CSM 153 Circuit Theory: Unit 2

• CSM 153 Circuit Theory focuses on direct circuit analysis.
• The notes cover Unit Two.
• Unit Two covers Ohm's and Kirchhoff's laws, series and parallel circuits, and methods of analysis.

Ohm's Law

• Electric current through a metallic conductor is proportional to the potential difference, assuming constant temperature and physical factors.
• Mathematically, Ohm's Law is V = IR, with R as the constant resistance.
• A conducting material obeys Ohm's Law if its resistivity is independent of the magnitude and direction of the applied electric field.
• A conducting device obeys Ohm's Law if its resistance is independent of the magnitude and polarity of the applied potential difference.
• Homogeneous materials obey Ohm's Law within a range of electric field values.
• Departures occur if the electric field becomes too strong.
• A resistor provides a specific resistance (R) in a circuit.
• A resistor has a specified resistance value that remains constant regardless of the magnitude/polarity of the potential difference.
• Device resistance is independent of the magnitude and polarity of the potential difference.

Series Circuits

• Two elements are in series if connected at a single point with no other current-carrying connections.
• Current is the same in all parts of a series circuit.
• The sum of voltages (V₁, V₂, V₃) equals the total applied voltage (V).
• From Ohm's law: V₁ = IR₁, V₂ = IR₂, V₃ = IR₃, and V = IR, where R is the total resistance.
• In a series circuit, total resistance is the sum of individual resistances: R = R₁ + R₂ + R₃.

Parallel Circuits

• Elements are in parallel if they have exactly two nodes in common.
• Parallel elements have the same voltage across them.
• The sum of currents through parallel branches (I₁, I₂, I₃) equals the total current (I).
• The source potential difference is the same across all resistors in parallel.
• From Ohm's law: I₁ = V/R₁, I₂ = V/R₂, I₃ = V/R₃, and I = V/R, where R is the total resistance.
• For a parallel circuit: 1/R = 1/R₁ + 1/R₂ + 1/R₃
• For two resistors in parallel: 1/R = (R₂R₁) / (R₁ + R₂)

Kirchhoff's Laws

• A node is a connection point for two or more elements.
• A branch is a single path in a network between two nodes.
• A loop is a closed path in a circuit where no element/node is encountered more than once.
• Electrical networks are systems of electrical conductors.
• Gustav R. Kirchhoff extended Ohm's Law into two laws for electrical networks in 1847.
• Kirchhoff's Laws enable calculation of current in any part of an electrical network.
• Kirchhoff's Laws are applicable to all circuits, regardless of series/parallel connections.
• The total current entering a junction equals the total current leaving it.
• The algebraic sum of currents into/out of a circuit junction must be zero (I₁ + I₂ + I₃ = 0).
• Kirchhoff's first law applies to any point or junction in a network.
• ∑Qin = ∑Qout; The total charge flowing to the junction equals the total charge flowing out.
• There is no accumulation or depletion of charge at a junction.
• First Law is a statement of conservation of charge for steady flow of charge/current.
• Charge is neither created nor destroyed but transferred from point to point.
• The algebraic sum of currents directed in/out of a junction must be zero: ΣI = 0
• Around a closed loop, the algebraic sum of emfs is equal to the algebraic sum of voltage drops: ΣE = ΣIR (Kirchhoff's 2nd Law)

Voltage and Current Divider Theorems

• In a series circuit, the voltage across a part is equal to its resistance multiplied by the total voltage, divided by the equivalent resistance:
• Vx = V(Rx / Req).
• The voltage division allows calculating the fraction of total voltage dropped across a resistor in a series.
• In a parallel circuit, the current in a branch equals the current entering the circuit multiplied by the equivalent resistance, divided by the branch resistance.
• Current division allows calculating the fraction of total current flowing through a resistor in parallel.
• Ix = IT(Req / R

Nodal and Mesh Analysis

• Mesh Analysis determines independent current variables.
• Steps include selecting appropriate number of independent current variables; express dependent current variables using KCL at nodes; apply KVL around loops; solve for independent currents.
• Nodal Analysis determines independent voltage variables.
• Steps include selecting an appropriate number of independent voltage variables; express dependent voltage variables using KVL; apply KCL at nodes; solve for independent voltages.
• Choose a ground node connected to maximum elements/sources as reference.
• Voltage is assumed to be zero at the ground node.