Unit 2: Electrical Circuits, Laws, and Measurement Standards

Fundamentals of Electrical Quantities

The volt is a fundamental unit of measurement defined as the potential difference across two points of a wire when it carries a current of $1\,A$ and dissipates $1\,W$ of energy. It is frequently referred to as potential difference, representing the potential capacity to perform work. In technical terms and within Ohm’s Law, the symbol for the volt is $E$, which stands for electromotive force, or EMF. This force serves as the electrical pressure that enables current to flow by pushing electrons through a medium. A single volt will provide the electrical pressure necessary to cause $1\,C$ ($6.25 \times 10^{18}$ electrons) to perform $1\,J$ of work. Alternatively, it is defined as the pressure required to move $1\,A$ of current through $1\,\Omega$ of resistance.

The ampere, represented by the symbol $I$ for intensity, measures the number of electrical charges passing a specific point in one second. One ampere is equal to one Coulomb passing a point per second, which consists of approximately $6.25 \times 10^{18}$ electrons, or exactly $6,250,000,000,000,000,000$ electrons. This movement of particles is what is harnessed to perform electrical work.

Electrical resistance is the inherent property of a substance to limit the flow of current. The unit of measurement for resistance is the Ohm, symbolized by the Greek letter omega (̐). One Ohm is the amount of resistance that limits current flow to exactly $1\,A$ when an electromotive force of $1\,V$ is applied. All matter possesses electrical properties because electrons govern chemical behavior; consequently, every substance is electrical in nature and can conduct electricity if subjected to sufficient voltage. In mathematical expressions, resistance is denoted by the letter $R$.

Power represents both the force and the flow rate of electricity as it passes through a device. It describes the rate at which electrical energy is converted into a usable form, such as heat or motion. In algebraic terms, power is expressed as either $P$ or $W$ for Watts. The standard formula for power is $P = E \times I$, meaning that wattage is equal to the voltage multiplied by the amperage. To successfully utilize electricity, the force or the quantity of flow must be varied and controlled.

Standards of Measurement and the Metric System

Communication of technical information requires established measurement standards to ensure universal meaning. A failure to adhere to these standards can have catastrophic consequences, as seen on September 23, 1999, when the Mars Climate Orbiter was lost. Engineers confused metric and imperial measurements, resulting in a error costing $125$ million dollars. While that probe remains tumbling through space, it serves as a reminder of the necessity for precise unit conversion.

Historical units of measure in the Imperial system were often based on inconsistent or arbitrary physical references. A "Chain" was a surveying tool consisting of $100$ links, totaling $66$ feet. A "Rod" was defined by the length of $16$ men standing one behind the other, equaling $16.5$ feet. A "Yard" was established as the distance from the tip of a king's nose to the tip of his outstretched hand, equaling $3$ feet. A "Foot" was derived by dividing the rod by $16$, resulting in $12$ inches. The "Inch" was historically defined as the length of three round, dry barley corns taken from the center of the ear and laid end to end. These variations often led to confusion and economic disputes.

In the 1790s, France developed the metric system, a base-ten system that remains the global standard. The SI (Système International) consists of seven basic units: the Meter ($m$) for length, Kilogram ($kg$) for mass, Second ($s$) for time, Ampere ($A$) for electric current, Kelvin ($K$) for temperature, Candela ($cd$) for light intensity, and Mole ($mol$) for molecular substances. Derived units include the Liter ($l$) and Cubic Meter ($m^3$) for volume, Square Meter ($m^2$) for area, Newton ($N$) for force, Meter per second ($m/s$) for speed, Joule ($J$) for energy, and Watt ($W$) for power.

Standard prefixes are used to modify these units based on factors of ten. Large factors include Exa ($E, 10^{18}$), Peta ($P, 10^{15}$), Tera ($T, 10^{12}$), Giga ($G, 10^9$), Mega ($M, 10^6$), Kilo ($k, 10^3$), Hecto ($h, 10^2$), and Deka ($da, 10^1$). Small factors include Deci ($d, 10^{-1}$), Centi ($c, 10^{-2}$), Milli ($m, 10^{-3}$), Micro ($μ, 10^{-6}$), Nano ($n, 10^{-9}$), Pico ($p, 10^{-12}$), Femto ($f, 10^{-15}$), and Atto ($a, 10^{-18}$).

Ohm’s Law and Circuit Analysis

Ohm’s Law defines the proportional relationship between voltage, current, and resistance in an electrical circuit, expressed as $E = I \times R$. This relationship implies that if one quantity changes, at least one other must change immediately. The formula can be rearranged depending on which value is unknown: $I = \frac{E}{R}$ to find current, and $R = \frac{E}{I}$ to find resistance. For example, if a circuit has a current of $2\,A$ and a resistance of 50\,̐, the voltage must be $100\,V$ ($2 \times 50$). If the current drops to $1\,A$ while resistance remains 50\,̐, the voltage must drop to $50\,V$, demonstrating that less pressure is required to move less current through the same resistance.

Kirchoff’s Laws, established in 1847, further describe circuit behavior. Kirchoff’s Voltage Law states that the sum of all voltage drops across components in a circuit will exactly equal the total supply voltage. Kirchoff’s Current Law states that the total current entering a circuit must be equal to the total current leaving it. These rules are essential for understanding both series and parallel configurations.

Series Circuit Characteristics and Calculations

In a series circuit, there is only one possible path for electron flow. Because there are no branches, the same current ($I$) must pass through every device in the circuit. For this reason, control devices like switches and fuses are always wired in series with the load. Voltage is dissipated as it passes through each device, resulting in a "voltage drop." The sum of these individual drops equals the source voltage. Series circuits can also function as voltage dividers, providing specific desired voltages at different points relative to a common reference ground when measured in parallel.

Solving a series circuit involves four primary steps. First, calculate the total resistance ($R_T$) by summing all individual resistors: $R_T = R_1 + R_2 + R_3 + ...$. Second, use the total resistance and source voltage ($E_T$) to find the total current ($I_T$) using $I = \frac{E}{R}$. Third, calculate individual voltage drops for each resistor using $E_n = I_T \times R_n$. Finally, calculate the power dissipated at each resistor or the total power using $P = E \times I$.

In a sample series circuit with a $24\,V$ supply and three resistors (R_1 = 200\,̐, R_2 = 300\,̐, and R_3 = 160\,̐), the total resistance $R_T$ is 660\,̐ ($200 + 300 + 160$). The total current $I_T$ is approximately $0.03636\,A$, or $36.36\,mA$ ($24 / 660$). The voltage drops are calculated as follows: $E_1 = 0.03636 \times 200 = 7.273\,V$; $E_2 = 0.03636 \times 300 = 10.91\,V$; and $E_3 = 0.03636 \times 160 = 5.818\,V$. Summing these drops ($7.273 + 10.91 + 5.818$) equals the $24\,V$ supply. Total power for this circuit is $P_T = 24 \times 0.03636 = 0.87264\,W$, or $872.6\,mW$.

Parallel Circuit Characteristics and Calculations

In parallel circuits, electricity follows multiple paths. Unlike series circuits, the voltage remains constant across every branch; thus, $V_{\text{total}} = V_1 = V_2 = V_3$. However, the current divides among the branches. The total current ($I_T$) is the sum of the individual branch currents: $I_{\text{total}} = I_1 + I_2 + I_3 + ...$. Because voltage is constant, branch current is determined by branch resistance: $I_n = \frac{E}{R_n}$. If a $24\,V$ source powers three parallel resistors of 200\,̐, 300\,̐, and 160\,̐, the branch currents are $0.12\,A$, $0.08\,A$, and $0.15\,A$ respectively, totaling $0.35\,A$ or $350\,mA$.

Total resistance in a parallel circuit is always less than the resistance of the smallest individual resistor because the additional paths make it easier for current to flow. There are three methods to calculate $R_T$. First, if all resistors are equal, use $R_T = \frac{R}{N}$, where $R$ is the resistance value and $N$ is the number of resistors. For example, four 500\,̐ resistors in parallel result in an $R_T$ of 125\,̐. Second, for two resistors, use the "Product over Sums" method: $R_T = \frac{R_1 \times R_2}{R_1 + R_2}$. For more than two resistors using this method, calculate the result for the first two, then use that result with the next resistor, and so on.

The third and most universal method is the Reciprocal Formula: $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ...$. After summing the reciprocals of all resistors, the final value must be inverted (using the $\frac{1}{x}$ key on a calculator) to find $R_T$. For example, with resistors of 100\,̐, 78\,̐, 67\,̐, and 350\,̐, the calculation is $\frac{1}{100} + \frac{1}{78} + \frac{1}{67} + \frac{1}{350} \approx 0.0406021$. The reciprocal of this value gives an $R_T$ of approximately 24.63\,̐. When using the reciprocal formula, rounding to 4 or 5 decimal places is recommended to maintain accuracy.

Practical Measurement for Electricians

Electricians primarily measure voltage ($V$), resistance in Ohms (̐), and amperage ($A$). Voltage is always measured in parallel with the load. To measure the potential difference at various points in a voltage divider circuit, one lead of the voltmeter is connected to a common reference point while the other probes the desired location. Amperage is measured in series with the load, requiring the meter to become part of the current path.

In industrial applications, signals often involve very small quantities, such as the common $4\text{-}20\,mA$ signal. A milliamp ($mA$) is $1/1000$th of an ampere, and a millivolt ($mV$) is $1/1000$th of a volt. Conversely, capitalized prefix letters indicate larger, potentially hazardous quantities, such as Kilovolts ($KV$), Megawatts ($MW$), or Megohms (M̐). Polarity is a critical consideration during measurement. If a digital meter is connected with reverse polarity, it will display a negative number. However, an analog meter connected with reverse polarity may be physically damaged as the needle attempts to move below zero.