Here, we take a closer look at the energy and power in electrical circuits. But first let's refresh our minds about the energy and power in general. Do you remember Newton? One of the greatest scientists that I really like. He showed first how force accelerates a mass. The unit of force is called after him: Newton. When a force shown as 'F' (in Newton) accelerates a mass shown as 'm' (in Kilogram), it results in acceleration that is shown as 'a' (meter/second2). The formula relating these is the famous one below:
Now for any object to move, energy is required and every moving object as some energy associated to it. Energy is shown with the letter 'W' and the unit of energy is called Joule or shorted as 'j'. Basically, when a force of F moves a mass for a distance of x meters, the energy that the force F spends on this mass is equal to:
Therefore the more distance a force moves an object, the more energy is spent. There are two kind of energies: Kinetic and Potential. The kinetic energy is the energy of a moving object. When a mass of m is moving with the speed of v, its energy is calculated from the formula below:
The potential energy is the energy of an object that is not moving yet, but has the potential to move if its energy is released. An example of such energy is a mass of m placed h meters higher than the earth ground. It has the potential energy to fall with a value calculated from the formula below:
where g is the earth gravity acceleration equal to approximately 9.8 Newton/meter (N/m).
Power on the other hand is the amount of energy spent in the unit of time (second) shown as 't'.
As you see from the formula, power is shown by 'P' and the unit of power is Watt, named after the famous James Watt, the father of renaissance, creator of the steam engine. In an electrical circuit, electrons move inside a wire. Of course they need energy to move. So let's put formulas together and find the energy and power of electricity. As mentioned before in the voltage section, voltage is energy used to move an electrical charge of q coulombs. One volt is one Joule of energy used to move one Coulomb of electrical charge, or more accurately:
Voltage is a potential energy, whenever it is released on an electrical circuit, it moves the electrons around creating electrical current. From the electrical current section we also know that the electrical current is equal to the mount of electrical charge passing in the unit of time:
Therefore using the voltage and current formulas we get:
Up to now we assumed constant values for our voltage and current and got the formula above for energy. But what if they are changing by time? To understand the final formula, let's assume a very short period of time as Delta t. In that short period of time we can assume that the voltage and current are constant. Of course the smaller the time is, the more constant they get. So we can write that the energy difference or Delta W is calculated from the formula below:
For small period of time approaching zero, we show Delta t as dt and Delta W by dW. Therefore to calculate the energy consumed between the time of t1 and t2 we integrate the formula and get:
The power is calculated from the formula below as it is the energy it the unit of time:
Or in the case of variable voltage and current the power is:
Later on you will see that there is a kind of power named Real Power, dissipated in the circuit as heat. There is also Imaginary Power that doesn't result in heat dissipation. This returns to complex numbers where there are real and imaginary parts, here as parts of power. For now stick with the real power, as this is the greatest concern of electrical designers. Real power is what basically we are charged for as our electricity consumption. It is the factor that determines the life time of a battery under different loads. It is always economical to keep the real power as low as possible in any thing, as power doesn't come cheap.
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