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Your step-by-step guide — add cosigner calculated
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Add cosigner calculated
welcome to electron line here's a really good example of how you take two voltages in the time domain and first convert them to the phasor domain before we go ahead and add the two together because it would be a lot more difficult to do it like this and if we're going to convert it from the time domain to the phasor domain we should write both voltages in terms of the cosines that makes it a lot easier so on the first one we need to first convert that to the cosine and the first thing we probably want to do is make it from a minus sign to positive sign by adding 180 degrees so this can now be written as a positive 10 times the sine of Omega t minus 30 degrees and plus 180 degrees so when we combine that together this is equal to 10 times the sine of Omega T plus 150 degrees and now we want to convert that to the cosine we can do that by subtracting 90 degrees so this can be written as 10 times the cosine of Omega T plus 150 degrees minus 90 degrees because that converts from a scientific cosign that combine these two this becomes equal to 10 times the cosine of Omega T plus 60 degrees so now we have V 2 which is expressed like this and we have V 1 which is expressed like this so now we can convert them to the magnitude phase formats that would be to the phaser format as we call it so V 1 can now be written as the magnitude which is 10 times the phase angle which would be positive 60 degrees and V 2 is equal to the magnitude of 20 times the phase angle of 45 degrees so that would be the phaser format of the two voltages now we want to add those we have to write those in terms of the real and imaginary parts so this can be written as 10 times the cosine of 60 degrees plus J times the sine of 60 degrees and this can be written as 20 times the cosine of 45 degrees plus J times the sine of 45 degrees and now we can write this as complex numbers in terms of the real and imaginary parts the cosine of 60 well that's equal to 0.5 times 10 that would be 5 plus the sine of 60 take the sine times 10 that would be eight point six six so plus J times eight point six six and for the V to the coast on a 45 that would be 0.7 or seven times 20 which is 14 point fourteen fourteen point fourteen for the real part and that would be for the imaginary part would be J times fourteen point fourteen now we can add them together now we can go v1 plus v2 and they could tell that's a lot easier now that would be equal to 19 point one four plus when we add this together that's 22 point eighty so twenty two point eight let's see that's eight that's yep that is correct and of course I can't forget to J because that's the imaginary part there we go so we don't really want to leave it like that because our voltages were expressed at the time domain so we want to reconvert that to the dime time domain so it can then say that v1 plus v2 is going to be equal to v-max times the cosine of Omega T plus the phase angle where v-max can be found v-max is equal to the square root of the sum of the squares of the components nineteen point one four squared plus twenty two point eight squared and the phase angle can be found by taking the inverse tangent of D that would be the imaginary part which is twenty two point eight divided by the real part which is nineteen point one four you see how much easier it is to convert to the phasor domain in order to find the sum of the voltages so let's go ahead and do this so this is a 19-point 1/4 squared plus 22.8 squared equals take the square root that gives us 29 point let's run off to one decimal place so this is equal to twenty nine point eight and on the phase angle twenty two point eight tooth out of by nineteen point one four let's see here and take the inverse tangent that gives us forty nine point nine nine so this is equal to forty nine point nine nine degrees which is basically fifty degrees all right so let's go ahead and plug that in here so v1 plus v2 is equal to the magnitude which is twenty nine point eight and times the coop times the cosine of Omega T plus a phase angle of 50 degrees and this would then be the resultant of adding those two voltage together so again go ahead and take the voltages in the time domain write them in the form of the cosines once you've done that you can easily convert them to the magnitude and the phase angle which means now we're converted into the phasor domain or the frequency domain and then we convert to the real and imaginary parts and the real parts together add the imaginary parts together and that reconvert back to the frequency I should say not the frequency but the time domain so they can have the same representation that's what you started with that's how it's done
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