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

• FredericChopin
In summary: The household AC voltage is not constant, it changes between 162 V and -162 V, but the average power it supplies to the electric devices is the same as a 230 V DC voltage. In summary, the relationship between AC power and DC power depends on the type of load and the values used for calculations. However, for a purely resistive load, the average power of an AC circuit can be equated to the power of a DC circuit, and the rms values of AC voltage and current can be equated to the voltage and current of a DC circuit. This is because the rms values represent the hypothetical values of voltage and current that would dissipate the same average power on the same resistor

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.

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

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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?

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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

Thank you very much.

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

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... :tongue2:

ehild

ehild said:
The voltage changes from 170 to -170 in every cycle... :tongue2:

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

What is the mathematical relationship between AC power and DC power?

The mathematical relationship between AC power and DC power is based on the concept of power, which is the rate at which energy is transferred. In both AC and DC systems, power is calculated as the product of voltage and current. However, the key difference between the two is that AC power is constantly changing, while DC power remains constant.

How is AC power different from DC power?

AC power, or alternating current power, is a type of electrical current that periodically reverses direction, changing polarity. This means that the voltage and current are constantly changing in magnitude and direction. On the other hand, DC power, or direct current power, flows in one direction only and the voltage and current remain constant.

Can AC power be converted to DC power?

Yes, AC power can be converted to DC power using a device called a rectifier. A rectifier is an electrical device that converts AC power to DC power by allowing current to flow in one direction only. This conversion is commonly used in electronic devices that require DC power to function.

What is the main advantage of using AC power over DC power?

The main advantage of AC power over DC power is that it can be easily transformed to different voltages using a transformer. This allows for the efficient transmission of electricity over long distances. In contrast, DC power cannot be easily transformed and is generally used for shorter distance applications.

How does the frequency of AC power affect its energy consumption?

The frequency of AC power does not affect its energy consumption. The energy consumption of AC power is determined by its voltage, current, and power factor. The frequency only affects the rate at which the power is delivered, but not the total amount of energy consumed.