What Is The Mathematical Relationship Between AC Power And DC Power?

[FredericChopin]

Is there a formula that shows the relationship between AC power and DC power?

Something on the lines of, for example:

Power of AC ... = Power of DC ... ?

Thank you.

 

[ehild]

What do you mean by AC power and DC power?

ehild

 

[FredericChopin]

What do you mean by AC power and DC power?

ehild

I mean power dissipated; as in P = W/t. The power dissipated by an AC and the power dissipated by a DC.

 

[Andrew Mason]

Is there a formula that shows the relationship between AC power and DC power?

Something on the lines of, for example:

Power of AC ... = Power of DC ... ?

Thank you.

For a DC circuit with constant voltage V and resistance R, the power is P = V²/R = VI

For AC, you have to define what you mean by voltage since it is changing all the time. You also have to provide details of the load.

If the load is purely resistive (no capacitors or inductors) the power is determined by P = V_rms²/R = V_max²/2R where V_rms is the root-mean-square value of the voltage over one cycle - that is to say that the square root of the average/mean of the squares of all the voltages over one cycle.

AM

 

[FredericChopin]

For a DC circuit with constant voltage V and resistance R, the power is P = V²/R = VI

For AC, you have to define what you mean by voltage since it is changing all the time. You also have to provide details of the load.

If the load is purely resistive (no capacitors or inductors) the power is determined by P = V_rms²/R = V_max²/2R where V_rms is the root-mean-square value of the voltage over one cycle - that is to say that the square root of the average/mean of the squares of all the voltages over one cycle.

AM

So... assuming the load is purely resistive, there is no formula that shows the relationship between DC power and AC power?

 

[ehild]

The power supplied by one source is not related with the power supplied by an other source. DC and AC sources are different sources.
You can speak about instantaneous power and average power. The instantaneous power is product of the instantaneous voltage and current. P(t) = U(t)*I(t).
In case of alternating voltage/current, it has more sense to use the average power, the average of P(t) for a cycle. For sinusoidal voltage and current, P_av=IoVo/2 where Io and Vo mean the maximum current/voltage.

ehild

 

[FredericChopin]

The power supplied by one source is not related with the power supplied by an other source. DC and AC sources are different sources.

ehild

Hm... Ok. So would it be wrong to say:

P_RMS of AC = P of DC

, assuming the load is purely resistive, or:

I_RMS of AC = I of DC for that matter?

Let me elaborate.

Let's say, for example, there is a DC circuit with a 2 Ω resistor which dissipates 100 W of power. There is also another circuit - an AC circuit - with the same resistor dissipating the same power.

Using the power equation, we know that there is 7.07 A of current passing through the DC circuit. Is it incorrect to say, therefore, that the average (RMS) current passing through the AC circuit is equal to 7.07 A?

 

[ehild]

The average of an AC current is zero. RMS means the square root of the time average of the square of the quantity, current or voltage.

We do not use the term "rms power", but average power. On a resistor R, and current I(t)=Io sin(wt), the power is P(t)=(Iosin(wt))²R, and the average of the current is Io²/2. Plot it and you will see. So the average power is P_av=RIo²/2. You can introduce the rms current which is I_rms=Io/√2, then you have the same formula for the average power you would get in case of I_rms=Io/√2 DC current. P_av=I_rms²R.

It is said that the rms value of the AC current is equal to that DC current which would dissipate the same power on a resistor as the AC current in average.

ehild

 

[FredericChopin]

The average of an AC current is zero. RMS means the square root of the time average of the square of the quantity, current or voltage.

We do not use the term "rms power", but average power. On a resistor R, and current I(t)=Io sin(wt), the power is P(t)=(Iosin(wt))²R, and the average of the current is Io²/2. Plot it and you will see. So the average power is P_av=RIo²/2. You can introduce the rms current which is I_rms=Io/√2, then you have the same formula for the average power you would get in case of I_rms=Io/√2 DC current. P_av=I_rms²R.

It is said that the rms value of the AC current is equal to that DC current which would dissipate the same power on a resistor as the AC current in average.

ehild

Ah... I understand.

So I_RMS of AC = I of DC

, for a current flowing through a resistor.

Can you extend that, therefore, and say:

So P_AVERAGE of AC = P of DC

, and:

V_RMS of AC = V of DC in the same situation?

 

[ehild]

, for a current flowing through a resistor.

Can you extend that, therefore, and say:

So P_AVERAGE of AC = P of DC

, and:

V_RMS of AC = V of DC in the same situation?

Well, it is about that. The rms value of the AC voltage/ current is the value of a hypothetical voltage/current that would dissipate the same average power on the same resistor.

So you can use P=UI=I²R=U²/R both for DC and AC, only in the case of AC U and I mean the rms values, and P is the average power.

The voltage of AC sources are given with their rms value. In my country, the household supply is 230 V, the frequency is 50 Hz. That means maximum voltage of 230√2=162 V. The time dependence of voltage is U(t) = 162 sin(100πt).

ehild

 

[FredericChopin]

Thank you very much.

 

[Andrew Mason]

So... assuming the load is purely resistive, there is no formula that shows the relationship between DC power and AC power?

Just to add to what ehild has said, power is the rate at which energy is delivered by electrical means. What you want to ask is not about the relationship between DC and AC power. Rather, it is about how to measure the rate at which energy is delivered by an AC source.

If you have a sinusoidal AC voltage with maximum (peak) voltage V_max applied to a purely resistive load R, then the rate at which energy is delivered to that resistive load is V_max²/2R. We refer to an AC voltage not by its maximum value but by its average power value, which is the Vrms value where:

V_rms² = V_max²/2

So, for example, when we refer to a household voltage being 120 VAC we really mean it is 120 Vrms. The voltage actually ranges from about 170 volts to 0 twice every cycle.

AM

 

[ehild]

So, for example, when we refer to a household voltage being 120 VAC we really mean it is 120 Vrms. The voltage actually ranges from about 170 volts to 0 twice every cycle.

AM

The voltage changes from 170 to -170 in every cycle...

ehild

 

[Andrew Mason]

The voltage changes from 170 to -170 in every cycle...

ehild

Yes. I was referring to the magnitude of the voltage. I should have said the magnitude of the voltage ranges from 170 volts to 0 twice in every cycle.

AM