# Electrical Circuits and Power

The word “electricity” is derived from the Greek word electron, meaning amber. The Greeks knew that a piece of amber rubbed against a cat’s fur would collect charges. Furthermore, they observed that identical objects touching the amber repelled each other. The same thing happened if the amber and fur were replaced with another pair of materials, glass and silk. However, if glass was brought next to amber, or if fur was brought next to silk cloth, the pairs would attract.

Benjamin Franklin (1706-1790) suggested that an imbalance results when objects having an excess of an invisible fluid (amber, glass) were brought next to objects deficient in this fluid (fur, silk); in an attempt to reach equilibrium, this invisible fluid would move from one object into the other and the two would attract. We had to wait another century until it was discovered that this fluid was, in fact, a stream of electrons that flows from one object to another when they rub. However, Franklin erroneously presumed a lack of the invisible fluid rather than a surplus of electrons; opposite to his assertion, electrons transferred from fur and silk to amber and glass (a).

Today, we understand the phenomenon by noting the existence of a charge imbalance between different atoms. As we discussed in previous chapters, atoms consist of negatively charged electrons orbiting a nucleus composed of positively charged protons and neutrons with no charge. For an atom to be neutral, the total number of protons and electrons is equal, resulting in a net zero charge. We also know that, just like with energy, electric charges are conserved. Electric charge can transfer from one material to another, but no new charge is created or destroyed.

Electric charges may remain stationary or flow through an electrical conductor. The study of charges at rest is called electrostatics. Electrostatic charges may be caused by friction (shuffling one’s foot across a carpeted floor), by contact (touching a metal door-knob), or by induction (lightning). Electrostatic charges are not of concern in this study and will not be discussed further. When motion of charges is involved, electric currents are produced that flows through circuits, powering devices ranging from simple battery-operated toys and home appliances to state of the art electronics.

## Electrical Circuits

To have a current between two points, we need a difference in electrical potential. This electrical potential can be supplied either by a battery (between the two electrodes), or by an electric utility company (between the two slots of a house’s electrical outlet). As long as there is no path between the two points (open circuit), or there exists a path that is non-conductive (such as connecting two point charges with a glass rod), no charge is transferred and no current flows. When the path is established by flipping a switch or by inserting a load, such as an electric motor or a light bulb, electricity flows. Unless a supply of charges from an external source (a battery or electric outlet) maintains the electric potential, the current eventually stops. In a battery, electric potential is maintained as it discharges through a chemical reaction. Household electrical outlets maintain electric potential by electrical generators driven by wind, water, gas, or steam turbines.

The amount of current that can flow through a conductor (wire) depends, not only on the electric potential across the wire, but also on its resistance along the path. This is given by Ohm’s Law, which is expressed as:

V = I.R (1)

where V is the voltage, I is the current, and R is the resistance.

The unit of electric current is amperes (A), defined as the quantity of charge in coulombs (C) flowing in one second. One coulomb is equal to the amount of charge possessed by 6.2x1018 electrons. In other words, one ampere of current represents the flow of 6.2x1018 electrons per second.

1 A = 1 C/s

Similar to gravitational potential energy, which is a measure of the work needed to raise a weight above the ground, the volt (V) is potential electric energy and is equal to the work needed to raise the level of one coulomb of charge from its ground level.

1 V = 1 J/C

The resistance is measured in ohms (W) and is the attribute of a conductor that controls the amount of current that can flow.

1 W = 1 V/A

More conductive wire material, bigger cross sectional area, and shorter length mean less resistance to electric flow and greater capacity to carry a current.

A current can either be uniform (direct current or DC) or varying in time (alternating current or AC). Direct current is the direct flow of electrons through a conductor. Alternating current results from the back and forth movement of electrons in a wire. The electrons themselves move very little, but their periodic motions result in the transfer of energy to adjacent electrons. This is similar to a longitudinal wave along a spring coil. Although atoms of the spring move only a little, a wave is propagated downstream, carrying the energy from one end of the spring to the other end. The electrical potential of an alternating current is created by moving a coil of wire in a magnetic field.

Usually the voltage provided by an electric utility company cannot be easily modified. Domestic electric service delivered by utility companies to US homes is 240 volt/60 hertz (b). This voltage is required for powering electric cooking ranges, clothes dryers, air conditioners, and other large appliances. The 240 V service is split at the power panel into the familiar 120 V circuits used for most household appliances which are plugged into wall outlets. Europe and Asia use 220 volt/50 hertz and appliances that are designed accordingly.

## Electrical Power

The electrical energy supplied by a generator or battery is either converted to work (as in an electric motor) or is dissipated to heat (as in an electric heater or an iron). The rate at which the energy is converted is called power, and is given by multiplying the rate at which electron flows (current) by the energy that is carried by each electron (voltage):

P = V . I (2)

Substituting for voltage from equation 1 we get:

P = R I² = V² / R (3)

These equations show that energy dissipated as heat by an electrical conductor increases as the square of the current (c). In a resistance heater, the goal is to maximize the amount of heat that can be dissipated, so we would prefer to draw high current. Since voltage is constant, equation 3 suggests we need to minimize resistance. The easiest way to control current in a particular circuit is to insert special components, called resistors, in various series or parallel configurations. Resistors are made of materials with low electrical resistivity but which can withstand high temperatures and are commercially available for a large range of resistances – from a fraction of an ohm to many megaohms (d).

If electric potential and current are given in volts and amperes, power will be in watts. One watt equals the energy used, in joules, in one second.

1 W = 1 J/s

One kilowatt is 1,000 W, and 1 megawatt is 1,000 kW or 106 W.

When discussing electricity consumption and production, it is customary to express power in kilowatts and time in hours. Since energy is the product of power and time, electrical energy is often expressed in kilowatt-hours (e). Table 1 shows power ratings for several household appliances.

 Table 1. Power Consumption for Several Household Appliances Appliance Power, W Appliance Power, W ClockRadioTelevision (color)Freezer/RefrigeratorHair DryerWasher (clothes)Vacuum CleanerHome Computer 270 - 40065 - 200600 - 10001200 - 1800350 - 5001000 - 1400100 - 400 Air ConditionerCoffee MakerIronDishwasherToasterMicrowave OvenDryers (clothes)Oven range 800 - 1000900 - 12001200 - 2400850 - 1400800 - 1400750 - 11001800 - 500010000 - 12000 Source: US DoE Office of Energy Efficiency and Renewable Energy, http://www.eere.energy.gov.

Example: Calculate the power dissipated by the lamp in the circuits.

Solution: Case (a): I = V/R = 18/3 = 6 A; P = V x I = 18 x 6 = 108 W Case (b): I = V/R = 36/3 = 12 A; P = V.I = 36 x12 = 432 W Note that increasing the battery voltage by a factor of 2, from 18 V to 36 V, increased the current by a factor of 2 and power by a factor of 2 x 2 = 4.

Example: A 100-W light bulb operates in an American household for 4 hours every night.

a. How much current is drawn by the light bulb?

b. What is the resistance in light bulb filament?

c. How much power is consumed by the light-bulb?

d. What is the monthly cost of operating this light bulb if the electricity is charged at a rate of \$0.10/kWh?

Solution: American electrical power is delivered at 120 volts.

a. The current drawn is: I = P/V = 100 W/120 V = 0.833 A

b. The resistance is calculated as: R = V/I = 120 V/0.833 A = 144W

c. The power is calculated as: P = VI = 120 Vx0.833 A = 100 W

d. Monthly energy consumption by the light bulb is: E = (0.1 k W)x(4 hr/day)x(30 days) = 12 kWh. The monthly electrical cost is 12x0.10 = \$1.20

Question: In the example above, which draws more current: the wire leading to the light bulb, or the bulb filament? Which gets hotter?

Answer: The flow of electrons is exactly the same in all parts of the circuit. As the electrons try to overcome the greater resistance in the thin filament, they heat it. The wires connecting to the outlet are significantly thicker than that of the filament and therefore barely warm as they allow the same amount of current to pass through.

## Series and parallel Circuits

Different electrical devices can be assembled in a circuit in series, in parallel, or in a combination of the two. In a series configuration, the output current of one device will be the input current of a second device, and so on; the same current passes through all devices (Figure 1a). Since the total load increases for the same voltage potential, according to Ohm’s law, the current must decrease. In a series circuit, resistances add up:

Rtotal = R1+ R2+… Rn (4)

Since the current passes through both resistances, the voltage across the two is the sum of the voltages across the individual resistances.

V=V1+V2.+… Vn (5)

The main disadvantage of a series configuration is that if one of the devices burns out we have an “open circuit,” in which current stops and none of the devices in the series circuit will work.

Example: Typically, Christmas tree decoration lights are arranged in a series configuration. Assume a strand of 50 miniincandescent bulbs is connected to a 120 V outlet; calculate the current drawn by the strand and the total power consumed. A typical bulb has a resistance of 8 ohms.

Solution: The voltage across each bulb is 1/50th the voltage drop across the strand, or 120/50 = 2.4 volts. The current drawn is I = V/R = 2.4/8 = 0.3 A. The power consumed is P = 50.VI = 50x2.4x0.3 = 36 W. As explained above, when the bulbs are put in series, if one bulb burns out the entire strand will go out. Today’s Christmas lights, however, will stay on even if one or more lights are burned out. The trick is that new bulbs have an internal shunt resistance connecting two poles. The shunt is coated with a high-resistance material that prevents shorting the filament. When a light burns out, the heat causes the coating to melt thus reducing the shunt resistance and allows the current to pass through the rest of the strand so the remaining lights will stay on.

In a parallel configuration, the same voltage potential is applied to all devices. The current necessarily divides. Since the same voltage is supplied to all electrical devices, each device works independently of others and a burned out device does not affect the operation of other devices. In a parallel circuit, currents add up:

Itotal = I1+ I2+…. In (6)

In other words, parallel resistances act to divide the current. It is left as an exercise to show that several resistances placed in parallel can be replaced by an equivalent resistance equal to:

1/Rtotal = 1/R1 + 1/R2+...1/Rn (7)

Example: A 1,200-W electrical heater, ten 100-W electric light bulbs, a 300-W refrigerator, and a 1,500-W microwave oven operate simultaneously. Assuming that the household circuit is 120 V and all appliances are placed in parallel, what is the total current drawn?

Solution: Applying Ohm’s law (I = P/V) for each device and using power ratings given in Table 1, we can write:

Electric Heater: 1200/120 = 10.0 A Ten 100-W light bulbs: 10x100/120 = 8.33 A Refrigerator: 300/120 = 2.50 A Microwave Oven: 1500/120 = 12.50 A

Total current drawn: 33.33 A

This is a large current and may be a fire hazard. Houses are usually equipped with 10-20 A circuit protection. If current exceeds these values, a fuse will burn out or a circuit breaker will trip, disconnecting the circuit. In house wiring, several circuits are used, each equipped with a separate circuit breaker. In the example above, at least two circuits each capable of carrying 20 A maximum are needed if all appliances are to be used simultaneously. Indeed, US electric codes require a microwave, garbage disposal, and refrigerator to have a 20 A circuit separate from lighting circuits and wall outlets for other appliances.

Example: A three-way light bulb can be constructed by putting two resistances in parallel. Whether the first, second, or both resistances are placed in circuit, the light can be dim, bright, or very bright. What are the values for the two resistances if the light bulb can be operated to dissipate 50/100/150 W of electricity?

Solution: The value of resistances and currents are calculated as:

1/R = 1/R1 + 1/R2 = 0.0124

R = 80.67W

P = V² / R = 110²/80.67 = 150 W

The total current is: I=I1+I2=0.45+0.91=1.36 A

At the brightest position, we have

R1 = V² / P1 = 110²/50 = 242 W; I1 = V / R1 = 110 / 242 = 0.45 A

R2 = V² / P1 = 110²/100 = 121 W; I2 = V / R2 = 110 / 121 = 0.45 A

as expected.

## References

(1) Toossi Reza, "Energy and the Environment:Sources, technologies, and impacts", Verve Publishers, 2005

(a) When Benjamin Franklin made his conjecture regarding the invisible fluid, he assumed that fluid flowed from glass to silk. In reality, silk has an excess of electrons and glass is deficient in electrons. If we had designated electrons as having positive charges, the flow of current would have been from higher to lower concentration of electrons, in line with the assertion that excess fluid moved from one object to the other.

(b) Hertz is the unit of frequency (cycles/s).

(c)The equations above are for direct currents. The same equations work with alternating currents, except current and voltage values must be substituted with “root-mean-squared” values.

(d) Note that resistivity is not the same as resistance. Resistivity is a property of material, whereas resistance represents the impedance to the flow of electricity and depends not only on the resistivity of the material, but also on size and geometry.

(e) 1 kWh = 1,000 W x 3,600 s = 3.6x106 J.

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The Electricity Journal, Direct Science Elsevier Publishing Company, This journal addresses issues related to generating power from natural gas-fired cogeneration and renewable energy plants (wind power, biomass, hydro and solar).

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National Council on Electricity Policy (http://www.ncouncil.org).

Southern California Edison (http://www.sce.com).

Pacific Gas and Electric (http://www.pge.com).