Electrical Engineering Fundamentals

This deck offers a friendly introduction to the core ideas every electrical engineering student or hobbyist encounters early on. It walks through the foundational relationships between voltage, current, resistance, and power, then expands into circuit analysis with Kirchhoff's Current and Voltage Laws. You'll also find cards that distinguish between direct and alternating current and explain how RMS and peak values relate to one another, which is essential for understanding real-world power supplies.

The cards are well suited for beginners who are just starting a course in circuits or electronics, as well as self-learners who want a quick refresher on the basics. If you're preparing for an introductory exam, building a small electronics project, or simply curious about how household wiring works, these questions will help you build a solid mental framework. The mix of definitions and short calculation prompts makes it easy to move from memorizing concepts to applying them.

Because much of this material builds on a small set of equations, try to focus on understanding the relationships rather than just memorizing formulas. When you review, work through the derivation cards alongside the definition cards so you can see how the same ideas appear in different contexts. Spacing your study sessions over several days is especially effective here, since repeated exposure helps the connections between Ohm's law, power, and Kirchhoff's laws become second nature.

Ohm's law is the cornerstone of circuit analysis, expressing the linear relationship between voltage, current, and resistance in a resistive element. Mathematically, \(V = IR\), where \(V\) is the voltage across the component in volts, \(I\) is the current flowing through it in amps, and \(R\) is the resistance in ohms (\(\Omega\)), a unit named after Georg Simon Ohm. The law can be rearranged into two useful forms: \(I = V/R\) for computing current from voltage and resistance, and \(R = V/I\) for computing resistance from measured voltage and current.

Electrical power, measured in watts, combines voltage and current through the relationship \(P = VI\). By substituting Ohm's law into this expression, two additional forms emerge that are often more convenient in circuit calculations: \(P = I^2 R\) when current and resistance are known, and \(P = V^2 / R\) when voltage and resistance are known. These equations allow engineers to determine how much energy a device dissipates or delivers in any resistive portion of a circuit.

Kirchhoff's two laws extend Ohm's law to networks with multiple components. Kirchhoff's Current Law (KCL) states that the sum of currents entering any node equals the sum of currents leaving that node (\(\sum I_{in} = \sum I_{out}\)), a direct consequence of conservation of charge. Kirchhoff's Voltage Law (KVL) states that the sum of voltage drops around any closed loop equals zero (\(\sum V = 0\)), reflecting conservation of energy, since the total energy gained by charges traversing a loop must equal the total energy lost. Together, KCL, KVL, and Ohm's law form the mathematical foundation for analyzing even the most complex linear circuits.

Electrical current comes in two principal forms. Direct current (DC) flows steadily in one constant direction and is the type produced by batteries. Alternating current (AC) periodically reverses direction, typically in a sinusoidal pattern, and is the form delivered by power grids. In the United States, household AC operates at 60 Hz (cycles per second), while in Europe and many other regions the standard is 50 Hz.

Because AC voltage varies with time, it is characterized by several related quantities. The peak voltage is the maximum instantaneous value, while the root-mean-square (RMS) value represents the equivalent DC voltage that would deliver the same average power to a resistive load. For a sine wave, \(V_{rms} = V_{peak}/\sqrt{2} \approx 0.707 \times V_{peak}\). Conversely, a standard 120 V RMS household supply corresponds to a peak voltage of \(120 \times \sqrt{2} \approx 169.7\) V.

Resistors are the simplest passive components, and they combine predictably in circuits. In series, resistances add directly: \(R_{total} = R_1 + R_2 + R_3 + \dots\). In parallel, the reciprocals add: \(1/R_{total} = 1/R_1 + 1/R_2 + 1/R_3 + \dots\). A common resistor network, the voltage divider, exploits these rules by placing two resistors in series across an input voltage and tapping the output across one of them, yielding \(V_{out} = V_{in} \times R_2 / (R_1 + R_2)\). This simple configuration provides a convenient way to scale a voltage down to any desired fraction of the input.

A capacitor is a passive component that stores energy in an electric field established between two conductive plates separated by a dielectric insulator. The amount of charge stored on the plates, \(Q\), is proportional to the capacitance \(C\) and the applied voltage \(V\), according to \(Q = CV\). Capacitance is measured in farads (F), defined as one coulomb per volt. The energy stored in a charged capacitor is \(E = \tfrac{1}{2}CV^2\). In a DC circuit, once a capacitor is fully charged, current ceases to flow, and the component behaves as an open circuit.

Capacitors combine in networks with rules opposite to those for resistors. In parallel, capacitances add directly: \(C_{total} = C_1 + C_2 + C_3 + \dots\). In series, the reciprocals add: \(1/C_{total} = 1/C_1 + 1/C_2 + 1/C_3 + \dots\). In AC analysis, a capacitor presents frequency-dependent impedance given by \(Z_C = 1/(j\omega C)\), where \(j\) is the imaginary unit and \(\omega\) is the angular frequency. This means capacitive impedance decreases as frequency rises, making capacitors useful for blocking DC while passing AC signals.

An inductor is a passive component, typically a coil of wire, that stores energy in a magnetic field. Its inductance is measured in henrys (H), defined as one volt-second per ampere. The energy stored in an inductor carrying current \(I\) is \(E = \tfrac{1}{2}LI^2\). Unlike a capacitor, an inductor in DC steady state behaves as a short circuit, passing DC current with no impedance once the current has stabilized. The impedance of an inductor in AC circuits is \(Z_L = j\omega L\), which increases with frequency. Inductors in series combine by direct addition, \(L_{total} = L_1 + L_2 + L_3 + \dots\) (assuming no mutual coupling), while in parallel they combine by the reciprocal rule, \(1/L_{total} = 1/L_1 + 1/L_2 + 1/L_3 + \dots\).

When a circuit containing an energy-storage element is switched on or off, voltages and currents do not jump to their final values immediately but instead change exponentially over time. The characteristic timescale governing this transition is called the time constant. In an RC circuit, the time constant is \(\tau = RC\), the time required for a charging capacitor's voltage to reach approximately 63.2% of its final value. In an RL circuit, the time constant is \(\tau = L/R\), representing the time for the current through an inductor to reach about 63.2% of its final value.

These time constants appear throughout practical electronics. They govern how quickly an RC filter responds to a sudden input change, how long a smoothing capacitor in a power supply takes to charge, and how rapidly an inductor limits the inrush of current in a switching regulator. Understanding time constants allows engineers to design circuits with predictable timing behavior, such as delay circuits, oscillators, and debouncing networks for mechanical switches.

When an inductor and capacitor are combined in an LC circuit, energy oscillates between the electric field of the capacitor and the magnetic field of the inductor. The frequency at which this exchange occurs most naturally is the resonant frequency, given by \(f_0 = 1/(2\pi\sqrt{LC})\). At resonance, the inductive and capacitive reactances cancel, leaving only the resistance of the circuit. This principle underlies tuned circuits in radios, wireless power transfer, and many filter designs.

The transistor is a semiconductor device that can amplify or switch electronic signals and electrical power. The two principal families are the Bipolar Junction Transistor (BJT) and the Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET). A BJT has three terminals: the base, collector, and emitter, and exists in two polarities, NPN and PNP. The defining characteristic of a BJT is its current gain, \(\beta = I_C / I_B\), which represents the ratio of collector current to base current and typically ranges from 50 to 300.

A MOSFET also has three terminals: the gate, drain, and source. Unlike the current-controlled BJT, the MOSFET is voltage-controlled: the gate-to-source voltage modulates the current flowing between drain and source. The minimum gate-to-source voltage required to create a conducting channel is called the threshold voltage, \(V_{th}\). MOSFETs operate in three regions: cutoff (off), triode or linear (acting as a variable resistor), and saturation (delivering a nearly constant current, useful for amplification).

The operational amplifier, or op-amp, is a high-gain differential voltage amplifier with two inputs (inverting and non-inverting) and a single output. When configured with negative feedback, an ideal op-amp obeys two golden rules: no current flows into either input (implying infinite input impedance), and the voltage difference between the inputs is zero (\(V^+ = V^-\)). The inverting configuration has a gain of \(A_v = -R_f / R_{in}\), while the non-inverting configuration has a gain of \(A_v = 1 + R_f / R_{in}\). A special case, the voltage follower or buffer, ties the output directly to the inverting input, giving unity gain and providing impedance matching between stages. A practical limitation is the gain-bandwidth product, a constant equal to the open-loop gain multiplied by the bandwidth, meaning that higher closed-loop gains reduce available bandwidth. Other useful op-amp circuits include the summing amplifier, which produces a weighted sum of inputs, and the comparator, used without feedback to detect which of two inputs is greater.

Digital electronics rests on Boolean logic gates that operate on binary signals. The AND gate outputs 1 only when all of its inputs are 1, expressed as \(Y = A \cdot B\). The OR gate outputs 1 when at least one input is 1, expressed as \(Y = A + B\). The NOT gate, or inverter, outputs the complement of its input, \(Y = \overline{A}\). These three gates form the basis of Boolean algebra, but additional gates simplify circuit design: NAND (the complement of AND), NOR (the complement of OR), and XOR, which outputs 1 when its inputs differ (\(Y = A \oplus B\)).

The NAND gate holds a special place in digital design because it is called a universal gate: any Boolean function, including AND, OR, and NOT, can be implemented using only NAND gates. Two fundamental identities, De Morgan's theorems, govern the manipulation of complemented expressions. The first states that the complement of a product equals the sum of the complements: \(\overline{A \cdot B} = \overline{A} + \overline{B}\). The second states that the complement of a sum equals the product of the complements: \(\overline{A + B} = \overline{A} \cdot \overline{B}\). These identities are essential tools for simplifying and transforming digital logic designs.

Beyond combinational logic, digital systems require memory. The flip-flop is a bistable circuit that stores a single bit, holding one of two stable states until triggered to change. Flip-flops are edge-triggered, responding to clock transitions, while similar level-triggered devices are called latches. Larger digital systems rely on components such as multiplexers (MUX), which select one of many input signals and route it to a single output based on select lines. Bridging the analog and digital worlds are the analog-to-digital converter (ADC), which samples a continuous signal and represents it as discrete values, and the digital-to-analog converter (DAC), which performs the reverse transformation.

Most of the world's electrical power is generated, transmitted, and distributed as alternating current. Three-phase power systems use three AC voltages offset from one another by 120 degrees, providing more efficient transmission and a more constant power output than single-phase systems. Compared with single-phase, three-phase delivery requires less conductor material for the same power capacity and produces a non-pulsating power flow, which is especially valuable for driving motors and heavy industrial loads.

A transformer is a device that transfers AC electrical energy between circuits via electromagnetic induction, allowing voltage levels to be stepped up for efficient long-distance transmission or stepped down for safe consumer use. The voltage ratio is set by the turns ratio of the transformer's windings: \(V_1/V_2 = N_1/N_2\), where \(N\) represents the number of turns in each coil. More turns on the secondary winding yield a higher output voltage; fewer turns yield a lower output voltage.

In AC power systems, the relationship between voltage and current is characterized by the power factor, \(PF = \cos(\theta)\), where \(\theta\) is the phase angle between them. Real (active) power, \(P = VI \cos(\theta)\), is the actual power consumed by the load and is measured in watts. Reactive power, \(Q = VI \sin(\theta)\), is the power alternately stored and released by inductors and capacitors, measured in volt-amperes reactive (VAR). Apparent power, \(S = VI\), is the product of RMS voltage and current, measured in volt-amperes (VA), and is related to the other quantities by \(S^2 = P^2 + Q^2\). Power factor correction, often implemented by adding capacitors to offset inductive loads, reduces transmission losses and improves efficiency in industrial and commercial power systems.

Signal processing relies on the Nyquist sampling theorem, which states that to accurately reconstruct a signal, it must be sampled at least twice as fast as its highest frequency component. Sampling below this rate causes aliasing, a form of distortion in which high-frequency components masquerade as lower frequencies. To analyze signals in the frequency domain, engineers use the Fourier transform, which decomposes any signal into its constituent sinusoids. The discrete version, the DFT, can be computed efficiently via the FFT algorithm, reducing computational complexity from \(O(N^2)\) to \(O(N \log N)\).

Filters selectively pass or attenuate signals based on frequency. A low-pass filter passes frequencies below a cutoff and attenuates higher ones, while a high-pass filter does the opposite. A simple RC low-pass filter has a cutoff frequency \(f_c = 1/(2\pi RC)\), at which the output power drops to half its input power, corresponding to the -3 dB point, where voltage is approximately 70.7% of the maximum. A band-pass filter combines these properties to pass only a specific range of frequencies, and the quality factor \(Q = f_0/\Delta f\) describes how sharply the filter selects around its center frequency, with higher \(Q\) indicating greater selectivity.

Underlying all of electrical engineering are the laws of electromagnetics. Coulomb's law describes the force between point charges as \(F = k q_1 q_2 / r^2\). Gauss's law relates the electric flux through a closed surface to the enclosed charge, \(\oint \mathbf{E} \cdot d\mathbf{A} = Q_{enc}/\epsilon_0\). Faraday's law states that a changing magnetic flux through a loop induces an EMF, \(\mathcal{E} = -d\Phi_B/dt\), where the negative sign embodies Lenz's law, the principle that induced currents oppose the change that created them. Ampère's law, extended by Maxwell to include changing electric fields, is \(\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 (I_{enc} + \epsilon_0 \, d\Phi_E/dt)\). Maxwell's four equations, namely Gauss's law for electricity, Gauss's law for magnetism (which forbids magnetic monopoles), Faraday's law, and the Ampère-Maxwell law, together predict the existence of electromagnetic waves traveling in vacuum at speed \(c = 1/\sqrt{\mu_0 \epsilon_0} \approx 3 \times 10^8\) m/s.

A suite of powerful theorems simplifies the analysis of linear circuits. Impedance \(Z = R + jX\) generalizes Ohm's law to AC, combining resistance and reactance. The Thévenin equivalent reduces any linear circuit to a single voltage source \(V_{th}\) in series with a resistance \(R_{th}\), while the Norton equivalent reduces it to a current source \(I_N\) in parallel with a resistance \(R_N\), with \(V_{th} = I_N \times R_N\) and \(R_{th} = R_N\). Superposition states that the response to multiple sources is the algebraic sum of the responses to each source acting alone. The maximum power transfer theorem says that a load receives maximum power when its resistance matches the source resistance, or in AC circuits, when its impedance is the complex conjugate of the source impedance. Finally, the Wheatstone bridge, a network of four resistors balanced when \(R_1/R_2 = R_3/R_4\), provides a classical means of measuring an unknown resistance with high precision.