Study Notes on Basics of Circuit Definitions and Laws

1.0 Introduction

• This chapter establishes the fundamental principles of Applied Electricity required for engineering circuit analysis.
• Key focuses include the governing laws of linear circuits: Ohm’s Law, Kirchhoff’s Current Law (KCL), and Kirchhoff’s Voltage Law (KVL).

1.1 Circuit Elements and Topology

• An electric circuit is an interconnection of elements forming at least one closed path for charge flow.
• Network Parameters:
  • Branch: A single element (e.g., resistor, source) with two terminals.
  • Node: A junction point where two or more branches meet.
  • Loop: Any closed path in a circuit formed by starting at a node and returning to the same node without passing through any intermediate node twice.

1.1.1 Classification of Elements

1. Active Elements: Elements capable of generating energy or providing power gain (e.g., Voltage sources, current sources, batteries, generators).
2. Passive Elements: Elements that consume or store energy (e.g., Resistors, Capacitors, Inductors).

1.1.2 Electric Current and Potential

• Electric Current ($i$): The time rate of change of charge ($Q$) passing through a specific area.
  • $i = \frac{dQ}{dt}$ [Amperes, A]
• Electromotive Force (emf): The energy provided by a source to move unit charge around a circuit.
• Potential Difference (V): The work done in moving a unit charge between two points.

1.2 Passive Element Characteristics

1.2.1 Resistance ($R$)

• The physical property of a material to oppose current flow.
• Factors affecting Resistance:
  • Directly proportional to length ($l$).
  • Inversely proportional to cross-sectional area ($A$).
  • Dependent on resistivity ($\rho$) and temperature.
  • Expression: $R = \rho \frac{l}{A}$ [Ohms, \Omega]
• Conductance ($G$): The reciprocal of resistance. $G = \frac{1}{R}$ [Siemens, S].

1.2.2 Inductance ($L$) and Capacitance ($C$)

• Inductor: Stores energy in a magnetic field. $v = L \frac{di}{dt}$
• Capacitor: Stores energy in an electric field. $i = C \frac{dv}{dt}$

1.3 Fundamental Circuit Laws

1.3.1 Ohm’s Law

• At constant temperature, the current through a conductor is directly proportional to the potential difference across it.
  • $V = IR$

1.3.2 Kirchhoff’s Current Law (KCL)

• Based on the Law of Conservation of Charge.
• The algebraic sum of currents at any node is zero.
  • $\sum<em>{k=1}^{n} i</em>k = 0$
  • Sum of entering currents = Sum of leaving currents.

1.3.3 Kirchhoff’s Voltage Law (KVL)

• Based on the Law of Conservation of Energy.
• The algebraic sum of all voltages around a closed loop is zero.
  • $\sum<em>{k=1}^{n} v</em>k = 0$

1.4 Network Analysis Techniques

1.4.1 Series Circuits

• Current: Remains constant through all elements. $I<em>T = I</em>1 = I_2 = …$
• Equivalent Resistance: $R<em>{eq} = R</em>1 + R<em>2 + R</em>3 + …$
• Voltage Division Rule (VDR): The voltage across a specific resistor $Rn$ is: $V</em>n = \frac{R<em>n}{R</em>{eq}} V_T$

1.4.2 Parallel Circuits

• Voltage: Remains constant across all branches. $V<em>T = V</em>1 = V_2 = …$
• Equivalent Resistance: $\frac{1}{R<em>{eq}} = \sum \frac{1}{R</em>n}$
• Current Division Rule (CDR): For two resistors in parallel, the current through $R1$ is: $I</em>1 = \frac{R<em>2}{R</em>1 + R<em>2} I</em>T$

1.5 Power and Energy in DC Circuits

• Electric Power ($P$): The rate at which energy is dissipated or absorbed.
  • $P = VI = I^2R = \frac{V^2}{R}$ [Watts, W]
• Electric Energy ($E$): The total work done over time.
  • $E = P \times t = VIt$ [Joules, J or kWh]
  • Note: $1 \text{ kWh} = 3.6 \times 10^6 \text{ J}$

1.6 Conclusion

• Mastery of these foundational definitions and laws is prerequisite for Advanced Circuit Analysis (Nodal and Mesh Analysis) and AC Theory in subsequent modules.