Unit 1: Fundamentals of D.C. circuits

A D.C. circuit is a closed conducting path driven by sources whose voltage and current do not vary with time, so all transients have settled and inductors/capacitors reach steady state. This unit builds from circuit variables and element laws (Ohm, Kirchhoff) up to systematic and shortcut analysis techniques and the major network theorems.

• Governing idea: every analysis reduces to applying element relations plus the two conservation laws (charge at nodes, energy around loops).
• Steady-state D.C. convention: inductors behave as short circuits (0 V), capacitors as open circuits (0 A).
• Passive sign convention: current enters the + terminal of an element that absorbs power; a product v·i > 0 means absorbing, < 0 means delivering.
• Reference node: one node is chosen as 0 V datum (ground) for all potentials.
• Ideal source assumption: voltage sources have zero internal resistance, current sources infinite internal resistance unless stated.

II. Circuit Variables and Passive Elements

The quantities measured and the three elements that store or dissipate energy.

A. Voltage, current, power and energy concepts

Define the four basic quantities and how they interrelate.

• Current (i): rate of charge flow, i = dq/dt, unit ampere (A); conventional current flows opposite to electrons.
• Voltage (v): energy per unit charge between two points, v = dw/dq, unit volt (V).
• Power (p): rate of energy transfer, p = v·i, unit watt (W); for a resistor p = i²R = v²/R.
• Energy (w): w = ∫p dt, unit joule (J); utility billing uses kilowatt-hour, 1 kWh = 3.6×10⁶ J.

B. Resistance

Opposition to current that dissipates electrical energy as heat.

• Definition: R = ρL/A, unit ohm (Ω); ρ = resistivity (Ω·m), L = length, A = cross-section.
• Energy behaviour: purely dissipative, w = ∫i²R dt; no energy stored.
• Conductance: G = 1/R, unit siemens (S).

C. Inductance

A coil that stores energy in its magnetic field and opposes change of current.

• Element law: v = L (di/dt), unit henry (H); L depends on turns and core.
• Energy stored: w = ½Li² joules.
• D.C. steady state: di/dt = 0 so v = 0 → acts as a short circuit.

D. Capacitance

Two conductors storing energy in an electric field, opposing change of voltage.

• Element law: i = C (dv/dt), unit farad (F); C = εA/d.
• Energy stored: w = ½Cv² joules.
• D.C. steady state: dv/dt = 0 so i = 0 → acts as an open circuit.

III. Fundamental Circuit Laws

The algebraic relations that every analysis technique rests upon.

A. Ohm's law

The voltage across a resistor is proportional to the current through it.

• Statement: V = I·R at constant temperature for an ohmic conductor.
• Rearrangements: I = V/R, R = V/I.
• Limitation: holds only for linear elements; diodes and lamps are non-ohmic.

B. Kirchhoff's laws

Two conservation laws governing nodes and loops.

1. Kirchhoff's Current Law (KCL): charge conservation at a node — sum of currents in = sum out, Σi = 0. Example: if 3 A and 2 A enter a node, 5 A must leave.
2. Kirchhoff's Voltage Law (KVL): energy conservation around a loop — algebraic sum of voltages = 0, Σv = 0; rises counted positive, drops negative (or vice versa) consistently.

IV. Methods of Circuit Analysis

How the laws are organised into procedures, from formal to intuitive.

A. Basic method of circuit analysis

Direct, exhaustive application of the element and Kirchhoff laws.

• Procedure: label every branch current, write KCL at nodes and KVL around loops, substitute V = IR, solve the simultaneous equations.
• Trait: always valid but produces many equations for large networks, motivating the shortcut and systematic methods below.

B. Intuitive method of circuit analysis — series and parallel simplification

Collapse resistor groupings into one equivalent to shrink the network.

1. Series: same current, resistances add, R_eq = R₁ + R₂ + ….
2. Parallel: same voltage, conductances add, 1/R_eq = 1/R₁ + 1/R₂ + …; two in parallel R_eq = R₁R₂/(R₁+R₂).

C. Voltage division rule

Splits a source voltage among series resistors.

• Formula: across Rₖ in a series string, Vₖ = V_s · Rₖ/ΣR.
• Example: 12 V across 2 Ω and 4 Ω in series gives V₄ = 12·4/6 = 8 V.

D. Current division rule

Splits a total current among parallel branches.

• Formula (two branches): I₁ = I_total · R₂/(R₁+R₂); the larger current takes the smaller resistance.
• General: Iₖ = I_total · Gₖ/ΣG.

E. Star–delta transformation

Converts a three-terminal Y (star) network to Δ (delta) and back when no two elements are simply series/parallel.

• Δ→Y: R₁ = (R_ab·R_ca)/(R_ab+R_bc+R_ca) and cyclically for R₂, R₃.
• Y→Δ: R_ab = (R₁R₂+R₂R₃+R₃R₁)/R₃, cyclically.
• Use: unlocks bridge networks that have no series/parallel reduction.

V. Sources and Systematic Analysis

Modelling the drivers of a circuit and the two matrix-style solution methods.

A. Introduction to dependent and independent sources

Distinguish fixed sources from those controlled by another circuit variable.

1. Independent source: value set externally and fixed — ideal voltage source (constant V) or current source (constant I).
2. Dependent (controlled) source: value proportional to a circuit voltage or current; four types — VCVS, VCCS, CCVS, CCCS — essential for modelling transistors and amplifiers.

B. Mesh and nodal analysis

Two systematic frameworks that minimise unknowns.

1. Mesh analysis (KVL-based): assign loop currents to independent meshes, write KVL per mesh, solve for currents; best when branches ≫ nodes.
   • Equation form: for mesh 1, Σ(R)·I₁ − (shared R)·I₂ = ΣV.
2. Nodal analysis (KCL-based): choose a reference node, write KCL at remaining nodes in terms of node voltages, solve; best when nodes are few.
   • Equation form: (ΣG)·V₁ − (shared G)·V₂ = Σ(source currents).

VI. Thevenin's Theorem

Any linear two-terminal network reduces to one source and one resistor.

A. Statement and conditions

Replace the network seen from two terminals by V_th in series with R_th, valid for linear bilateral networks.

• V_th: open-circuit voltage at the terminals.
• R_th: resistance looking in with independent sources deactivated (voltage sources shorted, current sources opened).
• Load current: I_L = V_th/(R_th + R_L).
• Example: a 12 V source with 4 Ω internal and open-circuit terminals gives V_th = 12 V, R_th = 4 Ω.

VII. Norton's Theorem

The current-source dual of Thevenin.

A. Statement and conditions

Replace a linear two-terminal network by current source I_N in parallel with R_N.

• I_N: short-circuit current at the terminals.
• R_N: identical to R_th (same deactivation procedure).
• Source transformation: V_th = I_N·R_N, R_th = R_N, so the two models interconvert.

VIII. Maximum Power Transfer Theorem

The load that extracts the most power from a source.

A. Statement and conditions

Maximum power is delivered to the load when load resistance equals the source's Thevenin resistance.

• Condition: R_L = R_th.
• Maximum power: P_max = V_th²/(4R_th).
• Efficiency: only 50 % at match (half the power lost in R_th), so power systems deliberately avoid this condition and favour low source resistance.

IX. Superposition Theorem

Linear networks respond to each source independently.

A. Statement and conditions

In a linear circuit with multiple independent sources, any voltage or current equals the algebraic sum of the responses to each source acting alone.

• Procedure: keep one independent source active, deactivate the rest (short voltage sources, open current sources), compute the response, then sum all contributions.
• Restriction: applies to currents and voltages, not to power, since p = i²R is non-linear; dependent sources are left in place throughout.
• Example: with a 10 V and a 5 V source present, find the branch current due to 10 V alone, then due to 5 V alone, and add them.