# Interactions Between Electrical and Nonelectrical Systems

## Introduction

An electrical system is often used to drive a non-electrical system (in an electric stove, for example, electric energy is converted to heat). Interactions between electrical and non-electrical systems are often described in terms of power. Electrical power associated with a particular circuit element is the product of the current passing through the element and the voltage difference across the element. This is often written as:

(1.)

$p(t) = v(t) \cdot i(t)$

Where p(t) is the instantaneous power at time t, v(t) is the voltage difference at time t, and i(t) is the current at time t. Units of power are watts, abbreviated W.

#### Note

• Power, either electrical or mechanical, is the rate at which work is being done.

• In electrical systems, voltage is an energy difference between two points. We can do work (or equivalently, transfer energy) by moving a unit of charge from one voltage level to another. Current is the rate at which charge is moving. Therefore, the rate at which we are doing work is the product of voltage and current.

• It also follows that the units of power, watts, is equivalent—for electrical systems—to volts multiplied by amperes. Since power is a rate of energy transfer, watts are also equivalent to joules/second.

## Sign Convention

Systems can either generate or absorb power—depending on whether its energy is decreasing or increasing. For example, if we lift a weight in a gravitational field, the weight is absorbing power, since we add energy to the weight from some other source to lift it. Conversely, water flowing downhill can turn a turbine1—the water is generating power, since its energy is being transferred to the turbine.

#### Analogy:

• A bow and arrow also illustrates these points. When we pull the bowstring back, the bow is storing energy—it is absorbing power. Releasing the bowstring transforms the bow into a power generator—the energy it has stored is transferred to the arrow. Of course, now the arrow is absorbing power!

• To take the analogy a bit further, when the arrow strikes its target, it will generate power by transferring its energy to the target (possibly by creating heat from friction or maybe by compressing the target material).

• Energy is never destroyed; we just move it from place to place. Unfortunately, sometimes we move it to a place where we can't get it back! Usually, we try to avoid this—regenerative braking systems on hybrid cars, for example, attempt to recover the energy usually “lost” during braking by using that energy to run a generator which re-charges the battery.

Electrical power can be either absorbed by a circuit element or generated by a circuit element. In electrical systems, the determination as to whether the element is absorbing or generating power can be made by the relative signs of the values of voltage and current. These sign conventions are an important issue, since it's vital to know where our energy is going.

Equation (1). above defines power as the product of the voltage times current:

$P = vi$

The power is positive if the signs of voltage and current agree with the passive sign convention—that is, if positive current enters the positive voltage polarity node. If the power is positive, the element is absorbing power. The power is negative if the signs of voltage and current disagree with the passive sign convention—that is, if positive current enters the negative voltage polarity node. If the power is negative, the element is generating power.

#### Example:

In Fig. 1(a) below, the element agrees with the passive sign convention since a positive current is entering the positive voltage node. Thus, the element of Fig. (a) is absorbing energy. In Fig. (b), the element is absorbing power— positive current is leaving the negative voltage node, which implies that positive current enters the positive voltage node. The element of Fig. (c) generates power; negative current enters the positive voltage node, which disagrees with the passive sign convention. Figure (d) also illustrates an element which is generating power, since positive current is entering a negative voltage node.

## Conservation of Power

As we noted above in our bow-and-arrow analogy, we never really create or destroy energy, we just move it around. This observation leads us to the law of conservation of energy, which basically states that any power generated somewhere is absorbed somewhere else, and vice-versa.

A more official statement of this law is that, for any closed2 system:

• The total power absorbed is equal to the total power generated.

Since, in our previous subsection, we defined the power absorbed as being positive and the power generated as being negative, we can state this more mathematically as:

• The sum of the power in a closed system is zero.

#### Example:

Determine the power absorbed or generated by the circuit element A in the circuit below.

#### Solution:

Since we are given voltages and currents in each of the other elements in the circuit, we can use conservation of power to solve the problem. If PA is the power for element A, we can write:

$\left( { - 2A} \right)\left( {6V} \right) + \left( {2A} \right)\left( {4V} \right) + \left( {3A} \right)\left( {4V} \right){\rm{ + }}{P_A} = 0$
• So: ${P_A} = - 8W$

• So element A is generating 8W of power3.

#### Important Points:

• In electric circuits, power is the product of voltage times current.

• The sign of the power is determined based on the passive sign convention. If the voltage and current agree with the passive sign convention (positive current enters the positive voltage terminal) the power is positive. If the voltage and current do not agree with the passive sign convention (if positive current enters the negative voltage terminal) the power is negative.

• The sign on power indicates whether power is absorbed or generated. If the power is positive, power is being absorbed. If the power is negative, power is being generated.

• In any closed system, power must be conserved. This means that the total power absorbed must equal the total power generated. Another way to state this—since the sign of the power indicated whether it is absorbed or generated—is that the sum of all the power in the circuit must be zero.

1. For the circuit elements below, determine the power generated or absorbed by the element. State whether the power is generated or absorbed.
2. For the circuit elements below, determine the power generated or absorbed by the element. State whether the power is generated or absorbed.
3. In the circuit below, determine the power absorbed or generated by the 10V source. State whether the power is absorbed or generated.

• 10W (absorbed)
• 3W (generated)
• 30W (absorbed)
• -4W (generated)

• -6W (generated)
• -4W (generated)
• 6W (absorbed)
• 4W (absorbed)

1. P = (2A)(10V) = 20W (Absorbed, since the voltage and current agree with the passive sign convention. The battery is apparently being charged by the circuit.)

• 1 Or turn a millwheel, if you are really old-fashioned.
• 2 A closed system doesn't allow energy to leave or enter it. This makes the definition sort of circular, since the law of conservation of power is for closed systems and closed systems are systems which obey the law of conservation of power. Defining a closed system is sort of a problem for engineers—our “closed” systems usually consist of the system we're analyzing and “everything else”. For example, if we are trying to design a heating system for a house, we need to account for the head added to the house by our furnace, the temperature of the house, and the heat lost to the environment, which includes everything outside the house. The earth, by the way, doesn't count as a closed system—we are constantly receiving energy from the sun.
• 3 It is not necessary to use conservation of power to solve this problem. We could use KVL and KCL to determine the voltage and current for element A and then find the power from its voltage and current. However, it's always good to have more than one possible approach available to solve a problem—that allows us to cross-check our results. This can be important, if for example, peoples' lives depend on the problem being done correctly.