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### Course: Electrical engineering > Unit 2

Lesson 5: AC circuit analysis- AC analysis intro 1
- AC analysis intro 2
- Trigonometry review
- Sine and cosine come from circles
- Sine of time
- Sine and cosine from rotating vector
- Lead Lag
- Complex numbers
- Multiplying by j is rotation
- Complex rotation
- Euler's formula
- Complex exponential magnitude
- Complex exponentials spin
- Euler's sine wave
- Euler's cosine wave
- Negative frequency
- AC analysis superposition
- Impedance
- Impedance vs frequency
- ELI the ICE man
- Impedance of simple networks
- KVL in the frequency domain

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# Negative frequency

The concept of negative frequency is perplexing, but it makes sense when you think about rotating complex exponentials.

## Want to join the conversation?

- If negative frequencies makes waves as positive frequencies, then if someone was on the radio on a positive frequency and someone else made the same waveform with a negative frequency, could they come in contact?(3 votes)
- Negative frequency is a property of complex exponentials that have imaginary exponents (what I call 'spinning numbers'). When you create a real radio wave from a real transmitter, the two complex exponentials (conjugates of each other) combine into a real sine wave. The sine wave has a frequency that's always a positive number.(4 votes)

- When it comes to visualising how the opposite complex exponential spins (e^(jx) and e^(-jx)) actually amount to the same outcome, the only difference between cosine and sine is that for sine, the negative spin (or frequency) term e^(-jx) has a minus sign in front.

(2j sin(x) = e^(jx)-e^(-jx))

That means it's starting point is actually at pi or 180°.

So when projected to the Im axis, the two "originals" still form a single "projection" - a sine wave.

Is this correct?

It's not difficult to visualise how the two opposite c. e. spins form a single cosine wave, since they naturally start at 0 pi or 0°. A bit trickier with the sine wave, because of that minus sign.

Still very straightforward when looking at equations, though.(2 votes)- These two visualizations may contribute to an intuitive understanding of Euler's equations.

https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-ac-analysis/v/ee-eulers-sine-wave

https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-ac-analysis/v/ee-euler-cosine

The plus or minus sign up in the complex exponents sets the direction of spin.

The minus sign between the two exponential terms in Euler's sine equation gets the sine function to start in the right place.(1 vote)

- What if we projected the sine wave, instead of the cosine wave? Then the two projections would be mirrored of each other, i.e. the projection for positive and negative frequency would be different?(2 votes)
- At1:30, you plot e^-jwt as the opposite of e^jwt. But e^jwt and e^-jwt are complex conjugates, so the drawing looks confusing. Is that deliberate?(1 vote)
- Sorry for the confusion. I drew the two arrows at different values of time, so it looks like one points up-right and the other left-down. The key point was they rotate in opposite directions.

If you view the video "Euler's sine wave" or "Euler's cosine wave" you will see the two short vectors plotted with the same time value. Notice that the rotating vectors (spinning numbers) have the same real (horizontal) value and opposite (conjugate) imaginary (vertical) values.(2 votes)

- Can you guide me to some research if done on negative frequency for the purpose of increasing the available bandwidth for communication?(1 vote)
- Hello Rajab,

Could you elaborate. I'm not sure what you are asking.

Normally communications is dictated by the Shannon-Hartley limit. Ref:

https://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theorem

You can reduce the noise, increase the power, or spread the signal to increase the channel capacity.

I can't over stress this simple equation. You will see it time and time again in communications. For example, this company talks about their waveform and how close it gets to the Shannon limit. Please look all the way down on the page at http://www.scs-ptc.com/en/PACTOR-4.html

Regards,

APD(2 votes)

- If my understanding is correct;

In a circuit even when a non-sinusoidal function is inputted, a circuit involving linear elements will produce sinusoidal responses.

Non-linear elements like diodes can effectively cut output values off, creating square-wave like output behaviours.

Now, these non-sinusoidal outputs would be modelled by the summation of infinite number of sinusoids (which can be truncated)

Now, considering the many number of different signals that are produced and broadcasted, wouldn't it be possible or say necessary to broadcast in some instances a negative frequency component?

Would this negative frequency component in all circumstances have a conjugate part that would turn it into a real part only time signal?(1 vote)- Real sinusoidal signals have only positive frequency. Negative frequency is an idea associated with complex exponentials. A single sine wave can be broken down into two complex exponentials ('spinning numbers'), one with a positive exponent and one with a negative exponent. That one with the negative exponent is where you get the concept of a negative frequency.(1 vote)

- Are there any practice exercises for us -the learner- to practice calculating with phasors anywhere?(1 vote)
- There are currently no practice problems for phasors. I found these on the web, on page 6... http://web.eecs.umich.edu/~aey/eecs206/lectures/phasor.pdf(1 vote)

- I don't know if I'm getting it right.. so the negative frequency has "no effect" plotting the real axis but only the imaginary axis because it rotates in the opposite direction?(1 vote)
- The idea of negative frequency arises from Euler's Formula for sine/cosine.

I used the cosine function in this video to illustrate where negative frequency comes from.

For cosine the signal appears to emerge from the horizontal axis. If I had chosen the sine function for the discussion the sine wave emerges from the imaginary vertical axis.

Both sine and cosine are composed of two opposite-rotating exponential terms combined in slightly different ways. One of those exponentials has a positive frequency and the other has a negative frequency.(1 vote)

- What is the real world significance of Negative Frequency in the Frequency spectrum in
**Communication**?

If none why should I consider the positive frequency accountable?

Thanks!(1 vote)

## Video transcript

- [Voiceover] I wanna
talk a little bit about one of the quirkier ideas
in signal processing, and that's the idea of negative frequency. This is a phrase that initially
may not make any sense at all, what does it mean
to be a negative frequency? Could there be a sine
wave that goes up and down at a rate of minus 10 cycles per second, what on earth does that mean? Or could there be a radio
station of minus 680 kilohertz? What is that, that doesn't
sound like it means anything. There is a sense in
which negative frequency is understandable and
we're gonna just quickly talk about that here. You remember, we can
describe a cosine like this, cosine of omega t equals
1/2 e to the plus j omega t, plus e to the minus j omega t. And each of these components, each of these complex exponentials here, we can draw as a rotating complex number. So for this first one, we
could actually draw it, if we wanted to, we
could draw it like this, we could draw a complex
number out here in space, and think of it as a rotating vector, and that would be e to the j omega t. This one has a plus sign. Now the other thing, this term over here, would look like a similar thing, it would be sum vector and space, there's that number right
there, e to the minus j omega t, and this one's rotating
in the negative direction. So that means this is rotating this way. So the idea of a negative frequency, when we talk about rotating
vectors, makes good sense. If we are rotating in the
positive direction, like this, you could say that's a positive frequency. And if we're rotating in the
negative direction, like this, you could say that's a negative frequency. So omega t gives us the speed, and this sign right here,
and this sign right here, give us the direction. The frequency here is plus omega t, and the frequency here is minus omega t. So in this sense of rotating vectors, negative frequency seems
like a pretty simple idea. Okay, so let's go where
it's not a simple idea, and that's when we do
this thing we did before where we projected these
vectors onto a cosine wave, it had spread out the time
axis linearly, like this, going down the page for
the cosine, remember, we did this, we drew a
line here on this side, we're gonna do plus omega t. So we're gonna plot e to
the plus j omega t here. This will be the real axis,
this will be the imaginary axis, and down here, this'll
be the voltage axis, and this'll be the time axis. So when we start out at
time equals zero, we have, we project it down here, and
we got that value right there. And then as time goes on,
if we tip our arrow up, like that, to that point,
then we project down to the cosine curve right here. If we let our arrow go all
the way to the other side, it projects like this down, it projects down to this
point on the cosine curve. And as we go farther and farther, let's go straight down for a second, that one projects to right there. and as we come over here, it projects to that point right there. And eventually, when we
get back to home again, when we get back to zero, the projection is right
to this point here. And so with one rotation
we've carved out one cycle of the cosine. So that seems pretty clear. Omega t is there, omega t is down here. Okay, so let's do it again, but this time we'll go over on this side, and we'll plot e to the minus j omega t. So this again is the real axis,
and this is the imaginary. And this is the voltage axis,
and this is the time axis. So let's start out again,
we're going straight sideways to time equals zero, so
let's project that down. And okay, that's pretty good,
that's the same as before, we got the same v here. Now let's tip it down, let's say we rotate this way a little bit, and here's our new position of our vector. And that projection, after
a little bit of time, goes to right here. And then if we go over this way, eventually, rotate some more, we'll project to this point here. And when we're straight sideways, it'll project to this peak right here. And you can see what's
happening is basically the exact same thing is happening
as happened on the left. Which is these points
carve out a cosine wave, just as we'd expect, and when this vector gets all
the way back around to zero, we've done one cycle, we're back home, and now we're projecting
to this point here. What happened here is we
took two different vectors rotating in opposite directions, this one clearly had a positive frequency cause it was going counter-clockwise. This one had a negative frequency cause it was going clockwise. And both of them carved out
the exact same cosine wave, when we got done. So in the vector world where
we're spinning vectors around, it seems very natural to talk
about plus and minus frequency but when we cast this back
into say a real-world v of t, notice that the idea of
negative frequency just sort of, it melts away, it evaporates,
it's not really there anymore. And it's been removed by
this process of projection. So when the idea of
negative frequency comes up, and it seems like it
doesn't make sense in this time to main view of the signals, but then just remember that
when we go back up here, and we look at the rotating vectors, that it just means which
way the vector's spinning.