KVL for Rms voltages

[Teszla]

A resistance of R = 1kΩ is conencted in series with a capacitor and they are connected to a tone generator. The frequency of the sinusoid signal from the tone generator is 100 Hz. With multimeter the rms value of the voltage from the tone generator is measured to 1 Vac, the voltage over R to 0.7 Vac and the voltage over the capacitor to 0.7 Vac.

a) The measurements seem to go against KVL. Please explain why.
b) What is the value of the capacitor? You can express C with R and ω.

For a) I understand the actual voltage values vary in amplitude, so when the voltage over C is maximum it will be lower for R and vice versa. But how can you explain this in a mathematical way?

 

[FvM]

AC quantities are vectors in a two-dimensional vector space, usually respresented by complex numbers. In the present case both voltages are orthogonal, summing like the legs of a right-angled triangle.

 

[Teszla]

Aren't vectors summed like a parallellogram rather than a triangle?

 

[FvM]

Aren't vectors summed like a parallellogram rather than a triangle?

Possibly. Thus I said "In the present case both voltages are orthogonal".

I see that you already know about two-dimensional vector problems, that AC vectors don't add like scalars. I'm sure you can work out the different cases.

 

[KerimF]

You may like reading "adding two sine waves" in the time domain at:

**broken link removed**

It shows an example about how AC signals of the same frequency could be presented by polar vectors (magnitude and phase).

 

[Ratch]

Teszla,

For a) I understand the actual voltage values vary in amplitude, so when the voltage over C is maximum it will be lower for R and vice versa. But how can you explain this in a mathematical way?

You understand wrong! It definitely says that both the voltages are the same at 0.7 volts.

First of all, voltages are not vector quantities. Voltage has magnitude and phase, but not direction, so they are phasors. In a resistor, the current and voltage are in phase in a series circuit. In a capacitor, the current leads the voltage by 90°. That means the capacitor voltage lags the resistor voltage by 90°. Adding the phasors we get for a magnitude sqrt(0.7^2+0.7^2) = 1, which is what is given. Problems like this are covered extensively in elementary text books on AC circuit theory. You should read them. You can get the value of C by solving 1k=1/(C*omega) .

Ratch

 

[FvM]

Saying AC voltages are not vectors is a bit sophistic, because the artificial word phasor is just a combination of phase and vector.

Cheers!

 

[Ratch]

FvM,

Re: KVL for Rms voltages
Saying AC voltages are not vectors is a bit sophistic, because the artificial word phasor is just a combination of phase and vector.

As I said before, voltage does not have a direction like a vector does. A phasor does not have a dot product, box product, or a cross product that a vector has. A phasor has a vector like similiarity with respect to addition, but that does not make it a vector.

Ratch

 

[FvM]

I hear from your explanation that you are associating vectors with the euclidian vector space R3, which is a kind of limited view, I think.

Vector is a rather abstract term in mathematics, different kinds o vectors have different properties. It seems to me that the said dot product has a meaning for AC quantities, it can e.g. represent real power. Direction has a different meaning in two-dimensional space than in R3, but many phenomena in AC networks and electrical machines can be well understood as a 2D direction.

But again, I'm not speaking against your description of phasors, only against the - as I think arbitrary - claim that they are no vectors. In any case, this a matter of terms definition where most questions can't be answered with wrong or right rather than useful or meaningless.

 

[KerimF]

Isn't the domain of phasors a subset of the one of vectors?
I mean:
Are there some properties of phasors that cannot be applied on vectors?

 

[Ratch]

FvM,

I hear from your explanation that you are associating vectors with the euclidian vector space R3, which is a kind of limited view, I think.

I was taught that a vector consisted of magnitude and direction.

Vector is a rather abstract term in mathematics, different kinds o vectors have different properties.

Anything mathematical is abstract. Vectors are descriptions of physical properties. In order to have different kinds of vectors, one must define what a vector is.

It seems to me that the said dot product has a meaning for AC quantities, it can e.g. represent real power.

I will concede that point, if the quantities are voltage and current phasors.

Direction has a different meaning in two-dimensional space than in R3

I think the meaning is the same, but the scope is limited in R2 space.

but many phenomena in AC networks and electrical machines can be well understood as a 2D direction

Attributes like phase difference should not be confused with direction.

But again, I'm not speaking against your description of phasors, only against the - as I think arbitrary - claim that they are no vectors.

My position is that if they don't have a direction, they are not a vector.

In any case, this a matter of terms definition where most questions can't be answered with wrong or right rather than useful or meaningless.

Yes, things have to be defined before declarations can be made. I have my definition of a vector. What is yours?

Ratch

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Isn't the domain of phasors a subset of the one of vectors?
I mean:
Are there some properties of phasors that cannot be applied on vectors?

OK, let's determine that. What is the domain of a phasor and the domain of a vector?

To what properties of phasors are you referring?

Ratch