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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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# Euler's sine wave

A sine wave emerges from Euler's Formula. Music, no narration. Animated with d3.js. Created by Willy McAllister.

## Want to join the conversation?

- this is just calming for some reason lol(6 votes)
- Magnitude of the vector is smaller but the amplitude of produced sine wave is larger. How this happens?(2 votes)
- Did you notice the vector add happening? It's represented by the dotted lines. The red and green short vectors are added together to get the full height of the sine.(6 votes)

- I absolutely love this video. Dropping the student into a quiet lounge and just laying out the conclusions from all the lessons is genius!(4 votes)
- Watching this illustration , it looks like the result sin figure could also be generated by just e^jwt (without the e^-jwt and /2i) => sinwt = e^jwt (magnitude = 1)

What am I missing?(2 votes)- Andrew is correct. The two conjugate exponential terms combine by vector addition to make the result fall on the imaginary axis. The 1/2 term is a scaling factor on each exponential to make the resulting sine come out with a magnitude of 1.(2 votes)

- I was wondering, can you represent the sine without the imaginary as: sin(wt) = (e^wt - e^-wt)/2?(2 votes)
- Sorry, no. If you drop the i out of the exponent the expression is two ordinary exponential terms. The one with the positive exponent, e^+wt, will rapidly become very large.(2 votes)

- I am in grade 6, do you think that I should learn this stuff?(0 votes)
- Sure! Enjoy browsing around this section with Euler's sine wave and cosine wave. But I wouldn't make this my beginning spot. Go to the introduction of Electrical Engineering and start there. I wish I had this to learn from when I started my engineering adventures in 6th grade.(5 votes)

- play at x2 speed it sounds different(2 votes)
- may i know what happens to the j in the denominator. eulers sine wave is defined as (e^jwt-e^-jwt) / 2j. im asking wht is the effect of this j in the numerator? please tell.(1 vote)
- An important identity is 1/j = -j.

Any time you see a j in a denominator you can bring it up to the numerator as -j.

The j factor causes the moving yellow dot to travel up and down the imaginary axis.(2 votes)

- Why is there j*sin(w*t), why do we need j? Is vector adding happening with j?(1 vote)
- Aren't the colors of the dashed portions of the vector addition graphic reversed? They are the same magnitude, so the answer remains the same, but shouldn't the parallel lines on opposite sides of the parallelogram (beautifully) illustrating the vector addition be the same color?(1 vote)

## Video transcript

(upbeat jazz music)