Comprehensive Study Notes on Electrical Quantities and Circuits

Fundamentals of Basic Electrical Quantities

The volt is defined as the potential difference between two specific points on a wire that is carrying $1\,A$ and dissipating $1\,W$ of energy. This quantity is commonly referred to as potential difference, representing the potential to perform work. The volt is also known as electromotive force, or EMF, and is represented in algebraic formulas such as Ohm’s Law by the symbol $E$. Functionally, the volt acts as the force that enables current to flow by pushing electrons through a circuit. It creates a form of electrical pressure that drives activity; specifically, $1\,V$ will cause $1\,\text{Coulomb}$ ($6.25 \times 10^{18}$ electrons) to perform $1\,\text{Joule}$ of work. Furthermore, $1\,V$ is the exact amount of electrical pressure needed to cause $1\,A$ of current to flow through a resistance of $1\,\Omega$.

An ampere is a measure of the specific number of electrical charges passing a given point in one second. To perform work that can be harnessed, a minimum number of particles must move, defined as $1\,\text{Coulomb}$, which equates to approximately $6.25 \times 10^{18}$ electrons, or exactly $6,250,000,000,000,000,000$ electrons. In mathematical expressions, the ampere is represented by the capital letter $I$, which stands for intensity. This serves as a measure of the amount of charges passing through a space within a designated timeframe of $1\,\text{second}$.

Electrical resistance is the inherent property of a substance to limit the flow of electrical current. This quality is quantified using the unit Ohm ($\Omega$). An Ohm is defined as the amount of resistance required to limit current flow to $1\,A$ when an electromotive force of $1\,V$ is applied. Because the electrons within matter govern all chemical properties, all matter possesses inherent electrical properties. Consequently, every substance has the capacity to conduct electricity if a sufficient amount of voltage is applied. In Ohm’s Law, resistance is represented by the symbol $R$.

Power represents the force and flow rate of electricity as it passes through a component. To harness the energy of electricity, it must be slowed down and converted into a usable form, which is achieved by varying the force or amount of electricity. Electrical power is expressed in Watts ($W$) and is calculated by multiplying the applied voltage by the current flow, expressed as the formula $P = EI$. In algebraic terms, power may be represented by either the letter $P$ or the letter $W$ for watts.

Standards of Measurement and Historical Context

Establishing a set of standards is essential for the effective communication of measurement data. The absence of such standards can lead to catastrophic failures, as evidenced on September 23, 1999, when the Mars Climate Orbiter was lost because engineers confused metric and imperial measurements. This error resulted in a loss of approximately $125\,\text{million}$ dollars, with the probe presumably continuing to tumble through space. Historically, various imperial units were used, often based on inconsistent physical benchmarks.

Imperial measurements included the Chain, a surveying instrument consisting of $100\,\text{links}$ totaling $66\,\text{feet}$. The Rod was defined as the length of $16$ men standing one foot behind the other, totaling $16.5\,\text{feet}$. The Yard was traditionally the distance from the tip of a king's nose to the tip of his outstretched hand, totaling $3\,\text{feet}$. The Foot was derived by dividing a Rod by $16$, totaling $12\,\text{inches}$. The Inch was defined as the length of three round, dry barley corns taken from the center of the ear and laid end to end, which is now standardized as $1/12$ of a foot. The inconsistency of these measures often led to confusion and economic disputes.

In the 1790s, France developed the metric system, a base-ten system comprising seven basic units. These basic units include 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 substance. Derived units in the metric system 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 and Unit Comparisons

Metric prefixes utilize factors of ten to denote scale. Large scale prefixes 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 scale prefixes include Deci ($d, 10^{-1}$), Centi ($c, 10^{-2}$), Milli ($m, 10^{-3}$), Micro ($\mu, 10^{-6}$), Nano ($n, 10^{-9}$), Pico ($p, 10^{-12}$), Femto ($f, 10^{-15}$), and Atto ($a, 10^{-18}$).

When comparing Standard International (SI) units to Imperial units, specific pairings emerge. Length uses Meters ($m$) versus Feet ($ft$). Area uses Square meters ($m^2$) versus Square feet ($ft^2$). Volume uses Cubic meters ($m^3$) versus Cubic feet ($ft^3$). Mass uses Kilograms ($kg$) versus Pounds ($lb$). Force uses Newtons ($N$) versus Pounds. Pressure uses Pascals ($N/m^2$) versus PSI ($lb/in^2$). Time and Frequency are standardized in both systems as Seconds ($s$) and Hertz ($Hz$). Electrical units such as Voltage (Volt), Resistance (Ohm), Current (Ampere), Capacitance (Farad), and Inductance (Henry) are generally the same across systems.

Electrical Measurement for Electricians

Electricians primarily focus on measuring three phenomena: Voltage ($V$) for potential difference, Resistance in Ohms ($\Omega$) for the difficulty electricity faces passing through a substance, and Amperage ($A$) for the number of charged particles (electrons) passing through. In industrial settings, very small measurements are common, such as the $4\text{-}20\,mA$ communication signal. A milli-amp ($mA$) is $1/1000^{\text{th}}$ of an amp, and a milli-volt ($mV$) is $1/1000^{\text{th}}$ of a volt.

Capitalization in measurement symbols often indicates larger, more dangerous quantities. Common units include micro volts ($\mu V, 10^{-6}$), Kilovolts ($KV, 10^3$), Mega-Watts ($MW, 10^6$), Megohms ($M\Omega, 10^6$), and Kilo-ohms ($K\Omega, 10^3$). Understanding these relationships and formulas allows for the resolution of problems in series, parallel, and combination circuits.

Ohm’s Law and Kirchhoff’s Laws

Ohm’s Law expresses the proportional relationship between voltage, current, and resistance through the formula $E = IR$. If one of these quantities is altered, the others change immediately to maintain this relationship. The law can be derived to find any unknown value if two others are known: $I = \frac{E}{R}$ and $R = \frac{E}{I}$. For example, if current is $2\,A$ and resistance is $50\,\Omega$, the voltage is calculated as $2 \times 50 = 100\,V$. If current is reduced to $1\,A$ with the same resistance, the voltage is $1 \times 50 = 50\,V$, demonstrating that half the current requires half the voltage to be pushed through the same resistance.

Kirchhoff’s Laws, developed in 1847, describe the behavior of voltage and current. Kirchhoff’s Voltage Law (KVL) states that the sum of all voltage drops across devices in a circuit must equal the total supply voltage. Kirchhoff’s Current Law (KCL) states that the total current entering a circuit must be equal to the current leaving the circuit.

Solving Series Circuits

In a series circuit, there is only one path for current flow, meaning the same current ($I_{Total}$) must pass through every device in the circuit. Consequently, switches and fuses are always wired in series with the load. Voltage is dissipated as it passes through each device, and the sum of these drops equals the supply voltage. To solve a series circuit, one should follow four steps: first, find the total resistance ($R_{Total}$); second, use $R_{Total}$ to find the current through each resistor; third, calculate the specific voltage drops across each resistor; and fourth, use those voltage drops in the power equation ($P = EI$) to solve for power.

Consider a series circuit with resistors $R_1 = 200\,\Omega$, $R_2 = 300\,\Omega$, and $R_3 = 160\,\Omega$ powered by a $24\,V$ supply. The total resistance is the sum of individual resistances: $200 + 300 + 160 = 660\,\Omega$. Using Ohm's Law ($I_T = \frac{E_T}{R_T}$), the total current is $24 / 660 = 0.03636\,A$, or $36\,mA$. The individual voltage drops are then calculated: $E_1 = 0.036\,A \times 200\,\Omega = 7.2\,V$, $E_2 = 0.036\,A \times 300\,\Omega = 10.91\,V$, and $E_3 = 0.036\,A \times 160\,\Omega = 5.818\,V$. The sum of these drops ($7.273 + 10.91 + 5.818$) equals the supply voltage of $24\,VDC$.

Total circuit power is found by $P_{Total} = E_{Total} \times I_{Total}$, resulting in $24\,V \times 0.03636\,A = 0.87264\,W$ or $872.6\,mW$. Individual power for resistors is calculated similarly: $P_1 = 7.273\,V \times 0.03636\,A = 0.26445\,W$ ($264.45\,mW$); $P_2 = 10.91\,V \times 0.03636\,A = 0.3966876\,W$ ($396.6876\,mW$); and $P_3 = 5.818\,V \times 0.03636\,A = 0.21154248\,W$ ($211.54\,mW$).

Voltage Dividers and Parallel Circuits

Voltage drops in a series circuit allow for different potential levels at various points. By using a series circuit and a common reference point, one can create a voltage divider to obtain any desired voltage from a source. To measure these voltages, a meter must be connected in parallel to the component being measured.

In parallel circuits, voltage has multiple paths to follow. Because the voltage does not pass through any resistors before reaching the loads, the voltage across every resistor in a parallel circuit is the same: $V_{Total} = V_1 = V_2 = V_3 = \dots$. However, current is calculated individually for each branch using $I = \frac{E}{R}$. The total current is the sum of these branch currents: $I_{Total} = I_1 + I_2 + I_3 + \dots$. For a $24\,V$ circuit with resistors of $200\,\Omega$, $300\,\Omega$, and $160\,\Omega$, the branch currents are $I_1 = 0.12\,A$, $I_2 = 0.08\,A$, and $I_3 = 0.15\,A$, equaling a total current of $0.35\,A$ or $350\,mA$.

Total resistance in parallel circuits is always less than the value of the smallest resistor because the current has multiple paths. There are three methods to find $R_{Total}$. 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 $500\,\Omega$ resistors in parallel result in $R_T = 125\,\Omega$. Second is the Product over Sums method for two resistors: $\frac{R_1 \times R_2}{R_1 + R_2}$. Third is the Reciprocal Formula: $\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots$. When using reciprocals, it is necessary to round values, which can introduce inaccuracies; using 4 or 5 decimal places is generally recommended for precision.

Practical Measurement Tips

When conducting measurements in any circuit, specific placement is required for accuracy and safety. To measure voltage, the voltmeter must be connected in parallel to the load. To measure current, the ammeter must be connected in series with the load. Polarity is critical, especially with analog meters. If a digital ammeter or voltmeter is connected with reverse polarity, it will display negative numbers. However, if an analog meter is connected with reverse polarity, it can be physically damaged as the needle attempts to move below zero.