagine I take out Some of the more functional words So if I take out this I take out preposition I take out an article So I take out the functional the grammar words if you like and I said meet me 11 p.m Next old cinema Would you understand what I mean if I said those just those words to you meet me 11 p.m Next old cinema sure you would understand exactly what I mean because these words have meaning But let's try it the opposite way around. Let's remove the the meaning words Let's remove these words and just leave the more functional words in let's get rid of that Let's get rid of that. Let's get rid of that Let's just leave the prepositions and the articles in and if I said to you add to the Would you understand what I mean? You would have no idea because all the meaning is gone So when I take out the words when I take out the important words the vocabulary Then you're going to really really struggle So this is to demonstrate that not only is it important for Expressing meaning when you're writing and speaking, but if you did not understand those words Imagine you had no idea what that meant that meant That meant or that meant you would have no idea how to how to understand it if you were doing the reading and the listening test So it's extremely important that you are improving your vocabulary Every single day and it's extremely important that you're not doing the wrong thing So lesson one today, we're going to look at myths and BS I'm sure you know what BS is most of the advice out there related to vocabulary is BS And we're going to look at how that will actually lower your score And we're going to look at the the two most popular YouTube videos on IELTS vocabulary and how they are actually BS Lesson two we're going to show you what actual band seven eight and nine vocabulary looks like So we're going to show you some essays and because a lot of people talk about high level vocabulary in simple words and complex words But they don't really know what those what those are or what those look like So we're going to show you that on Wednesday and then on Friday in lesson three We're going to give you a step-by-step vocabulary improvement plan so that you can learn Vocabulary effectively and more importantly use vocabulary effectively on test day. So let's look at the myths and the BS so if you enter IELTS vocabulary into YouTube and you sort by page views or video views the two most popular Videos on YouTube related to IELTS vocabulary are these use these five words to score high and IELTS speaking 2.4 million views so 2.4 million people have watched that video and Memorize 1800 words in 18 hours nearly a million people have watched this now first of all do not go to these videos and Be nasty to the people who made them or leave nasty comments or say anything negative This is not about being negative and it's not about bashing people. It's not about that It's about giving you guys the right information and the two most popular videos on IELTS vocabulary and YouTube Perfectly illustrate the two main problems when it comes to learning vocabulary So let's look at the first one use these five words to score high in the IELTS speaking test So if I was speaking to the person who made this video I would simply ask them these questions because what they are saying is learn five words And you'll get a high score on the IELTS speaking test So what I would say to them is number one is the examiner of fool Is the examiner an idiot? I don't think they are I think it's their job to do speaking tests day in day out and are you going to fool them with five special words so imagine this imagine you go into the speaking test and Your vocabulary is okay Let's say it's at a band five band six level and then you say these five words Is the examiner are going to be fooled into thinking you have an amazing vocabulary? No, because they're going to be basing your vocabulary score on all of the words you use not just the five words So it's just a basic misunderstanding of how the examiners are thinking about your speaking score Will you use more than five words? Hopefully all right hopefully in the IELTS speaking test You're going to use more than five words and you're being judged on all of the words you're using not five So is it a good idea to just learn five good words? No, it would be a silly idea Will you be asked about a range of different topics? Yes, and the examiner will be thinking and listening for topic-specific words Are you able to use topic-specific words for a range of different topics? Just learning five words is not going to help you with that and it's saying a word evidence of a wide vocabulary No memorizing a word just thinking about a word and using a word is not evidence that you have a wide-ranging Vocabulary you need to know how to use a range of different words not just five words So two point four million people who follow this video and if you look at the comments Everybody is saying thank you and how amazing it is because people are desperate They really want some sort of magic system That's going to mean that they don't have to do any work and get the score that they need and that's just BS Because there are no high-level words. There are no magic words. There are no diamond words There are no golden words band-line words or impressive words if you are watching videos that talk about these then just switch them off and They are not showing you the right way to do things If you are at a school and I've seen this many times where schools will get a booklet or like Like they'll create their own textbook and call it like diamond words or band-line words and give it to their students It's just a list of words That school is more interested in your money than there are in your IELTS results. There are Appropriate words. There are accurate words and that's it. That's the only way to think about it.  Did you go into that conversation and think? How many diamond words can I use or how many high-level words can I use? No, you just use the words that were appropriate and hopefully you use them accurately And those are the only two things that you think about when you're speaking in your native language Or you're writing in your native language So why do you think this is any different when you're doing IELTS test? Really comes from people looking for an easy way out a short cut a magic solution Because IELTS is very hard and it's a very high-stakes test People are pretty desperate and they will listen to people who are basically lying to them If somebody's talking about these things that they're lying to you. They're they're marketing to you They're not interested in helping you And this is a very good quote.  I can take a student who is at a band five and I can give them five words and Even teach them how to use those five words perfectly, but they're still a band five student They're not going to move from a band five to a band seven until their whole vocabulary Moves up to that level teaching them like five words is like putting lipstick on a pig Still a pig and by the way Do not again do not go to the person who made that video and start saying nasty things to them That's not what it's about and it's about helping you guys as much as possible so that one is Nonsense all right. How about this one memorize 1800 words in 18 hours? So you can probably tell from the title that this might not be that reliable. So Is learning a list of words a good idea? So ignore the fact that this this video is about learning 1800 words in 18 hours Most of the advice out there most of the schools out there are teaching IELTS vocabulary in this way Learning some sort of list of words. So either a list of words with the meanings in English or a list of words with the translations in it Does this actually help? So these are the questions that I would ask those those people making those videos Number one, is it a memorization test? Is IELTS a memorization test? Are they testing memorization? No, it's a language test. You cannot improve your language ability You cannot improve your ability to produce language and discuss topics by memorizing stuff It's just not a memorization test. It's like trying to put a square peg into a round hole Do you have to use the word? Yes, the key there is use There's a massive difference between learning a list of words and learning the meanings and being able to use those words You have to understand way more than that for example spelling pronunciation style use grammar collocations None of these are taught When you learn a list of words because you're learning those words out of context You're not actually learning them in the way that that most people learn new vocabulary my son is four years old and I read to him every night one of the reasons why I read to him every night is to improve his vocabulary What I don't do is go home and say Tom. Here's a list of ten words to read Please memorize these and I've even seen this with some kids like they should some parents show their kids these like Videos that just say like duck ball like trying to teach them English like that's not really helping them and Does it take time to learn vocabulary? Yes, most people Many many studies academic studies have shown most people can only learn between 10 and 20 words a day Some people even fewer than that. Do you think you could learn 1800 words in 18 hours? No, and why do people believe that again? They're desperate. They want a shortcut. They want an easy way of doing things and Another quote a little knowledge is a dangerous thing like imagine that I got this camera I could Take this camera and give it to a stranger walking past my office and say watch one YouTube video on how to photography like they could get that little bit of knowledge and Would they be able to then become a professional photographer? No Learning lists of words is a dangerous thing. A little bit of knowledge is a very very dangerous thing So the top three ways to summarize the top three ways to lower your score That's the whole reason for this first video if you're doing these things you are lowering your score number one focus on a small number of high-level or band nine or High-level words or band diamond words or whatever your school calls them is a terrible idea You are not improving your vocabulary and you're probably using those words incorrectly Memorizing lists of words is a terrible idea because again, you're not using those words correctly You have to use the words not memorize them and number three focusing on speed Trying to do things as quickly as possible Trying to apply that to any other job that you're trying to do is doing things as quickly as possible A good way to do things not when you're trying to learn how to do things properly So instead of focusing on speed focus on trying to do things effectively So those are the myths those are the things that are BS and What we're going to look at in lesson 2 is a lot of you talk about band 7 band 8 band 9 vocabulary What does that actually look like? Well? We've already talked about in this video that there's no such thing really as band 9 vocabulary It's not like you look at an essay and every single word is like this There's like amazing vocabulary band 9 essays have some simple words some complex words some High-level words some lower-level words We're going to look at what that actually looks like and then on Friday We're going to look at our vocabulary improvement plan Hopefully you enjoyed that if you need any extra help or you want extra information on vocabulary Straightaway, you don't want to wait until Wednesday go to isle-to-advantages.com vocabulary you'll be able to find everything that you need there and you'll help with writing speaking and all for free and You'll find a lot of like Free PDFs and free gifts and stuff there All you have to do is just go there and have a look if you don't have to enter that you can just Google IELTS advantage or Google IELTS advantage vocabulary and Google will take you there if you need extra help if you want Some advice or you want to work with us or you want to help and some extra help you can send me an email and either myself or Ruth here will reply to you or one of our other team will reply to you and And they will get back to you probably within 24 hours We are very busy at the moment though a lot of people emailing us so might be tomorrow before we get back to you And hopefully you find that interesting Let me just check the comments To see if any of you guys have any questions because a lot of the stuff we talked about today It's probably the opposite to what you have been taught or what you've learned so let me Have a look first quit is there any book for vocabulary as I've already said most vocabulary books Are simply lists of vocabulary the best way to learn vocabulary is actually just reading English just reading books and there's no particular book that you want to read just read in English every day and we'll talk a lot about that on Friday when we give you the Vocabulary improvement plan a lot of you are asking for a book about IELTS or a book about vocabulary No, there aren't any there are some but it's not an effective way to improve your vocabulary See It's only showing me the last sort of 10 comments so can't read the other ones, but Muhammad says we need more of these videos go to our YouTube channel And you'll find a lot more of the videos or go to our website and you'll find a lot more videos Okay, so Thank you to everybody who's saying thank you. Sorry. I can only read Facebook will only show me the last like 10 comments and most of them are just thank you, but Thank you very much for saying thank you. Hopefully you find that lesson useful We're going to look at real Essays and talk about the band scores that they would get on Wednesday at the same time 12 p.m UK time and I'll see you then. Thank you very much guys And if you need any help feel free to get in touch. Bye. Bye


Hello everyone, welcome to another lecture of basic electrical circuits. In the previous lecture we looked at second order systems and their transient response and so on. What we just started to do at the end of the lecture was to determine sinusoidal steady state response of linear circuits. Essentially what it means is that we want to determine the steady state solution to an input like cos omega t. Of course there can be a phase shift it could be cos omega t plus phi. Now it is possible to substitute that in the differential equation and substitute the known form of the solution which is some combination of cos and sin and work out the answers. But it turns out there is a lot easier way and that is what I was describing. I first showed it with the case of a resistor which seems rather trivial but its importance becomes very clear when we use that analysis for circuits containing capacitors and inductors. So that is what we are going to do in this lecture. If you have any questions about the previous lecture please go ahead and ask and I will discuss that before going on with the topics of this lecture. So let me start with our lecture today. What we mean by steady state response is that the transient response has already died out. Now that is a topic for another set of lectures in itself but you have to also verify whether the transient response dies out. Now in this course what we will assume is that for all the circuits that we consider with real R cell and C's the transient response indeed will die down if you wait long enough. So after that what you evaluate will be the steady state response. So that is why it is also useful to calculate this because the total response always consists of the transient and steady state response. After a certain period of time for a class of circuits known as stable circuits the transient response will die out. So the total response will be equal to the steady state response. That is why it is useful to calculate this. If the transient response was always dominant there is not much point calculating the steady state response but it turns out that the steady state response will dominate in a lot of practical cases after a long time. How long is long depends on the time constants of the circuit. So now let me go back to the case with the capacitor. So what I mean by this is response to some stimulus of the form cos omega t. So now I will assume that just like I did at the end of the previous lecture that the voltage across the capacitor is cos omega t. So let me call this V1. I1 then will be minus omega c sin omega t. Now let me apply sin omega t. I will get a current I2 equals c times the derivative of V2 which is omega c cos omega t. Now clearly these are linear components the capacitors here is the same capacitor. So now if I apply some superposition of V1 and V2 I will get the same superposition of I1 and I2. I hope this part is clear that I take some superposition of V1 and V2. What do I mean by that? I will take some alpha 1 times V1 plus alpha 2 times V2. Then the total response will be alpha 1 times I1 plus alpha 2 times I2. So let me do that and the particular linear combination I will take will be this multiplied by 1 and this multiplied by j. So V will be cos omega t plus j sin omega t and the response to that will be the same superposition. I have to multiply this by 1 and multiply this by j. So I would get omega c times minus sin omega t plus j cos omega t. This will be the total current I. So far I have not done anything profound. I just for a single capacitor given a voltage I can easily find the current by differentiating the voltage and now I have also taken some particular superposition of voltages in two cases and the currents in two cases. Now the important thing is that from basics of complex numbers we know what this is. This is equal to exponential j omega t. So essentially this superposition is like taking a capacitor C and applying a voltage V equals exponential j omega t to it. Now previously I first applied cos omega t and then sin omega t and consider that superposition. Now what I will do is I will take this exponential j omega t and try to calculate the current directly. Now this also is trivial in case of a capacitor. It is C times the time derivative of exponential of j omega t which is basically j omega c exponential j omega t. It does not look like I have done anything significant here. We calculated the response to exponential j omega t directly by differentiating and also by taking superposition of cos and sin omega t. Now but the most important thing which I pointed out last week as well is that in this case remember V was exponential j omega t. So the current for this particular value of V turns out to be j omega c times V. Now of course this is not a general rule. The general rule is that I is C times the derivative of the voltage. But if the voltage is an exponential j omega t then you clearly see that the current is proportional to voltage. So this is a very very important thing and in fact there is a whole body of work on linear circuit analysis based on this. Now we will not go into the depth of this but I will only point out that this exponential j omega t if you apply to any linear system the steady state response will consist of exponential j omega t times something. So the output will be proportional to the input in this particular case. This is true even if you have a differential equation like you have capacitors and inductors in a circuit. So you will have the circuit described by a differential equation not an algebraic equation. Even then the output will be proportional to input as long as the input is exponential j omega t. So now the point is that this the relationship looks similar to that of a resistor. For a resistor if you apply any V let us say V r the current through the resistor is always proportional to V r. I r is G times V r or 1 over r times V r. Now for a capacitor also if the voltage is exponential j omega t the similar type of relationship holds. This j omega t c is analogous to conductance of a resistor. So the point is now we have earlier had complicated circuits using resistors and we were able to analyze them. We have used nodal analysis, mesh analysis etc to set up the equations and we can invert some matrix and get the solution. Whereas it was a little more complicated for circuits with capacitors and inductors we ended up with differential equations. But now what I am saying is as long as the excitation is exponential j omega t the response will also be exponential j omega t times something. So the relationship looks similar to that of a resistor and every technique that we have used for calculating quantities in a resistive circuit can be used here. Any trick we have all the circuit's theorems we had for resistive circuits is also useful for these circuits. Is this clear or there any questions about this? This is the most important thing. The algebra I have showed so far is rather trivial but the important thing to realize is that for exponential j omega t the response is exponential j omega t times something. In other words the response is simply proportional to the stimulus. Any questions about this? Now this exponential j omega t is some fictitious input I created by superposing cos and sin. My actual input is cos omega t. Now I have to find the response to that. Now it turns out that if you calculate the response to exponential j omega t finding the response to cos omega t is very easy. This is because we generate this exponential by taking cos plus j times sin. The response also is the response to cos omega t plus j times the response to sin omega t. In this particular set of calculations and this is true of any other calculation involving real circuits and real inputs all the coefficients everything will be real. The only place where this j or square root of minus 1 appears is in this when you take the superposition. So each of the responses will be real numbers and the total response to exponential j omega t is response to cos plus j times response to sin. So the response to cos omega t can be simply obtained by taking the real part of the total response and this is what we are interested in. Now this itself is some real quantity that means that it has 0 imaginary part and this part also is a real quantities. So the response to cos omega t can be just obtained as real part of response to exponential j omega t. So now that is the convenient part of it. The response to exponential j omega t is easy to calculate because you get this proportionality between v and i for every component including capacitors and inductors. I have showed it for the capacitor I will quickly show it for the inductor also. So the analysis of any circuit is like analyzing a circuit with resistors and you do that and to calculate response to cos omega t you simply calculate real part of the response to exponential j omega t. This is much easier than applying cos omega t to your circuit and finding the response. So that is why this is extremely widely used and this analysis is known as sinusoidal steady state analysis and this will also lead to phasor analysis. Any questions? Now of course you do not always want to calculate response to cos omega t. What I mean is the stimulus may not always be just cos omega t. It may be cos omega t plus phi or it could be sin omega t which can also be expressed as cos omega t minus phi by 2 radians. So in general you may have a phase shift but that is very easy to accommodate here because if you have exponential j omega t plus phi that will be the response to cos omega t plus phi and plus j times response to sin omega t plus phi. So the response to cos omega t plus phi would simply be the real part of response to exponential j omega t plus phi. So we will see examples of this. Any questions about the principle or the basic idea of sinusoidal steady state analysis? So there is a question about how do we visualize a complex resistance? The term is impedance it is not called resistance anymore I will come to that. Now what is the visualization for it? What it really means is that in a resistor if you apply cos omega t as a voltage the current will be cos omega t. There will be no phase shift. I mean a capacitor or an inductor or in general a circuit containing capacitors and inductors. If the stimulus is cos omega t the response will be cos omega t plus some phase shift. So the complex impedance essentially has two numbers the magnitude and phase.  So you need two numbers whereas in case of resistive circuits you only need to tell how much the amplitude is scaled because the phase is not going to be changed at all. I hope that is clear. So let me quickly do it for an inductor. If I have I 1 of cos omega t plus phi I will take cos omega t plus phi directly in this case forced into an inductor. What will be the voltage across this? The inductor value is L. What is the value of V 1? Please let me know it would be L time derivative of I 1 which is minus omega L sin omega t plus phi that is correct. And similarly if I have sin omega t plus phi the voltage V 2 would be L time derivative of I 2 which will be omega L cos omega t plus phi. So again if I form a new input which is I which is exponential j omega t plus phi this you can think of as being formed by I 1 times 1 plus I 2 times j. Voltage V will be L time derivative of I which is j omega L exponential j omega t plus phi. Now we can quickly verify whether what I said earlier is true this V is nothing but j omega L times cos omega t plus phi plus j sin omega t plus phi.  If you take the real part that is what we will get and also like I said the voltage will be j omega L times I when I is an exponential. So this works for inductors and capacitors and also here like I said this response to cos omega t is minus omega sin omega t and that is exactly the same as the real part of this one. So if you take the real part you will get minus omega c sin omega t. So in general if you want the response to cos omega t plus phi you find the response to exponential j omega t plus phi and take the real part. And the reason we do this seemingly round about thing is that the response to exponential j omega t plus phi is lot easier to calculate than for sin omega t or cos omega t. It does not involve any differentiation or integration or finding the or finding the solution to a differential equation. Now here when I did it I showed it by differentiation but you see that the voltage is just proportional to the current. So by finding the proportionality constant we will be able to solve it exactly as we did for resistors. So I hope that part is clear. So if for linear elements which is very important everything here uses linearity otherwise we could not have used superposition etcetera. If I is exponential j omega t we will be proportional to exponential j omega t. And this proportionality constant is in general a complex number. It is known as the impedance and it is in general a complex number. So if you take a resistance if I is exponential j omega t then V would be R times exponential j omega t. For C if I is exponential j omega t V would be 1 over j omega c exponential j omega t. This is the inverse of what we calculated. There we calculated i from V now we are calculating V from i as long as they are proportional it just becomes a reciprocal. And for l have j omega l exponential j omega t. So now the resistor of course was always proportional. So for a resistor V and i were always proportional and the proportionality constant was the resistance R. Now it is also proportional for C and l. So that is what I said and these numbers are nothing but the impedances of these components.  The impedance of a capacitor is 1 over j omega c and the impedance of an inductor is j omega l. So instead of going from i to V we can also go from V to i. So let us say if V is exponential j omega t for a linear element the current i will be some number times exponential j omega t. And that number naturally is also a complex number and it is denoted by y and that is known as the admittance and it is analogous to conductance of a resistor. And if we evaluate for these three components V is exponential j omega t then i will be quite easy to calculate i 1 by R exponential j omega t, j omega c exponential j omega t and 1 by j omega l exponential j omega t. These numbers 1 by R j omega c and 1 by j omega l are the admittances of these components ok. Is this clear? So now because of this proportionality relationship between V and i the analysis of any circuit reduces to that of resistive circuits. Any questions about this? Let us take a very simple example. What I want to calculate is the response of this circuit R c to V p cos omega t ok. Now I could go ahead and solve the differential equation but the easier way which is what I have been advertising all along is instead of this. So in this case I will get some V c of t this is what I want to calculate. I will go about it in a somewhat roundabout way but which is actually easier. So let me make it to be with general cos omega t plus phi and here it will be exponential j omega t plus phi ok. Now the point is that because the excitation is exponential j omega t plus phi the voltages and currents everywhere will be of the form exponential j omega t plus phi times some number ok. So I showed this for capacitors and inductors but it is true of any linear circuit. So then what I do is because I have exponentials everywhere I can use the proportionality relationship between V and i ok. Now in case of resistors the relationship is the same as what it always was V equals i R and in case of capacitors I will denote the capacitor by this box which stands for some impedance and the impedance of the capacitor is 1 by j omega c ok. So this is and the impedance of resistor is just R ok. So now because everything is like a resistor in the sense that the voltages and currents are proportional to each other. We analyze this circuit just like we analyze resistive circuits. The only difference is that when we had purely resistive circuits all the constants were real. Now we also have complex numbers as constants ok and we will interpret that properly. So let me call this Zc the impedance of the capacitor. So what will be the voltage here Vc what is this going to be? What is the Vc going to be in this circuit when I have exponential j omega t plus phi? Like I said you can analyze this exactly as you would analyze resistive circuits and that is because V and i are proportional to each other ok. I am sure you would be it would be very easy for you to analyze some circuit like this where I had R and then some R1 I asked you for the voltage here and this circuit is exactly the same except instead of R1 we have this impedance Zc ok. So please tell me the value of Vc what would be the value of Vc? So clearly you can use the voltage divider formula and this Vc is going to be Zc by Zc plus R times the applied voltage which is Vp exponential j omega t plus phi ok which can be further written as 1 by j omega c R plus 1 by j omega c Vp exponential j omega t plus phi which can be further simplified R plus 1 by j omega c. So 1 by 1 plus j omega c R Vp exponential j omega t plus phi ok this is ok. Now if I take the real part of this I will get Vc of t I will leave that to you to calculate it will be some sinusoid ok but if you take the real part you will get the response to Vc ok. Is this part clear? But it is not very difficult to do so let me do it and show it to you. So Vc of t in response to cos omega t plus phi is real part of ok. So in general for all these analysis you need to brush up your manipulation of complex numbers it is rather simple but you still need to know to use them fluently. So we will have first rationalize this part ok. So when you take the real part you will get one term from the multiplication of these two and another one from multiplication of those two ok and I will write out the answer right away which is that Vp times cross omega t plus omega C R sin omega t divided by 1 plus omega square C square R square ok. So this is what it is and you can always reduce this to single sinusoid with some phase shift. Is this ok? Any questions about any of this? In fact we will find a simpler way to write down Vc of t just in a moment. So the easier way to do that is response to exponential j omega t we saw was 1 by 1 plus j omega C R Vp exponential j omega t plus phi ok and in general it will be some number times exponential j omega t ok. So in this case I also have exponential j phi ok by the way when I say j omega t plus phi it j is multiplying phi as well ok. So everywhere I wrote j omega t plus phi without brackets but what it meant was that it is j times omega t plus phi ok the real part of that is cos of omega t plus phi that is what I had meant ok. So now this there will be some complex number multiplying this exponential j omega t which also it is easy if you write it in polar form ok that is let us say this number is rather let me do it only for this part of it can be written in the magnitude and phase form the magnitude of this is 1 over 1 plus omega C R square the square root of that and the phase of that is exponential some j theta where theta is minus tan inverse omega C R. So in general whatever complex numbers you have you write it in polar form ok. So if I call this complex number as h ok it can be written as the absolute value of h or the magnitude of h times exponential j angle of h ok. So once you have that the response becomes magnitude of h exponential j times angle of h times v p times exponential j omega t plus phi which of course can be written as magnitude of h times v p and exponential j omega t plus phi plus angle of h ok. So you can clearly see the role of the magnitude and phase earlier there was a question from Charmille about how do you interpret a complex impedance. Now in this case it is not an impedance but any complex number multiplying exponential j omega t should be interpreted like this ok. So the magnitude scales the amplitude of the sinusoid and the phase shifts the phase of the sinusoid because if I write it in this form taking the real part is quite trivial right.  So this modifies the magnitude and this modifies the phase ok. So and this is true for single elements also if you have let us say the voltage across an inductor being j omega l times the current through the inductor there is a modification of magnitude by omega l and modification of phase by 90 degrees ok. So you can see that this can be written as omega l exponential j phi by 2. So this v l becomes omega l times exponential j omega t plus phi by 2 and let me just for clarity say that the peak value of the current is I p then we will have I p here also and here. So you can see that the magnitude of the impedance omega l modifies the magnitude of the voltage or the amplitude of the voltage ok. So and then the phase of the impedance which is pi by 2 modifies the phase of the sinusoid ok. So if I take the real part of this I would get omega l I p which is the amplitude of the sinusoid of the voltage times exponential j omega t plus pi by 2 which is minus sin omega t ok. So I hope this is clear if not you just do a calculation with a couple of inductors and capacitors and it will become very clear ok. So if there are any questions on these I will take them now and now you see that everything it becomes as easy as analyzing a resistive circuit ok. Only thing is you have to handle complex numbers that is all and that is not very difficult. But what you should not lose out in all this algebra is the connection to reality which is that you are trying to find response to cos omega t plus phi that you do by finding the response to exponential j omega t plus phi and taking the real part. If you take the real part of that you will get some sinusoidal wave form which is the response to cos omega t plus phi ok. Any questions ok? So just to summarize and this implies analysis similar to analysis of resistive circuits and a response to cos omega t plus phi basically can be obtained by taking the real part of response to exponential j omega t plus phi and finally we should not forget why we did all of this where this is applicable and this is only for sinusoidal steady state. That means that you apply sinusoid which is cos omega t plus phi and then you find the steady state which is after the transients die out ok. I hope this part is clear. Now from here we it turns out we do not even have to write exponential j omega t everywhere because it appears everywhere in the in all the expressions. So we will omit that and the resulting stuff is what is known as phasor analysis ok. So let me take this RC circuit again and if I have I will just use exponential j omega t but whatever I say we will also apply when I have plus phi over there and I will just write out the solutions. You can verify these things yourselves here we saw the voltage was 1 over 1 plus j omega c r v p exponential j omega t and here it turns out it will be j omega c r by 1 plus j omega c r exponential j omega t ok. Now do not worry about the details what I want to point out is that if you have a loop ok where the input to the circuit. So I am assuming that we have a linear circuit and the inputs are of the form all inputs must be of the form exponential j omega t that is at the same frequency ok same omega if you have multiple inputs it is ok but they should all have the same frequency ok. So if you have multiple sinusoidal inputs of the same frequency the inputs all will be of this form ok there may be some multiplying constant but they will all be of this form ok. So then if you look at any loop every voltage across elements in a loop will have exponential j omega t right here I have illustrated this with exponential j omega t here there and there. Now this should be pretty clear because I said that any response in a linear circuit will be proportional to exponential j omega t if the input is exponential j omega t. So you take a loop and you have only applied exponential j omega t's so every branch voltage would consist only of exponential j omega t times something ok. So everything will be like this so the point of this is that what does KVL say Kirchhoff's voltage law it says that sum of voltages around the loop equals 0 ok. So let me call these voltage drops let us say V 1, V 2 in that polarity and V 3 ok V 1 plus V 2 plus V 3 is 0 in this loop this is the statement of KVL. Now I know that V 1 is something times exponential j omega t and V 2 is something else times exponential j omega t and V 3 is yet another thing times exponential j omega t and the sum is 0. Now because exponential j omega t is a common factor to all of these I can omit them and do my calculations only with these numbers that is I do not need exponential j omega t at all what I need are these numbers this, this and this. In fact I omitted VP here VP should be here as well ok. So in general these numbers multiplying exponential j omega t will be some complex numbers I will denote that by V 1 bar V 2 bar and V 3 bar V 1 V 2 V 3 bar will be some complex numbers and I can omit this multiplying factor exponential j omega t and write KVL only in terms of these complex multipliers of exponential j omega t. Now it is very clear that exactly the same holds for KCL as well because all currents will have exponential j omega t and KCL says that the sum of all currents entering a node or leaving a node is 0. Now because all currents contain exponential j omega t I can simply rewrite the currents in terms of the sum of the complex multiplying factors ok. So KVL which says V 1 of t plus V 2 of t plus I will take just 3 voltages equals 0 will reduce to some complex number V 1 exponential j omega t plus some complex number V 2 exponential j omega t plus V 3 exponential j omega t equals 0. Now this is common so I get rid of that I can simply write KVL in terms of the complex multipliers. Remember V 1 is not the voltage V 1 times exponential j omega t is the voltage but that appears as common everywhere so I will ignore that all together and I will write V 1 plus V 2 plus V 3 to be 0. Again the algebra is quite simple and I think you can learn it very easily but it is also important to figure out what is the meaning of this ok what do these complex numbers mean and it turns out that this is a very widely used analysis and these complex numbers which are multiplying exponential j omega t are known as phasors ok ok. And similarly just like this exponential j omega t is common to all voltages in the loop it is also common to all currents at a node and the statement of KCL which is I 1 of t plus I 2 of t plus I 3 of t equal to 0 becomes I 1 exponential j omega t plus I 2 exponential j omega t plus I 3 exponential j omega t equals 0 from which I will again remove the factors exponential j omega t. So, I 1 plus I 2 plus I 3 which are complex numbers are 0 and this I 1 bar I 2 bar I 3 bar are the phasors of currents ok. So, now what happens is that everything just looks like DC circuit analysis containing DC analysis of circuit containing only resistors ok. But once you see let us say a phasor I 1 to figure out what the voltages should take real part of I 1 times exponential j omega t ok. So, I can write KCL and KVL in terms of phasors basically omit exponential j omega t ok. So, which means that to find some voltage v k of t you should take real part of the phasor of v k times exponential j omega t and this omega is something that you need to know. The very important thing is that even if you have multiple sources all of them should be at the same omega that is when you will have all the voltages and currents being proportional to just exponential j omega t ok. If you have omega 1 and omega 2 you will have exponential j omega 1 t and exponential j omega 2 t in different proportions and you will not be able to cancel them. If at all you find a situation where you do have multiple frequencies let us say omega 1 and omega 2 because these are linear circuits what you can do is first you disable all the sources of omega 2 and do the analysis of only the sources at the frequency omega 1 and then you disable all the sources at omega 1 and analyze the circuit for frequency omega 2 and then you add up the results separately after finding out the real part of phasor times exponential j omega t ok this is fine. So, again I will show this with a very simple example first of all the element relationships these we have already worked out, but I will show it just for the sake of clarity a resistor if it has some voltage v r exponential j omega t and a current i r exponential sorry v r bar exponential j omega t and i r bar exponential j omega t we omit exponential j omega t and simply write i r and v r and the relationship of course is the familiar relationship of the resistor v r by i r equals r and if you have a capacitor ok will have phasors i c and v c the phasors will be related by 1 over j omega c and I will have the inductor current and the inductor voltage which are phasors times exponential j omega t the relationship between these two with only phasors would be v l by i l equals the impedance of the inductor which is j omega l ok. So, we can omit this exponential j omega t from everywhere do our analysis and only when you need the expression for the signal in the time domain as a function of time you need to take this phasor times exponential j omega t and take the real part ok. Any questions about this? So, let us take a simple example. So, let me take the series RLC circuit and I want to find the response to v p cos omega t in steady state ok.   So, if you have a source cos omega t v p cos omega t the phasor corresponding to that is v p alternatively if you had v p sin omega t the phasor would be what would be the phasor corresponding to v p sin omega t? What is the phasor? The phasor multiplied by exponential j omega t should give you the voltage. I worked it out for v p cos omega t if this has to be v p sin omega t what should be the phasor v s, v s bar please try and answer this question. So, clearly this will be minus j v p. In general if you need the phasor of v p cos omega t plus phi the phasor corresponding to that would be v p exponential j phi. So, this is what we will use as the source ok. So, I will take this v p cos omega t. So, the source itself would be represented by a phasor v p and the resistance will remain as the resistance and the inductance basically is an impedance j omega l and the capacitance is an impedance minus it is 1 by j omega c which can also be thought of as minus j by omega c. So, now each of these has some voltages across it which is represented by phasors v r, v l and v c and there will be a there is a single loop and there is a certain current i flowing through the loop. When I say a current i is i is the phasor and the actual current is exponential is i bar times sorry real part of i bar times exponential j omega t ok.  So, you will be able to find all of these things very easily and you see that this circuit is just as complicated to analyze as a single loop of 3 components, 3 resistors in series ok. So, that is all that is there to it this is how you do phasor analysis, but please do not forget the connection to reality which is that if you want the voltage across the capacitor you take real part of the phasor v c times exponential j omega t ok. Any questions these also can be found ok. There is a reason I took this second order circuit we will take up exactly what these things do to the input signal in later lectures ok. And just to for completeness I will give you another example ok. In this case let me take v s of t to be v p cos omega t plus phi and this time I will take a parallel RLC circuit as I mentioned earlier if you null the input source if LCR are in parallel it is a parallel RLC circuit ok. So, in terms of phasors I would have to replace v s by its phasor which is v p exponential j phi R will remain as an impedance R, L will be j omega L and C will be 1 by j omega C ok. So, now similarly everything can be calculated ok. So, for instance if you want the voltage across the inductor which is also the same as the voltage across the capacitor v c what do we have it is v s times the parallel combination of these two impedances ok. The parallel combination of these two impedances is j omega L parallel 1 over j omega C which is j omega L times 1 over j omega C divided by j omega L plus 1 by j omega C which can also be written as 1 minus omega square LC by taking j omega C to the top ok.  It is just an impedance divider instead of a resistive divider we have parallel combination of these two impedances and that divided by the total impedance which is R plus the parallel combination ok. So, let me just complete this before ending the lecture which I will take R to the numerator and we will get something like. Now, again what does this mean there is a complex number multiplying v s which will modify the amplitude and phase of the sinusoid. If v s is a sinusoid v c will be a sinusoid at the same frequency and its amplitude and phase will be modified how much will they be modified by that will depend on the amplitude and phase of this number the magnitude and phase of this complex number ok. So, I hope that part is clear that brings us to the end of this lecture. So, the summaries as follows if you apply exponential j omega T to a linear circuit every voltage and current will also be of the form exponential j omega T. This gives us a lot of simplification in analysis. Now, exponential j omega T itself is first of all a fictitious input in the lab we would apply sin omega T cos omega T etcetera, but the response to cos omega T can be obtained from the response to exponential j omega T by taking the real part ok. This is because all the coefficients all the component values everywhere are real and I showed by superposition that you can think of exponential j omega T as combination of cos omega T and sin omega T ok. So, that is number 1. Secondly, because for every element voltage and current will have exponential j omega T the voltage becomes proportional to current just like that for a resistor. So, even though you have inductors and capacitors those circuits can be analyzed in a way that we use for resistive circuits which you agree was very easy because we did not have any differential equations. The only caveat here is that this applies only when the input is of the form exponential j omega T that is for sinusoidal inputs and in steady state because the transient response has to die out and that we cannot calculate using this method ok, but all the usual circuits that we consider in this course will have transient responses that die out over time. And finally, because this exponential j omega T is common to everything all voltages and currents we start writing Kirchhoff's current loss and voltage loss by omitting the exponential j omega T all together. And we will consider only the complex multipliers for each of these voltages and currents and those are known as phasors. And this analysis using phasors is known as phasor analysis and it is very very convenient to use whenever you have sinusoidal inputs to a linear system ok. So, any questions about this I will answer now ok, there is a question that says do phasors have a relationship to Laplace Fourier transform ok. Now the Laplace transform analysis of systems is derived from a similar reasoning that if you have exponential st given as an input to a linear system the response will also be of the form exponential st times something ok. This is true of any linear differential equation if you have a linear differential equation with an input of exponential st the solution will also be exponential st times something the steady state solution ok, so that is the relationship. Now what is the exact relationship if you take Laplace transforms if you do Laplace transform analysis of a circuit and substitute s equal to j omega you will get this essentially this phasor analysis ok. Now the Laplace transform and also Fourier transform analysis can also be used for signals which are not sinusoids ok the way we have presented it is only for sinusoids. So, if you reduce the if you use the Laplace transform for analysis of sinusoidal signals essentially what you get is the phasor analysis. So, you can think of this as a subset of Laplace or Fourier transform. Now Laplace transform is used for circuit analysis the Fourier transform in particular is mainly used for signal analysis there is another thing which we have not touched at all now we have showed that you can analyze circuits with sinusoidal inputs using phasor analysis. There is also the Fourier transform which says that any signal can be thought of as a combination of sinusoids ok, so like a square wave or a triangular wave can be thought of as a combination of sinusoids. So, what the Laplace transform and Fourier transform analysis do is to combine these two ideas that the response of exponential j omega t is exponential j omega t and any signal can be decomposed as some combination of exponential j omega t with this basically you can analyze any circuit any linear circuit with any input ok using Laplace transform and also may be less conveniently using Fourier transform ok. So, the spacer analysis is a sort of sub case of Laplace or Fourier transforms when you restrict the input to be single frequency sinusoids ok I hope that answers the question ok. So, then we can break now it turns out that for all the institutions which were subscribed to the online lecture the exams have already started. So, this will be the last online lecture, but we will continue to post the lectures by recording it offline and we will also have the assignments and you will have about 3 weeks or a months to complete them. So, please watch out the web page for these details they will be up shortly ok.


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