Current, resistance and DC circuits

So far the charges have mostly stayed put. When they move, we have electric current, and with it the ability to carry energy from a power plant to a city, run motors and heaters, and process information in computers. This chapter is the foundation of circuit analysis, which electrical, computer and mechanical engineers use every day.

It defines current, resistance and resistivity, states Ohm's law, computes electrical power and energy costs, and develops the tools for analyzing circuits: series and parallel combinations and Kirchhoff's two laws. It ends with real batteries, voltage dividers and electrical safety.

Electric current

Current is the rate at which charge flows past a point:

By convention, current flows in the direction positive charge would move: out of the positive terminal of a battery, through the external circuit, and back into the negative terminal. In metal wires the moving charges are actually electrons drifting the other way, but for circuit analysis the convention gives identical results.

Typical currents: a phone charging at a few amperes, a household circuit breaker rated at 15 A, a car starter motor drawing 150 A or more, and a lightning stroke peaking around 30 000 A.

Drift velocity

Electrons in a wire move randomly at high speed, but their net drift along the wire is remarkably slow. If the wire has cross-sectional area \(A\) and \(n\) free electrons per cubic meter, the current is \(I = nqv_dA\), so the drift velocity is \(v_d = I/(nqA)\).

Resistance and Ohm's law

Pushing current through a material takes a voltage, because the moving charges collide with the atoms. The resistance of a component is the ratio of the voltage across it to the current through it:

For many materials, including metals at constant temperature, \(R\) is constant over a wide range of voltage. Such materials are called ohmic, and the relation \(V = IR\) is Ohm's law. Diodes, transistors and lamps with hot filaments are not ohmic: their current is not proportional to voltage.

Resistivity

The resistance of a uniform wire depends on its material, length and thickness:

where \(\rho\) is the resistivity of the material in Ω·m. Copper has \(\rho = 1.68 \times 10^{-8}\) Ω·m and aluminum \(2.65 \times 10^{-8}\) Ω·m; nichrome (used in heating elements) is about \(1.1 \times 10^{-6}\) Ω·m; glass and rubber are \(10^{10}\) Ω·m and higher. A longer wire has more resistance; a thicker one has less.

Temperature dependence

The resistance of metals rises with temperature, approximately as \(R = R_0[1 + \alpha(T - T_0)]\). For copper, \(\alpha = 0.00393\) per °C, so a winding that heats from 20 °C to 75 °C has 22% more resistance. This effect is used deliberately in resistance temperature detectors (RTDs), platinum sensors common in process plants.

Electrical power and energy

When current flows through a voltage, energy is transferred at the rate

In a resistor all this power becomes heat (Joule heating). That is useful in kettles, toasters and electric furnaces, and wasteful in transmission lines and motor windings. The three forms are equivalent for a resistor; choose the one that uses what you know.

Utilities bill energy, not power, in kilowatt-hours: 1 kWh is 1 kW for one hour, or 3.6 MJ. A 100 W lamp left on for 10 hours uses 1 kWh, which costs about 15 cents at a rate of $0.15/kWh.

Series and parallel resistors

Resistors in series carry the same current, and their voltages add, so their resistances add:

Resistors in parallel share the same voltage, and their currents add, so their conductances (\(1/R\)) add:

Parallel resistance is always less than the smallest resistor, because each extra path makes it easier for current to flow. For two in parallel, \(R = R_1R_2/(R_1 + R_2)\). Household outlets are wired in parallel, so each appliance gets the full voltage and switching one off does not affect the others.

Kirchhoff's laws

Many circuits cannot be reduced to series and parallel combinations, especially those with more than one source. For these, Gustav Kirchhoff's two laws always work:

• The junction (current) law: the total current flowing into any junction equals the total flowing out. This is conservation of charge.
• The loop (voltage) law: around any closed loop, the sum of the voltage rises and drops is zero. This is conservation of energy.

Two systematic methods build on these. In mesh analysis (which you met in Linear Algebra) you assign a current to each loop and write the loop law for each. In node analysis you choose a reference node (ground, 0 V), assign unknown voltages to the other nodes and write the junction law at each. Both give a linear system to solve.

A negative answer is not an error. It simply means the current flows opposite to the direction you assumed. Choose directions freely, solve, and let the signs tell you the truth.

Real sources and voltage dividers

A real battery is an ideal source of voltage \(\mathcal{E}\) (its emf) in series with a small internal resistance \(r\). When it supplies current \(I\), its terminal voltage drops to \(V = \mathcal{E} - Ir\).

A voltage divider is two resistors in series across a source, with the output taken across one of them: \(V_{\text{out}} = V_{\text{in}}R_2/(R_1 + R_2)\). Dividers set reference voltages, scale sensor signals for a microcontroller and read potentiometers. But attaching a load changes them: a 12 V divider made of 10 kΩ and 5 kΩ gives 4.0 V unloaded, but only 2.4 V if a 5 kΩ load is connected across the 5 kΩ resistor (the parallel pair becomes 2.5 kΩ). Keep the load resistance much larger than the divider's resistors.

Electrical safety

It is current through the body, not voltage, that causes harm. Around 1 mA can be felt, 10 to 20 mA can make it impossible to let go, and 50 to 100 mA through the chest can trigger ventricular fibrillation. Dry skin has a resistance of tens of kilohms, but wet skin can fall below 1 kΩ, which is why bathrooms and outdoor outlets need special protection.

• Fuses and circuit breakers protect wiring from overheating by cutting off excessive current. They protect the building, not you.
• Ground-fault circuit interrupters (GFCIs) compare the current going out and coming back and trip within milliseconds if more than about 5 mA is missing, because that current may be flowing through a person.
• Grounding the metal case of an appliance means a fault sends current to ground and trips the breaker, rather than leaving the case live.