It seemed to me that the standard form was to have the frequency expressed in radians/sec & the phase shift in degrees. So I went back to my first circuits textbook (Linear Circuits, by Ronald E. Scott, 1960, Addison-Wesley Publishing Co.) and found this in the chapter introducing AC circuits.

Expressing frequency in rad/s or in Hz are both very common. As is expressing angles in either degrees or radians. As long as you treat the units properly, there is no ambiguity at all. There is nothing in the quoted passage that is in bad form in any way, but neither is there really anything that indicates that θ is to be expressed in degrees. No more so than a passage in a physics text in which E=mgh uses h = 10 ft means that h should be assumed to be in feet.

In your example, they just happen to use degrees for the phase angle in that passage because they felt that conveyed the point better to their audience. It really doesn't establish what their normal convention was even within that text (though I would not be the least bit surprised for an engineering text to prefer degrees for angles as a rule).

The use of ω in the equation does not actually mean that it is in radians per second. Having said that, using ω to imply radians/second and using f to imply cycles/second is so widespread that it is a defacto standard regarding variable/symbol usage in contexts like this.

Another one is using ω = 2πf to convert between them. But this is actually not true.

If h1 is a distance in inches and h2 is the same distance in feet, then

h1 = h2

because they are the SAME distance.

36 inches = 3 feet

It is NOT the case that

h1 = 12*h2

This says that

36 inches = 12*3 feet = 36 feet

Similarly, if f = 50 Hz, then 2πf is 314 Hz.

Think about it.

If I have a frequency of f = 50 Hz, what is 6·f? It's 300 Hz. What is 7·f? It's 350 Hz.

Since 6 < 2π < 7, doesn't it follow that 300 Hz < 2πf < 350 Hz?

Or to look at it another way, are we not in agreement that 50 Hz is the same frequency (to three sig figs) as 314 r/s?

50 Hz = 314 r/s

So if 2πf = 314 r/s and this is the same frequency as f = 50 Hz, then we are claiming that

2πf = f

and thus π = ½

Of course, people would immediately say, "Well, that's not what I meant." Statements like that have no place in mathematical equations. If the equation doesn't mean what you meant, then the equation is the wrong equation because it doesn't mean what you meant it to mean. So use the right equation!

The correct way to write that expression is

ω = (2π radians/cycle)·f

Now when we plug in f = 50 Hz, we get 314 radians/second.

But this (wrong) formula, which came about purely because people couldn't be bothered to use units properly, is so widely established and understood that while we technically have to guess what was meant, the chance of our guessing wrong is negligible.

 

So are you really saying that all of the following have the same frequency?

(628 r/s)t
(628 Hz)t
(628 rpm)t
(628 deg/min)t

Not at all. I am saying the question can be answered with the assumption that the units are the same as the units of t. Thus you can say the it the frequency is 200 / (unit of t) since sin is periodic in 6.28... i.e 2π.

For your example saying that you cannot solve a problem that says the length is 96. That is nonsense.

If the length of A is 96 and the length of B is 3, what is the ratio what is the ratio of A to B?

Bob

 

Not at all. I am saying the question can be answered with the assumption that the units are the same as the units of t. Thus you can say the it the frequency is 200 / (unit of t) since sin is periodic in 6.28... i.e 2π.

And if I have ten people build ten different waveform generators all of which put out a signal that is

v(t) = 10 V · sin(628t)

Wouldn't it be nice if all ten of those generators put out pretty similar waveforms? Seems like it would be a useful and handy thing.

But Person A uses units of seconds for t, Person B uses units of minutes for t, Person C uses units of days for t, and Person D uses units of years. If they each make the assumption that you claim is so reasonable, are the resulting waveforms going to be similar? If not, perhaps it's not such a reasonable assumption.

For your example saying that you cannot solve a problem that says the length is 96. That is nonsense.

Okay, since it's nonsense, solve it.

Problem: Bob's height is 96. Is Bob more likely to be (A) an unusually tall professional basketball player, or (B) an unusually short kindergarten student?

If you can't solve it, then perhaps the claim that you cannot solve it isn't nonsense.

If the length of A is 96 and the length of B is 3, what is the ratio what is the ratio of A to B?

You have absolutely no idea.

You: "Hey Chuck, how long is that piece of wood you've got?"
Chuck: "Ninety-six."

You: "Hey Brenda, how long is that piece of wood you've got?"
Brenda: "Three."

You: "So Chuck's piece of wood is 32 times longer than Brenda's."
Brenda: "Are you blind? Clearly mine is much longer than Chuck's."

You: "Can't be. The ratio of the numbers is all the matters. Units are irrelevant."
Chuck: "What do you mean units are irrelevant."

You: "Since I'm taking a ratio, the units cancel out, so it doesn't matter."
Breand: "But what if they aren't the same units?"

You: "Doesn't matter, I can just assume that they are."

Brenda: "Hey Chuck, what units did you use?"
Chuck: "Millimeters. How about you?"
Brenda: "Feet."

You: "Doesn't matter. 96/3 is 32 and so Chuck's wood is 32 times longer than yours, Brenda, and that's all there is to it."

How about a real world example (made generic, but similar has happened a lot more than once).

• Person goes to the doctor's office and gets weighed by the nurse, who writes down 100 on the form.
• Person talks to a doctor diagnoses the malady and prescribes a particular drug at 10 mg/kg of body weight, twice a day for thirty days, and writes that on the form.
• Doctor gives the form to the physician's assistant who writes out a prescription for sixty 1000 mg tablets.
• Person goes home and takes medicine as prescribed.
• Person is rushed to the hospital three weeks later and dies.
• Medical examiner's paperwork shows that the deceased's body weight 98.3 lb.
• Cause of death found to be cumulative overdose from a prescription drug that was taken at more than twice the recommended dosage.

Medical professionals that make the kind of assumptions you are advocating can kill people. But, generally, they are limited to killing people one at a time. Engineers that make these kinds of assumptions can kill people in job lots.

 

The saga of the Gimli Glider is the classic story of units screw up on July 23, 1983.

Air Canada Flight 143 on a Boeing 767 from Toronto to Edmonton found its way cruising at 41,000 feet half-way into its flight and out of fuel.

Both ground crew and the captain used the wrong conversion factor of 1.77 pounds/litre instead of the correct 0.8kg/litre in order to determine the volume of fuel required for the trip.

How the plane managed to land safely at a decommissioned airstrip being used by the Winnipeg Sports Car Club that same day is a miracle.

https://www.cbc.ca/archives/when-a-metric-mix-up-led-to-the-gimli-glider-emergency-1.4754039

https://nationalpost.com/news/canad...air-canada-767-as-famed-plane-put-up-for-sale

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

 

I have seen the notations:
A*sin(w*t+pi/4)
A*sin(w*t+45)

both meaning the same thing, w is rads/sec and t is time in seconds and either pi/4 rads or 45 degrees is the angle. That's the normal interpretation. The only way to be sure is to look back at the earlier exercises in the course work where the interpretation would have been established already through other problems that have been solved already. The OP would have to know this.

 

Not at all. I am saying the question can be answered with the assumption that the units are the same as the units of t. Thus you can say the it the frequency is 200 / (unit of t) since sin is periodic in 6.28... i.e 2π.

For your example saying that you cannot solve a problem that says the length is 96. That is nonsense.

If the length of A is 96 and the length of B is 3, what is the ratio what is the ratio of A to B?

Bob

So many strawman arguments on one page! Amazing.

All I said is that it is possible to give an answer to the original question in such a way that the units do not matter. And I did so. The answer is:

200 cycles per unit of t. (approximately, taking Pi to be 3.14)

Bob

 

So many strawman arguments on one page! Amazing.

All I said is that it is possible to give an answer to the original question in such a way that the units do not matter. And I did so. The answer is:

200 cycles per unit of t. (approximately, taking Pi to be 3.14)

Bob

Where is there any mention of pi in the equation?

 

sin is a mathematical function that takes a unitless argument and is periodic in 2π.

sin(30°) implies a conversion = sin (30 * π / 180).

Actually I did make a mistake in that answer, it should be 100 cycles per unit of t.

Bob

 

One can assume what sin(30°) means.
One can infer what sin(30) means.
But what the heck is sin(1234) or sin(628)?

 

One can assume what sin(30°) means.
One can infer what sin(30) means.
But what the heck is sin(1234) or sin(628)?

Sorry but I disagree. Without the degree symbol, the argument of sin is a pure number and it is periodic in 2π.

Bob

 

If 628 as in sin(628) is a pure number, then what is 628t as in sin(628t)?

 

A pure number multiplied by a number with units produces a number with the same units.

3 time 4 inches = 12 inches. 3 inches times 4 inches = 12 square inches.

Having been a physics major there was never any ambiguity about what sin(x) meant. In later classes we usually did away with sin and cos altogether, preferring e^ix since it is easier to manipulate mathematically.

Bob

 

A pure number multiplied by a number with units produces a number with the same units.

3 time 4 inches = 12 inches. 3 inches times 4 inches = 12 square inches.

Having been a physics major there was never any ambiguity about what sin(x) meant. In later classes we usually did away with sin and cos altogether, preferring e^ix since it is easier to manipulate mathematically.

Bob

So according to you, 628 has no units.
If t has units of second then 628t has units of second.

But according to you sin(x) must take x with no units.

 

You are correct, the 628 there does have units.

In physics, when we write an equation with sin for a periodic (in time) function we would write is as sin (ωt), where ω is angular frequency with units of radians / second, which has units of 1 / seconds, and t which has units of seconds. So ωt is unitless. Sorry, it has been roughly 47 years since my last physics class.

So that was indeed a misstatement. 628 actually must have the units of 1 / (units of t). Then the argument to sin is a dimensionless number, as is the result of the sin function.


Bob

 

Anybody who does not recognize the variable 't' as units of time in seconds has GOT TO BE KIDDING or way overly PEDANTIC.
't' is time in seconds when there is no other information present.
For example, do you really think time 't' could be in units of time in centuries? If you say that's possible you have no common sense relative to electronic studies. Even if you say 't' is time in weeks that's still stretching it.
But hey it's up to you declare whatever you wish but you just make this much much more complicated than it needs to be.

 

The argument in sin(628t) must be dimensionless.
So we assume that if t is in second, then 628 ought to be in 1/second.
But who brought pi to the table?
I know we're driving this into the ground but what the heck, what else is there to do?

 

The argument in sin(628t) must be dimensionless.
So we assume that if t is in second, then 628 ought to be in 1/second.
But who brought pi to the table?
I know we're driving this into the ground but what the heck, what else is there to do?

I answered that many posts back.

sin(x) is periodic, with a period of 2 pi. That is where it comes in.

Bob

 

I have seen the notations:
A*sin(w*t+pi/4)
A*sin(w*t+45)

both meaning the same thing, w is rads/sec and t is time in seconds and either pi/4 rads or 45 degrees is the angle. That's the normal interpretation. The only way to be sure is to look back at the earlier exercises in the course work where the interpretation would have been established already through other problems that have been solved already. The OP would have to know this.

That second is not proper use of units. Relying on inferring what the units are based on the value of the number is ludicrous.

A*sin(w*t+1.2)

Is that radians or degrees (or some other measure of angles, such as grads, but that would be very unlikely)?

What is SO hard about using the proper units?!

A*sin(w*t+pi/4)
A*sin(w*t+45°)

 

Sorry but I disagree. Without the degree symbol, the argument of sin is a pure number and it is periodic in 2π.

Bob

Exactly. And since 628t is NOT a pure number (it has units of time), it CAN'T be the argument to sine or any other transcendental function.

 

That second is not proper use of units. Relying on inferring what the units are based on the value of the number is ludicrous.

A*sin(w*t+1.2)

Is that radians or degrees (or some other measure of angles, such as grads, but that would be very unlikely)?

What is SO hard about using the proper units?!

A*sin(w*t+pi/4)
A*sin(w*t+45°)

Well actually i forgot the little degrees symbol that is the real second form i meant.