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Unit 1: Fundamentals of D.C. circuits

ECE131 — Basic Electrical And Electronics Engineering 4 min read

I. Orientation: Principles and Conventions

A direct-current circuit is an interconnected network in which voltages and currents are constant or treated as constant under steady-state conditions. Its analysis rests on conservation of charge, conservation of energy, and the voltage-current relationships of circuit elements.

  • Circuit model: Ideal components are connected by conductors having zero resistance unless stated otherwise.
  • Steady-state assumption: After transients disappear, ideal inductors behave as short circuits and ideal capacitors as open circuits under D.C.
  • Reference directions: Current arrows and voltage polarities may be chosen arbitrarily; a negative result means the actual direction or polarity is opposite.
  • Passive sign convention: Power is positive when current enters an element's positive-voltage terminal; the element then absorbs power.
  • Ground reference: One node is assigned (0\text{ V}), and all node voltages are measured relative to it.
  • SI conventions: Voltage is measured in volts (V), current in amperes (A), resistance in ohms ((\Omega)), power in watts (W), and energy in joules (J).

II. Circuit Quantities and Elements

These quantities describe the electrical state of a D.C. network and the behavior of its passive components.

A. Resistance

Resistance is the opposition offered by a material or component to electric current.

  • Definition: For a linear resistor, resistance is the voltage-current ratio:
    TEXT
    R = V/I

    Here (R) is resistance in ohms, (V) is voltage in volts, and (I) is current in amperes.
  • Physical dependence: A uniform conductor has:
    TEXT
    R = ρl/A

    Here (\rho) is resistivity in (\Omega\text{m}), (l) is length in metres, and (A) is cross-sectional area in (\text{m}^2).
  • Interpretation: A larger (R) permits less current for the same applied voltage; (10\text{ V}) across (5\Omega) produces (2\text{ A}).

B. Inductance

Inductance is the property by which a changing current produces an opposing induced voltage.

  • Voltage relation:
    TEXT
    v = L(di/dt)

    Here (v) is in volts, (L) is inductance in henries, (i) is current, and (t) is time.
  • D.C. behavior: At steady state, (di/dt=0), so an ideal inductor has (v=0) and acts as a short circuit.
  • Energy storage:
    TEXT
    W = ½Li²

    Here (W) is magnetic energy in joules.

C. Capacitance

Capacitance is the ability of a component to store electric charge and energy in an electric field.

  • Charge relation:
    TEXT
    q = Cv

    Here (q) is charge in coulombs, (C) is capacitance in farads, and (v) is voltage.
  • Current relation:
    TEXT
    i = C(dv/dt)

    Here (i) is capacitor current and (t) is time.
  • D.C. behavior: At steady state, (dv/dt=0), so an ideal capacitor carries no current and acts as an open circuit.
  • Energy storage: The stored electric energy is (W=\tfrac12Cv^2), measured in joules.

D. Voltage

Voltage is the electric potential-energy difference per unit charge between two points.

  • Definition:
    TEXT
    V = W/Q

    Here (V) is voltage, (W) is energy in joules, and (Q) is charge in coulombs.
  • Polarity: (V_{ab}=V_a-Vb); therefore, (V{ba}=-V_{ab}).
  • Measurement: A voltmeter is connected in parallel and ideally has infinite internal resistance.

E. Current

Current is the rate at which electric charge passes through a cross-section.

  • Definition:
    TEXT
    I = dQ/dt

    Here (I) is current, (Q) is charge, and (t) is time.
  • Conventional direction: Current is taken as flowing from higher to lower potential through a passive resistor.
  • Measurement: An ammeter is connected in series and ideally has zero internal resistance.

F. Power and energy concepts

Power measures the rate of electrical energy transfer, while energy measures the total work transferred.

  • Power relation:
    TEXT
    P = VI = I²R = V²/R

    Here (P) is power in watts; the last two forms apply to resistors.
  • Energy relation:
    TEXT
    E = Pt

    Here (E) is energy in joules for constant power and (t) is time in seconds.
  • Sign: (P>0) indicates absorption; (P<0) indicates delivery.
  • Commercial unit: (1\text{ kWh}=3.6\times10^6\text{ J}).

III. Fundamental Circuit Laws

These laws connect element behavior with conservation of charge and energy.

A. Ohm's law

Ohm's law states that current through an ohmic conductor is proportional to its voltage when physical conditions, especially temperature, remain constant.

  • Equation:
    TEXT
    V = IR

    Here (V), (I), and (R) denote resistor voltage, current, and resistance.
  • Characteristic: A linear resistor has a straight-line (V-I) graph with slope (R).
  • Limitation: Diodes, transistors, filament lamps, and electrolytes may have nonlinear or temperature-dependent characteristics.

B. Kirchhoff's laws

Kirchhoff's laws express charge conservation at nodes and energy conservation around closed paths.

  1. Kirchhoff's Current Law (KCL): The algebraic sum of currents at a node is zero:

    TEXT
    ΣI = 0


    Currents entering may be positive and currents leaving negative; thus, (I_1+I_2=I_3).

  2. Kirchhoff's Voltage Law (KVL): The algebraic sum of voltages around a closed loop is zero:

    TEXT
    ΣV = 0


    Voltage rises and drops must use one consistent traversal convention; for a source (V_s) and two resistor drops, (V_s-I R_1-I R_2=0).

IV. Circuit Reduction and Direct Analysis

Direct analysis identifies circuit structure, simplifies valid combinations, and applies the fundamental laws.

A. Basic method of circuit analysis

The basic method converts a circuit diagram into equations whose unknown voltages and currents can be solved.

  • Procedure: Identify nodes, branches, loops, element values, source polarities, and required unknowns.
  • References: Assign current directions and voltage polarities before writing equations.
  • Equations: Apply Ohm's law, KCL, and KVL until the number of independent equations equals the number of unknowns.
  • Verification: Check units and confirm that total source power equals total absorbed power.

B. Intuitive method of circuit analysis

The intuitive method uses circuit behavior and symmetry to reduce calculation without replacing physical laws.

  • Open and short circuits: An open branch carries (0\text{ A}); an ideal short has (0\text{ V}).
  • Symmetry: Equal resistive paths connected to identical potentials carry equal currents.
  • Relative values: In series, larger resistance has the larger voltage drop; in parallel, smaller resistance carries more current.
  • Boundary checks: Replacing (R) by (0\Omega) or (\infty\Omega) can test whether a derived expression behaves plausibly.

C. Series and parallel simplification

Series and parallel combinations can be replaced by equivalent resistance without changing terminal behavior.

  1. Series resistors: They carry the same current and add directly:

    TEXT
    Req = R1 + R2 + ... + Rn

  2. Parallel resistors: They share the same voltage and add by conductance:

    TEXT
    1/Req = 1/R1 + 1/R2 + ... + 1/Rn
  • Two-resistor form: (R_{eq}=R_1R_2/(R_1+R_2)).
  • Condition: Components are series only if their common node has no other branch; they are parallel only if both terminals share the same two nodes.

D. Voltage division rule

The voltage division rule determines the voltage across a resistor in a series network.

  • Rule:
    TEXT
    Vk = Vs(Rk/Rtotal)

    Here (V_k) is the selected resistor voltage, (V_s) is total applied voltage, (Rk) is that resistance, and (R{total}) is total series resistance.
  • Example: Across (3\Omega) in a (3\Omega+6\Omega) series circuit supplied by (18\text{ V}), (V_k=18(3/9)=6\text{ V}).
  • Loading: A connected load changes the divider because it appears in parallel with part of the network.

E. Current division rule

The current division rule determines branch current in a parallel resistive network.

  • General rule:
    TEXT
    Ik = Itotal(Gk/Gtotal)

    Here (Ik) is branch current, (I{total}) is incoming current, (G_k=1/Rk) is branch conductance, and (G{total}) is total conductance.
  • Two branches: (I1=I{total}R_2/(R_1+R_2)); current divides inversely with resistance.
  • Check: The calculated branch currents must add to (I_{total}) by KCL.

F. Star-delta transformation

Star-delta transformation converts three-terminal resistor networks that cannot be simplified directly by series-parallel rules.

  • Delta to star: For delta resistors (R{ab},R{bc},R{ca}), let (S=R{ab}+R{bc}+R{ca}):
    TEXT
    Ra = RabRca/S
Rb = RabRbc/S
Rc = RbcRca/S
  • Star to delta: For star resistors (R_a,R_b,R_c), let (P=R_aR_b+R_bR_c+R_cR_a):
    TEXT
    Rab = P/Rc,  Rbc = P/Ra,  Rca = P/Rb
  • Equivalence: Both networks present identical resistance between each pair of external terminals.

V. Electrical Sources

Sources supply or control voltage and current in a circuit model.

A. Introduction to dependent and independent sources

Independent sources have prescribed values, whereas dependent sources are controlled by another circuit voltage or current.

  1. Independent sources: An ideal voltage source fixes terminal voltage; an ideal current source fixes branch current regardless of load.

  2. Dependent sources: Their diamond-shaped symbols represent VCVS, VCCS, CCVS, or CCCS sources.

    • A voltage-controlled voltage source may satisfy (v_s=\mu v_x), where (\mu) is voltage gain.
    • A current-controlled current source may satisfy (i_s=\beta i_x), where (\beta) is current gain.
  • Internal activity: Dependent sources remain active during equivalent-resistance calculations because their values are determined by circuit variables.

VI. Systematic Circuit Analysis

Mesh and nodal methods organize Kirchhoff equations for networks that resist direct simplification.

A. Mesh analysis

Mesh analysis assigns a current to each smallest loop of a planar circuit and applies KVL.

  • Setup: Choose all mesh currents clockwise for consistency.
  • Shared resistor: Its drop in mesh 1 is (R(I_1-I_2)), where (I_1) and (I_2) are adjacent mesh currents.
  • Supermesh: If a current source lies between two meshes, write KVL around both meshes and add the source constraint.
  • Scope: The method is efficient when the circuit has fewer meshes than essential nodes and is limited to planar networks.

B. Nodal analysis

Nodal analysis determines node voltages by applying KCL, usually with fewer equations than branch-current analysis.

  • Setup: Select a reference node and assign voltages (V_1,V_2,\ldots) to the remaining nodes.
  • Branch current: Between nodes (a) and (b), (I=(V_a-V_b)/R).
  • Node equation: Sum all currents leaving or entering each nonreference node and set the algebraic total to zero.
  • Supernode: A voltage source between two nonreference nodes forms a supernode; combine its KCL equation with (V_a-V_b=V_s).

VII. Network Theorems

Network theorems replace or decompose linear circuits to simplify terminal calculations.

A. Thevenin's theorem

Thevenin's theorem states that any linear two-terminal network can be replaced by one voltage source (V{th}) in series with one resistance (R{th}).

  • Voltage: (V_{th}) is the open-circuit terminal voltage.
  • Resistance: Deactivate independent voltage sources by shorting them and independent current sources by opening them, then find the terminal resistance.
  • Dependent sources: Keep them active and apply a test source, using (R{th}=V{test}/I_{test}).
  • Load current: For load (R_L), (IL=V{th}/(R_{th}+R_L)).

B. Norton's theorem

Norton's theorem replaces a linear two-terminal network by a current source (I_N) in parallel with resistance (R_N).

  • Current: (I_N) is the short-circuit current between the terminals.
  • Resistance: (RN=R{th}).
  • Source conversion:
    TEXT
    IN = Vth/Rth
Vth = INRN

    Here (IN), (V{th}), (RN), and (R{th}) are the Norton current, Thevenin voltage, Norton resistance, and Thevenin resistance.
  • Use: Norton form is especially convenient for parallel loads and current calculations.

C. Maximum power transfer theorem

Maximum power transfer occurs when a resistive load equals the Thevenin resistance seen from its terminals.

  • Condition:
    TEXT
    RL = Rth

    Here (RL) is load resistance and (R{th}) is source-network Thevenin resistance.
  • Maximum power:
    TEXT
    Pmax = Vth²/(4Rth)

    Here (P{max}) is maximum load power and (V{th}) is Thevenin voltage.
  • Efficiency: At maximum transfer, equal power is dissipated in (RL) and (R{th}), so efficiency is (50\%).
  • Application: The theorem is useful in signal and communication circuits but generally unsuitable for efficient power transmission.

D. Superposition theorem

Superposition states that the response in a linear circuit containing multiple independent sources equals the algebraic sum of the responses produced by each source acting alone.

  • Procedure: Retain one independent source at a time, solve its contribution, and add all signed voltage or current contributions.
  • Deactivation: Replace ideal independent voltage sources by shorts and ideal independent current sources by opens.
  • Dependent sources: Keep dependent sources active because they remain controlled by circuit variables.
  • Limitation: Superposition applies directly to voltage and current, not power, because power depends nonlinearly on (V^2) or (I^2).
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Unit 2