DC Circuit Theory - Network Theory (Electric Circuits) - Electrical Engineering

Introduction

• Basic DC circuit theory studies how electrical elements are interconnected to form circuits and how electrical current, the flow of charge, is driven around a closed path by a potential difference or voltage measured in volts (V).
• All matter is made of atoms. Atoms contain a nucleus (protons and neutrons) and electrons. Protons carry positive charge, neutrons are neutral, and electrons carry negative charge. Forces between the nucleus and outer electrons bind atoms together.
• When charges are separated they create a difference in electric potential between points. If a closed conducting path is provided, free electrons drift under the influence of that potential difference, producing an electric current. Materials oppose the flow of charge; this opposition is called resistance.
• The three fundamental, closely related electrical quantities in basic circuits are Voltage (V), Current (I) and Resistance (R, unit: Ω).

Electrical Voltage

• Voltage (V) is the electric potential difference between two points. It represents potential energy per unit charge and acts as the driving "push" that causes charges to move in a circuit.
• Voltage can be described as the work done (in joules) to move a unit charge (one coulomb) from one node to another. Mathematically, v = dw/dq where v is potential difference, dw is differential work and dq is differential charge.
• The difference in potential between two nodes is often called the potential difference or voltage drop across an element.
• Voltage is measured in volts (V). One volt is defined so that one volt will cause one ampere of current to flow through one ohm of resistance.
• A voltage source whose value does not change with time is a DC voltage. A periodically varying voltage is an AC voltage. Both are measured in volts; prefixes such as mV (millivolt), μV (microvolt) and kV (kilovolt) are commonly used.
• Batteries and power supplies provide typical DC voltages (for example, +5 V, -9 V, +12 V). The circuit symbol for an ideal DC voltage source (battery) usually shows polarity markings (+ and -). The symbol for an AC source is a circle with a sine wave inside.
• By analogy, a water tank at some height produces pressure at the outlet; similarly, the higher the potential difference the greater the capacity to do electrical work.
• Voltage can exist between two points even if no current flows. However, current cannot flow without a driving voltage.

Voltage Symbols

• Voltage is always measured between two points. The quantity is often denoted by the letter V (or lowercase v) on circuit diagrams. In some contexts the symbol E or ε denotes electromotive force (emf) of a source.
• Common voltage source symbols and their polarities should be learned so that sign and direction of voltages across components are correctly applied in analysis.

Electrical Current

• Current (I) is the rate of flow of electric charge. It is measured in amperes (A). One ampere equals one coulomb of charge passing a point per second, Q = I·t.
• In conductors the moving carriers are electrons; electrons physically drift from the negative terminal to the positive terminal of a source. For convenience and historical reasons, circuit analysis commonly uses conventional current which flows from positive to negative.
• Arrows on circuit diagrams indicate the assumed direction of current. Analysis must be consistent with that assumed direction; if the calculated current is negative, its actual direction is opposite to the assumed direction.
• Current can be expressed in different units via prefixes: mA (milliamps) and μA (microamps). Current may be positive or negative depending on direction relative to an assumed reference.
• Current that flows steadily in one direction is direct current (DC). Current that reverses direction periodically is alternating current (AC). AC values are often compared by their RMS (root mean square) value which gives the equivalent heating (power) effect of a DC current.
• Ideal current sources supply a specified current regardless of the voltage across them and therefore "prefer" short-circuit conditions; ideal voltage sources maintain a specified voltage and "prefer" open-circuit conditions.
• Using the water analogy, current is the flow rate of water through a pipe: higher flow rate corresponds to higher current for the same pipe geometry.

Conventional Current Flow

• Conventional current is the direction of positive charge flow, from the positive terminal of a source through the circuit toward the negative terminal.
• The convention was established historically before the electron was discovered. Many circuit symbols (for diodes, transistors, current arrows) are drawn using the direction of conventional current.
• Calculations using conventional current yield correct results provided the chosen directions are used consistently.

Electron Flow

• Electron flow is the physical motion of electrons from the negative terminal to the positive terminal of a source; this is opposite to conventional current direction.
• Both descriptions are valid; circuit analysis is independent of the physical carrier as long as a consistent sign convention is applied.
• In practical circuit design both conventions appear in literature. It is normally easier to use conventional current (positive→negative) for diagrammatic work and sign bookkeeping.

DC Circuit Theory - Resistance

• Resistance (R) quantifies how strongly a material opposes current flow. It is measured in ohms (Ω). Resistance is always non-negative for passive resistive materials.
• The circuit element that provides resistance is the resistor. Resistors are passive devices: they cannot supply energy but dissipate electrical energy as heat (or sometimes light).
• Conductors (for example copper, aluminium) have very low resistance; insulators (for example glass, rubber) have very high resistance. Semiconductors (for example silicon) have resistances intermediate between conductors and insulators and are the basis of diodes and transistors.
• Resistance can be linear or non-linear. A linear resistor obeys Ohm's law (voltage proportional to current). A non-linear resistor does not have a constant ratio of v to i.
• The reciprocal of resistance is conductance (G), measured in siemens (S). G = 1/R. Conductance is a measure of how easily current flows.
• For very small resistance values, it is often convenient to work with conductance. Using conductance, i = G·v.
• Power dissipated in a resistor is always positive and can be expressed in several equivalent forms:
  • P = V·I
  • P = I²·R
  • P = V² / R

Resistor Symbols and Types

• Common circuit symbols: the rectangular or zig-zag symbol for a resistor (depending on regional drawing conventions). Fixed resistors, variable resistors (potentiometers/rheostats) and special resistor types (thermistors, LDRs) have distinctive symbols and characteristics.
• Resistor value depends on material, length, cross-sectional area and temperature. Ohm's law, v = i·R, describes the voltage-current relationship for a linear resistor.
• Resistance is independent of frequency. For a pure resistor the AC impedance equals its DC resistance (Z = R, angle 0°).
• Conductance units were historically called mho (ohm spelled backwards) and are now the siemens (S). The symbol ℧ is sometimes used for mho.

Essential Laws and Circuit Concepts

• Ohm's Law: v = i·R. This fundamental linear relation links voltage (v), current (i) and resistance (R).
• Charge-Current Relation: Q = I·t. One coulomb equals one ampere-second.
• Power: instantaneous power p(t) = v(t)·i(t). For steady DC circuits, P = V·I.
• Kirchhoff's Current Law (KCL): The algebraic sum of currents entering a node (junction) is zero. KCL expresses conservation of charge.
• Kirchhoff's Voltage Law (KVL): The algebraic sum of voltages around any closed loop is zero. KVL expresses conservation of energy.
• Series and Parallel Resistances: For resistors in series, R_eq = R₁ + R₂ + ... ; for resistors in parallel, 1/R_eq = 1/R₁ + 1/R₂ + ... . These combinations simplify circuit analysis.
• Thevenin and Norton equivalents: Any linear network seen from two terminals can be replaced by a Thevenin equivalent (voltage source in series with R_th) or a Norton equivalent (current source in parallel with G_n or R_n). These are fundamental tools for simplifying circuits and analysing load behaviour.
• Sign conventions matter. When applying KVL and KCL, choose consistent reference directions for currents and voltage polarities; negative results indicate opposite direction to the assumption.

Worked Examples and Simple Applications

• Example - Using Ohm's law: If a resistor R = 10 Ω has a voltage v = 5 V across it, the current through it is i = v/R = 5/10 = 0.5 A.
• Example - Power dissipation: For the same resistor, P = V·I = 5 × 0.5 = 2.5 W. Alternatively, P = I²·R = (0.5)² × 10 = 2.5 W.
• Application - Voltage dividers: Two resistors R₁ and R₂ in series across a voltage source V produce an output voltage across R₂ equal to V_out = V · R₂ / (R₁ + R₂). This simple divider is widely used to obtain reference voltages and scale sensor signals.
• Application - Current limiting: A resistor in series with an LED limits the current so the LED operates safely. Choose R = (V_supply - V_LED)/I_desired.
• Measurement - Instruments: A voltmeter measures voltage across two points and has a high input resistance; an ammeter measures current in series and has a low resistance. Care is required when connecting meters to avoid changing the circuit conditions.

Summary

• DC circuit theory links Voltage, Current and Resistance through fundamental laws such as Ohm's law, KCL and KVL.
• In a linear resistor, increasing the applied voltage increases the current proportionally. Increasing the resistance for a fixed voltage decreases the current.
• Current is directly proportional to voltage (I ∝ V) and inversely proportional to resistance (I ∝ 1/R) for linear resistive circuits.
• Voltage is measured in volts (V), current in amperes (A), resistance in ohms (Ω), and conductance in siemens (S).
• Direct current (DC) flows in one direction; alternating current (AC) reverses periodically. RMS values of AC give equivalent heating power to DC values.
• Conventional current flows from positive to negative; electron flow is from negative to positive. Use either consistently in analysis.
• Resistors are passive elements that dissipate power; conductors have low resistance while insulators have high resistance. Semiconductors lie between and enable active devices.
• Practical circuits use combinations of series and parallel resistors, voltage dividers, current-limiters and equivalent-source transformations (Thevenin/Norton) to design and analyse behaviour.
• The visual and conceptual water-tank analogy remains a useful aid: voltage ~ pressure, current ~ flow rate, resistance ~ pipe restriction.