DC Circuits Unit-1 Comprehensive Notes

Electric Current

• Flow of free electrons is called electric current.

• When electric pressure (voltage) is applied to a conductor, free electrons move; this directed flow is the current.

• Conventional current direction is from the positive terminal to the negative terminal (opposite to electron flow).

• Strength of current I is the rate of charge flow: I = \frac{dQ}{dt} = \frac{Q}{t}

• Unit: ampere (A) = C/s. Definition: "One ampere is the current when 1 C of charge flows in 1 s".

Electric Potential & Potential Difference

• Electric Potential (V): ability of a charged body to do work by moving other charges (attraction/repulsion).

• Definition: V = \frac{W}{Q} = \frac{\text{Work done}}{\text{Charge}}

• Unit: volt (V) = J/C.

• Potential Difference (voltage): difference in potential between two points.

Resistance

• Opposition to current flow; measured in ohm (Ω).

• Definition: A wire has 1 Ω resistance if a 1 V potential difference causes 1 A to flow: R = \frac{V}{I}

• Key characteristics:

  • Proportional to length: R ∝ l

  • Inversely proportional to cross-sectional area: R ∝ 1/A

  • Depends on material (ρ, resistivity)

  • Depends on temperature

• Ohm’s law (in terms of material properties): If a conductor of length l, cross-section A, and resistivity ρ, then
  R = \rho \frac{l}{A}

• Resistors are used for voltage division and current limiting.

Electric Power

• Power in an electric circuit: rate of doing work:
  P = \frac{W}{t}

• In electrical terms: P = VI = I^2 R = \frac{V^2}{R}

Inductor

• An inductor is a coil of wire designed to exploit the relationship between magnetism and electricity; current through a coil produces magnetic flux.

• Core concepts:

  • Inductance L (Henries, H) relates current i to flux linkage NΦ: N\Phi = L i

  • Faraday’s Law for a single coil: V_L = N \frac{d\phi}{dt} = \mu \frac{N^2 A}{l} \frac{di}{dt} where:

  • N = number of turns

  • A = cross-sectional area

  • Φ = magnetic flux in Webers

  • μ = permeability of core material

  • l = length of the coil

  • Relationship between flux and current: N\Phi = L i

• Inductors oppose changes in current; they pass steady DC easily.

• Schematic symbol is a coil; core types include air, iron, ferrite; core type affects inductance.

Capacitors

• Two conducting plates separated by dielectric; stores electric charge.

• Capacitance definition: C = \frac{Q}{V}

• 1 Farad = 1 C/V.

• Capacitance proportional to plate area A and inversely proportional to separation d: C = \frac{\varepsilon A}{d} where ε is the permittivity of the dielectric.

Voltage and Current Sources

• Ideal Voltage Source: output voltage is constant regardless of load (zero internal resistance in ideal case).

• Ideal Current Source: output current is constant regardless of load (infinite internal resistance).

• In practice, sources have nonzero internal resistance; smaller internal resistance makes a source closer to ideal for voltage sources, larger internal resistance closer to ideal for current sources.

• Source Transformation:

  • Case (i): A voltage source with a series resistance can be converted to an equivalent current source with a parallel resistance.

  • Case (ii): A current source with a parallel resistance can be converted to an equivalent voltage source with a series resistance.

  • Polarity/direction of the equivalent source changes with the transformation when the source polarity/direction changes.

Kirchhoff’s Laws

• Useful for solving networks and determining equivalent resistance and branch currents.

Kirchhoff’s Current Law (KCL)

• At any junction, the algebraic sum of currents entering/leaving is zero: the total current entering equals total current leaving.

• Convention: incoming currents positive, outgoing currents negative. Often written as \sum I = 0.

• Example form: if currents I1, I2, I3 enter/leave a node, then I1 + I4 - I2 - I3 - I_5 = 0.

Kirchhoff’s Voltage Law (KVL)

• For any closed loop, the algebraic sum of the voltage drops across elements and the emfs is zero:
  \sum IR + \sum \varepsilon = 0.

• Voltage sign conventions:

  • Going from a positive to negative terminal across a source is a voltage drop (negative sign for emf).

  • Going from negative to positive is a rise (positive emf).

  • Across a resistor, going in the direction of current yields a voltage drop (negative sign); going opposite the current yields a rise (positive sign).

DC Circuits: Series, Parallel, and Series–Parallel

• DC circuits with direct current can be classified as:

  • a) Series Circuit

  • b) Parallel Circuit

  • c) Series–Parallel Circuit

1.9.1 Series Circuit

• Definition: resistors connected end-to-end, single path for current.

• Key relations:

  • Same current through all resistors: I = I1 = I2 = I3

  • Total voltage is sum of drops: V = V1 + V2 + V3 = IR1 + IR2 + IR3 = I(R1 + R2 + R_3)

  • Equivalent resistance: R{eq} = R1 + R2 + R3

• Voltage divider rule (for two resistors R1 and R2):

  • Current is same: I = V / (R1 + R2)

  • Voltages: V1 = I R1 = V \frac{R1}{R1+R2}, \quad V2 = I R2 = V \frac{R2}{R1+R2}

1.9.2 Parallel Circuit

• Definition: resistors connected across the same two nodes; multiple current paths exist.

• Key relations:

  • Voltage across each resistor is the same: V = V1 = V2 = \ldots

  • Total current is the sum of branch currents: I = I1 + I2 + \ldots

  • Reciprocal of total resistance: \frac{1}{R{eq}} = \frac{1}{R1} + \frac{1}{R_2} + \ldots

• Current divider rule (two resistors R1, R2):

  • Branch currents: I1 = I \frac{R2}{R1 + R2}, \quad I2 = I \frac{R1}{R1 + R2}

Junction, Node, Loop and Mesh

• Node/Junction: a point where two or more circuit elements meet.

• Junction: a node with three or more branches meeting.

• Loop: any closed path through a circuit where no node is encountered more than once.

• Mesh: a closed path with no other paths inside it; every mesh is a loop, but not every loop is a mesh.

Maxwell's Mesh Current Method

• A method where Kirchhoff’s voltage law is written for each mesh using mesh currents instead of branch currents.

• Steps:
  1) Assign a separate mesh current to each mesh (clockwise assumed).
  2) If a circuit element is shared by two meshes, the actual current is the difference of the mesh currents crossing that element.
  3) Write KVL for each mesh in terms of unknown mesh currents.
  4) If a mesh current comes out negative, its true direction is opposite to the assumed one.

• Note: Branch currents are real; mesh currents are fictitious quantities.

Nodal Analysis (Node Voltage Method)

• Based on KCL; used to determine node voltages with respect to a reference node.

• Advantages: fewer equations when the number of loops is large.

• Steps:
  1) Identify all nodes; choose a reference (ground) node.
  2) Assign node voltages with respect to the reference: V1, V2, …
  3) Apply KCL at each non-reference node, write equations in terms of node voltages.
  4) Solve the simultaneous equations to obtain node voltages; currents in branches follow from Ohm’s law.

Superposition Theorem

• Applicable for linear and bilateral networks with multiple independent sources.

• Statement: In a linear network with more than one source, the current in any branch is the algebraic sum of the currents produced by each source acting alone (with other sources replaced by their internal resistances).

• Procedure: For each source, deactivate others (V sources replaced by short circuits, I sources replaced by open circuits), compute the branch current due to that source alone, then sum results.

• Useful for analytical insight and solving networks with multiple sources.

Thevenin's Theorem

• Statement: Any two-terminal network can be replaced by a single emf V{TH} in series with a single resistance R{TH}.

• Definitions:

  • V_{TH} (Thevenin voltage) is the open-circuit voltage across the terminals with the load removed.

  • R_{TH} is the resistance seen at the terminals with all independent sources replaced by their internal resistances (voltage sources shorted, current sources opened).

• Application steps:
  1) Remove the load RL.
  2) Find V{TH} across the open terminals. 3) Find R{TH} seen from the terminals with sources deactivated.
  4) Draw Thevenin equivalent: V{TH} in series with R{TH}.
  5) Reconnect RL and compute IL = V{TH} / (R{TH} + R_L).

• Key note: Thevenin’s voltage is the open-circuit voltage; Thevenin resistance is the resistance seen looking back into the network with sources turned off.

Norton’s Theorem

• Norton’s Theorem is the dual of Thevenin’s Theorem.

• Statement: Any two-terminal network can be replaced by a current source IN in parallel with a resistance RN.

• Definitions:

  • I_N is the current through a short circuit across the terminals (i.e., the short-circuit current).

  • RN is the same as R{TH} (the resistance seen from the terminals with sources deactivated).

• Application steps:
  1) Short the load terminals and find IN (current through the short). 2) Find RN seen from the terminals with sources deactivated.
  3) Draw Norton equivalent: IN in parallel with RN.
  4) Reconnect RL and compute IL = IN × \frac{RN}{RN + RL}.

• Conversion between Thevenin and Norton: V{TH} = IN RN and R{TH} = R_N.

Worked Thevenin/Norton Examples (Representative)

• Thevenin with a 5 Ω load

  • Given a circuit, remove RL = 5 Ω, compute V{TH} and R{TH}.

  • Example results: V{TH} = 20.732 V, R{TH} = 1.463 Ω.

  • Load current: IL = \frac{V{TH}}{R{TH} + RL}} = \frac{20.732}{1.463 + 5} \approx 3.21\ \text{A}

• Thevenin with a 1.5 Ω load

  • Example results: V{TH} = 15 V, R{TH} = 4.5 Ω.

  • Load current: IL = \frac{V{TH}}{R{TH} + RL}} = \frac{15}{4.5 + 1.5} = 2.5\ \text{A}

• Thevenin with an 8 Ω load

  • Example results: V{TH} = 3.275 V, R{TH} = 2.23 Ω.

  • Load current: IL = \frac{V{TH}}{R{TH} + RL}} = \frac{3.275}{2.23 + 8} \approx 0.281\ \text{A}

• Norton with a 20 Ω load

  • Short-circuit current: I_N = 2.5 A (example).

  • Norton resistance: R_N = (10 \parallel 10) + 15 = 20\ \Omega

  • Load current: IL = IN \cdot \frac{RN}{RN + R_L} = 2.5 \cdot \frac{20}{20 + 20} = 1.25\ \text{A}

Numerical Techniques (Summary of Methods)

• Mesh Current Method (Maxwell's approach): assign mesh currents, write KVL for each mesh, relate branch currents to mesh currents, solve linear equations, interpret negative results as opposite-direction currents.

• Nodal Analysis (Node Voltage Method): set reference node, assign node voltages, apply KCL at each node to obtain equations in node voltages, solve for voltages, then compute branch currents with Ohm’s law.

• Superposition: handle one source at a time with others deactivated, sum the effects on the desired quantity.

• Current and Voltage Dividers (in Series/Parallel): use divider rules to compute partial voltages or currents.

Connections to Foundational Principles and Real-World Relevance

• DC Circuit basics build intuition for power systems, electronics, and signal processing.

• The concepts of R, L, C, and their dynamic behavior underpin AC circuits and transient analysis (beyond the scope of this unit).

• Thevenin and Norton equivalents simplify complex networks to enable quick analysis of how a change at the load affects current, voltage, or power.

• KCL and KVL embody conservation laws (charge and energy) in circuit form and are foundational in network analysis.

Quick Reference: Key Formulas (LaTeX)

• Current: I = \frac{dQ}{dt}

• Ohm’s Law: I = \frac{V}{R}, \quad V = IR, \quad P = VI = I^2 R = \frac{V^2}{R}

• Resistance: R = \rho \frac{l}{A}

• Inductance: L = \frac{\mu N^2 A}{l}, \quad V_L = N\frac{d\phi}{dt}, \quad N\Phi = Li

• Capacitance: C = \frac{Q}{V} = \frac{\varepsilon A}{d}

• Thevenin: V{TH},\; R{TH} with IL = \frac{V{TH}}{R{TH} + R_L}

• Norton: IN,\; RN\; (RN = R{TH}) with IL = \frac{IN RN}{RN + RL}

• KCL: \sum I = 0

• KVL: \sum IR + \sum \varepsilon = 0

Notes and Practical Implications

• Always check sign conventions when applying KVL to avoid sign errors.

• In series circuits, current is the same through all elements; in parallel, voltage is the same across all branches.

• For Thevenin/Norton conversions, ensure sources are deactivated correctly (voltage sources shorted, current sources opened).

• When solving networks with many loops, nodal analysis often reduces the number of equations compared to mesh analysis.

• The superposition approach is especially useful when multiple independent sources exist but the circuit remains linear; remember to deactivate sources properly for each case.

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