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

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SUMMARY

The mathematical relationship between AC power and DC power is defined primarily through the concepts of root-mean-square (RMS) values and average power. For a purely resistive load, the power dissipated in a DC circuit is given by P = V²/R, while for AC, it is Paverage = Vrms²/R, where Vrms is the RMS voltage. The RMS current (Irms) for AC can be equated to the DC current (I) that would dissipate the same power in a resistor. Thus, it is established that Paverage of AC equals P of DC under these conditions.

PREREQUISITES
  • Understanding of electrical power formulas, specifically P = VI and P = V²/R.
  • Knowledge of AC and DC circuit characteristics, particularly RMS values.
  • Familiarity with resistive loads and their behavior in electrical circuits.
  • Basic grasp of sinusoidal waveforms and their mathematical representations.
NEXT STEPS
  • Study the derivation and application of RMS values in AC circuits.
  • Learn about the differences between instantaneous power and average power in AC systems.
  • Explore the implications of reactive components (inductors and capacitors) in AC power calculations.
  • Investigate real-world applications of AC and DC power in household and industrial settings.
USEFUL FOR

Electrical engineers, physics students, and professionals working with power systems who need to understand the differences and relationships between AC and DC power in practical applications.

FredericChopin
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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.
 
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What do you mean by AC power and DC power?

ehild
 
ehild said:
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.
 
FredericChopin said:
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 = Vrms2/R = Vmax2/2R where Vrms 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
 
Andrew Mason said:
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 = Vrms2/R = Vmax2/2R where Vrms 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?
 
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, Pav=IoVo/2 where Io and Vo mean the maximum current/voltage.

ehild
 
Last edited:
ehild said:
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:

PRMS of AC = P of DC

, assuming the load is purely resistive, or:

IRMS 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?
 
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 Pav=RIo2/2. You can introduce the rms current which is Irms=Io/√2, then you have the same formula for the average power you would get in case of Irms=Io/√2 DC current. Pav=Irms2R.

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
 
ehild said:
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 Pav=RIo2/2. You can introduce the rms current which is Irms=Io/√2, then you have the same formula for the average power you would get in case of Irms=Io/√2 DC current. Pav=Irms2R.

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 IRMS of AC = I of DC

, for a current flowing through a resistor.

Can you extend that, therefore, and say:

So PAVERAGE of AC = P of DC

, and:

VRMS of AC = V of DC in the same situation?
 
Last edited:
  • #10
FredericChopin said:

, for a current flowing through a resistor.

Can you extend that, therefore, and say:

So PAVERAGE of AC = P of DC

, and:

VRMS 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
 
  • #11
Thank you very much.
 
  • #12
FredericChopin said:
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 Vmax applied to a purely resistive load R, then the rate at which energy is delivered to that resistive load is Vmax2/2R. We refer to an AC voltage not by its maximum value but by its average power value, which is the Vrms value where:

Vrms2 = Vmax2/2

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
 
  • #13
Andrew Mason said:
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... :-p

ehild
 
  • #14
ehild said:
The voltage changes from 170 to -170 in every cycle... :-p

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
 

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