Understanding Phase Angles

Help for ham radio Amateur Extra Class students; phase angles -- what they are and how to calculate them.

[sage]

Best teacher, you are great example of being a good teacher. You are enthusiastic and informative. Keep up the good work.

[cgsraRude]

You've got a great voice for this.

[mathaneshanrajagopal]

WOW, this is the best video I have ever watched. I can literally do the phase and phase diagram at ease

[dyientangchan]

wow...what a mind blowing narrative of the lesson? I like how this guy present with so much confidence and conviction!

[dahelmang]

Thank you! I'm not studying radios specifically, but this is the first video I've found thqt explains these concepts simply.

[stanleycates]

this man is an excellent teacher - 83 year old retired power EE

[vwsandvettes]

AC4LX here - I got my Amateur Extra ticket way back in like, '92 along with my FCC General Radio Telephone operator permit while in the USAF. Its good to get this refresher, and thank you for the video and clear and easy to understand presentation!

[johnishikawa]

This was very good! One thing though... that negative sign out in front of the tangent operator. I think that instead of indicating a reciprocal, as in 1/tan or something, it's more appropriate to call it the inverse tangent, or perhaps, it means the angle for which the tangent equals what is in the parentheses.

[wallyalshamari]

thank you so much for the insightful, yet entertaining lesson. Well done!

[elihotchkiss]

Thank you for this lesson. It was as entertaining as it was educational.

[shahriartanvir]

Sounded like the beginning of a Disney movie. Loved it <3

[caoweigejiba]

very good narrative. patient and enthusiastic. thank you for the lesson.

[jamesbrittain]

Cheers for the video, I love your narration, fantastic to listen to!

[RosaJimenez-pgmh]

Michael, thank you so much for your videos, this one included.
Your presentations and lectures are great.

There is a problem, however, at approximately 9:30 into the presentation as Michael describes the tan-1(x) function.

Michael states, "tan^-1(x), the reciprocal of the tangent ...." Ooops, that is incorrect.
The tan^-1(x) function is NOT the "reciprocal" of the tan(x) function.

Both tan^-1(x) and tan(x) are "inverse functions" of each other, completely different than reciprocals of each other.

A reciprocal is the "multiplicative inverse" of a quantity. The reciprocal of x is 1/x. Just as the reciprocal of 4 is 1/4. The reciprocal of the tan(x) function therefore, is 1/tan(x); the reciprocal of the tan(x) function is also known as the cotangent(x) function.

Let's recognize that tan^-1(x) is the "inverse function" of tan(x). An inverse function, is completely different than a reciprocal. So then, what is an inverse function?

We know how to take a number x, entering x into some function, where the outcome is f(x). For instance, let's use the number 4 as our x. We enter 4 into the function (x)^2, and we calculate (4)^2 = 16. We have started with some x, calculated, and obtained an f(x).

But what happens if we start with f(x) and we need some function to find x?
In this case, we may use the "inverse function" of f(x), known as f^-1(x), to obtain our x.

So, lets use our trig example, starting with the tan(x) function.
As in the video, if we have some angle x, and calculate the tangent function of angle x, we will obtain some number, tan(x). So let's do that. Let use negative angle -14.0362 and calculate tan(x). We calculate tan(-14.0362) and obtain the number -0.2500.

Now, let's start with our number -0.2500, using the "inverse function" of tan(x), a function called tan^-1(x) and also known as arctan(x), to calculate our original angle.

Working with the tan^-1(x), we calculate tan^-1(-0.2500), to obtain our angle, -14.0362.
We have just used the "inverse function" of tan(x), also known as function tan^-1(x) or arctan(x), to start with some number, with the final goal to obtain an angle.

The confusion with reciprocals and inverse functions often stems from confusing the accepted math notation. The reciprocal of x is 1/x. Great. We can also use exponents to represent the quantity 1/x, as x^-1. Just as 1/125 can also be written as 125-1.

The reciprocal of tan(x) can be written as 1/tan(x). Using exponents, 1/tan(x) also can be expressed as (tan(x))^-1. Here, the negative exponent ^-1 applies to the entire quantity tan(x). 1/tan(x) is equivalent to (tan(x))^-1. Of course, the reciprocal of tan(x), can be written simply as cotangent(x).

In math, we indicate the "inverse function" using a convention that appears like exponential notation, but it is NOT exponential notation! Instead, this notation indicates "inverse function."

We may have function f(x). The "inverse function" notation for f(x) is f^-1(x); the ^-1 here is NOT exponential notation, but is simply a convention that indicates the "inverse function" of f(x). The inverse function notation for tan(x) is tan^-1(x). Again, the ^-1 here has NOTHING to do with exponents, it is simply the notation convention that indicates "inverse function" of tan(x).

In contrast we can express reciprocals; f(x)^-1 means 1/f(x). Here, the ^-1 is a negative exponent. Likewise, if we have tan(x)^-1, we mean 1/tan(x). The "reciprocal" of f(x) is 1/f(x) or f(x)^-1. The "reciprocal" of tan(x) is 1/f(x) or tan(x)^-1. Choose your preference.

So, we use the "inverse function" of tan(x), which is tan^-1(x), to get back to our phase angle.
This minor attention to detail in math notation has real meaning.

On calculator keys, the trigonometry "inverse functions" are often noted as asin, acos, atan. On some calculator keys, the trigonometry "inverse functions" can sometimes be noted as sin^-1, cos^-1, tan^-1.

Whereas on a calculator, should you need to calculate a "reciprocal" of tan(x), you simply calculate 1/tan(x). Time for a coffee break. 73 to all. Many thanks for Michael's excellent work

[shamanking]

What is the best way to test and find the phase angle ???
what instrument can use to measure it ???.
Thanks.

[katfishzomby]

thank you. an art major trying to research for a electrical test tech exam.

[tonystark_]

ELI the ICE man
1.Voltage(E) leads current(I) in inductor
2.Current(I) leads voltage(E) in capacitor

[kucingbloon]

THIS IS GREAT! I DEFINITELY LOVED IT, THANKYOU!!!

[derykvaccaro]

Thanks for the lesson. Great voice. You should start doing audiobook narration!

[wiktorczapiewski]

Great video, no bs and easy to understand. Thanks a lot!