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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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# Lead Lag

Sine and cosine look similar, except they are out of phase. When we talk about sine and cosine as a function of time, the difference is called "lead" or "lag". Created by Willy McAllister.

## Want to join the conversation?

- At5:37, cosine lags negative sine by 90 degree. Why don't we say cosine leads negative sine by 270 degree ?(3 votes)
- You can say that. Once you get the idea of "lead" and "lag" as it relates to the relative timing of sinusoids you can use the terms as you wish. The two statements you quoted are equivalent.(6 votes)

- At2:08, why can't I say sine leads cosine by 90 degrees?

Why do we have to see the waves as if they are moving to the left?(2 votes)- "Lead" and "Lag" reflect the relative timing of two signals. If func1 has a peak
**sooner**than func2, we say func1 leads func2. In these graphs, time increases as you go to the right. So a time near the origin happens sooner than a time out to the right on the time axis.

If you look at the cosine function, it has its peak at t = 0.

The sine function has its peak at t = 90 degrees. That is later than the cosine peak.

So we say "cosine leads sine by 90", or equivalently, "sine lags cosine by 90".(6 votes)

- What is the phase difference between 3 phase AC sine waves and how can we calculate them?(2 votes)
- Hello Rajab,

The three phase are electrically offset from each other by 120°.

The mathematics for the calculations are not hard to do. Unfortunately, it can be hard to visualize the waveforms and keep track of what is happening. This is especially true of circuits involving reactive components (inductors and occasionally capacitors) and transformers.

There is an entire language electrical engineers master to handle these problems. If you are interested please take a look at phasors. Ref:

https://en.wikipedia.org/wiki/Phasor

Please leave a comment below if you would like to continue this conversation.

Regards,

APD(3 votes)

- At5:15you said that sin(theta) = cos(theta - 90)

but in textbooks I often see sin(theta) = cos(90 - theta).

I am pretty sure they are the same , but I am confused.(1 vote)- Your formula comes from the "complementary" angles inside a right triangle. The two angles other than the 90 angle are "complements" of each other. https://www.khanacademy.org/math/trigonometry/trigonometry-right-triangles/sine-and-cosine-of-complementary-angles/a/sine-and-cosine-are-cofunctions.

The term "co sine" means "complement of sine".

In the video I used the negative of the angle:

theta - 90 = -(90 - theta)

This version of the formula gives the same answer. The reason is because cosine is an "even" function. It has symmetry about 0. Even means:

cos(theta) = cos(-theta)

cos(+45deg) = 0.707 = cos(-45deg)

cos(90 - theta) = cos(theta - 90) = sin(theta)

Whew, I thought you had me there for a second. Sine and Cosine are so similar they have symmetry every which way you look.(4 votes)

- Sir so what is the involvement of this AC circuit(1 vote)
- why in inductor voltage leads the current?it is really hard for me to understand that what is going on.(1 vote)
- in case of pure inductive circuit current lags behind voltage by 90' What actually happening physically due to which current is lagging?(1 vote)
- The voltage across an inductor is highest when the
**slope**of the current is greatest. If you assume the current is a sinewave, find the steepest part (where it crosses 0) and that's where the voltage will be the highest. If you plot out a bunch of other voltage points you will end up drawing another sinusoid that is 90 degrees out of phase.(1 vote)

- I can't intuitively understand why a lead/lag would occur though. Maybe because i've been dealing with DC circuits for too long? I do understand that current and emf vary with time, but why would they be out of phase?

they each depend on each other and V=iR, so where is room for phase difference?(1 vote)- "They each depend on each other" ... very true.

"v = iR" ... also true --- for a RESISTOR.

For a capacitor that i-v dependence is different than a resistor,

i = C dv/dt ... for a capacitor the current depends on the**slope**of the voltage.

For an inductor the i-v dependence is different again,

v = L di/dt ... for an inductor the voltage depends on the slope of the current.

Exercise: Draw your favorite sinusoid (sine or cosine). Label that as the voltage on a Capacitor. Then, just by looking at your drawing, look at the slope of the voltage at every point and sketch in the current. The slope of voltage is zero at the top and bottom of the humps. The slope is greatest where the sinusoid crosses the time axis.

After you sketch current for one cycle, what does the current look like? (I hope you get a sinusoid with a 90 degree phase shift from the voltage.)

Do a similar exercise with the inductor equation. Start with a current sinusoid and derive the voltage.(1 vote)

## Video transcript

- [Voiceover] In this
video we're gonna introduce a couple of words to help talk about the relationship between sines and cosines, or different sinusoids that
have the same frequency, but a different timing relationship. So, what I've shown here, is a plot of a cosine, and a sine wave. And the axis here is in theta, in the angle, the radian angle of the cosine or sine. Now I can label these, the pi over two represents a 90 degrees change. Pi is 180, degrees. This is 270 degrees. And this is 360 degrees. Those are two equivalent, scales for the angle, the angle axis, in degrees or in radians. Now, one thing we notice
here is that sine and cosine, they look the same,
but they don't overlap. If I change this to a time, If I change this to a time, axis, what I can say is the cosine wave reaches its
peak at time equals zero, and the sine peak, the sine wave, reaches its peak at a later time, this is increasing time going this way. So, the sine is delayed
compared to the cosine. The peak here is delayed here. If I go down and look at these two peaks, we see the same relationship. This sine peak in orange,
is behind, is delayed, from the cosine. So when we have this timing relationship between two periodic waves, what we say is, in this case, we say that the cosine, leads, the sine wave. And the amount of lead
is the difference between these two points, and we can say the lead is 270 minus 180, in this
case, it'd be 90 degrees. So we say that cosine, leads sine by 90 degrees. Now I can take exactly the
opposite point of view. If I actually measure where
the sine is relative to cosine, I say it's behind, then
we would say it lags. So the phrase we hear would be sine lags cosine by 90 degrees. So that's the term lead and
lag, that's what those mean. Now these terms apply,
this idea of a delay, this only applies when these
frequencies are the same. If the frequencies are different, the relationship between
the two wave functions changes all the time. So, we use the word lead and lag when we know that the two signals
we are talking about are exactly the same frequency. One thing I want to be able to do is express sines and cosines
in terms of each other. So, if I have a sine wave, could I actually express this orange curve as a cosine wave? How would a do that? So what I've noticed, if I look at the value of sine right here, and this is sine at 90 degrees,
or sine at pi over two, if I look at this value here, what I notice is that,
this has the same value, which is the peak of one. Cosine has the same
value, the peak of one, but 90 degrees earlier, 90 degrees before, because it's a leading function. So this suggests a conversion factor. Any time I pick out a value of the sine, if I look back 90 degrees, I'll see the same value for cosine. So I can write something like this, I can say that sine of theta equals the cosine of theta, minus 90 degrees. If I go out to some value, let's say there on the sine curve, and if I back up 90 degrees, like that, I'll read the same
value on the cosine curve, so these two functions will
give me the same number. So I can write this
identity in reverse also. If I have a cosine, if I'm riding along on this cosine wave, what I notice is, if
let's say I'm right here, I'll notice I have my peak value here, and if I added, if I went later in time, or if I added 90 degrees, I would have the same value
on that orange sine curve. So if I look here, on cosine, if I want to know what that is,
in terms of a sine function, if I add 90 degrees to the argument, the sine function will
give me the same value. So what that says, is cosine theta, equals sine of theta, plus 90. So these are two
identities, we can use this to convert a something
expressed as a sine, into a cosine, or vice versa. Now I want to show you
two more identities, that are actually pretty useful. And here what I have is, I
sketched on in dash lines the negative of the orange curve, so this is a negative sine wave. You can see it the opposite of the original sine wave we had. So now I have the case
here where the cosine is trailing, or lagging,
the negative sine. Right, it comes later
in time, right there. So cosine lags negative sine. And what I'll do is, I'll write the same sort of identities here, but in terms of this negative sine, and those come out like this. Cosine of theta, equals minus sine, so that's the dashed one, of theta, minus 90 degrees. So what that means, if I want
to know the value of cosine, and I can flip that around the same way, and I can say that negative sine of theta equals cosine of theta plus 90. That's the same identity but in reverse. If I want to know the
value of negative sine, I just take that argument, I add it, I advance it by 90 degrees, and take the cosine,
it'll have the same value. So, this identity and, this identity, are pretty useful to have around. This one allows us to convert
sines and cosines together. This pair here is useful for
moving minus sines around. So that's what I wanted
to say about lead and lag. These are sort of slang or jargon, the nicknames of the relationship between two different wave forms
of the same frequency but different phase timing,
different phase delay. And then we worked out
some, some identities that are kind of useful to have around, to be able to convert these two wave forms from one to the other and back.