Circuit Analysis I – Basic Laws (Chapter 2)

2.1 Ohm’s Law

• Ohm’s law states that the voltage across a resistor is directly proportional to the current I flowing through the resistor.
• Mathematical expression for Ohm’s Law: $V = I\,R$
• Two extreme possible values of R:
  • R = 0 (zero) corresponds to a short circuit.
  • R = \infty (infinite) corresponds to an open circuit.
• Open and short circuits (concepts):
  • Open circuit: A two-terminal element is open if the branch current is zero (i.e., (i = 0)) for any branch voltage. The characteristic lies along the v-axis (since current is identically zero). In the diagram, this is represented by (R = \infty).
  • Short circuit: A two-terminal element is short if the branch voltage is zero (i.e., (v = 0)) for any branch current. The characteristic lies along the i-axis (since voltage is identically zero). In the diagram, this is represented by (R = 0).
• Conductance is the ability of an element to conduct current; it is the reciprocal of resistance:$G = \frac{1}{R}$
  • Measured in mhos or siemens (S).
• Power dissipated by a resistor: slide mentions this topic but does not display a formula on the transcript. (Common relations include $P = I^{2}R = \frac{V^{2}}{R} = VI$.)

2.2 Nodes, Branches and Loops

• Definitions:
  • A branch represents a single element (e.g., a voltage source or a resistor).
  • A node is a point of connection between two or more branches.
  • A loop is any closed path in a circuit.
• Example 1 (numbers stated): A circuit has 5 branches, 3 nodes, and 6 loops (3 of them independent).
  • Fundamental theorem of network topology relation (for a network): with b branches, n nodes, and l independent loops, the topology satisfies the loop relation (often written as) $l = b - n + 1$ (i.e., the number of independent loops is b - n + 1).
• Example 2: A question about whether a path should be counted as one branch or two branches (discussion of branch counting isn’t fully resolved in the transcript).

2.3 Kirchhoff’s Laws

• Kirchhoff’s Current Law (KCL) (1): The algebraic sum of currents entering a node (or a closed boundary) is zero. Formally around a node: $\sum I = 0$ (sum of currents entering, with sign convention, equals sum of currents leaving).
• Example 4 (from transcript): Determine current I for the circuit shown: $I + 4 - (-3) - 2 = 0 \Rightarrow I = -5\ \text{A}$
  • The negative value indicates the actual current direction is opposite to the assumed direction; the enclosed area can be treated as a single node.
• Kirchhoff’s Voltage Law (KVL) (3): The algebraic sum of all voltages around a closed path (loop) is zero. Formally: $\sum v = 0$
• Example 5 (KVL): Applying KVL to the circuit in the figure:
  • Given: $v<em>a - v</em>1 - v<em>b - v</em>2 - v_3 = 0$
  • Where: $v<em>1 = I R</em>1, \ v<em>2 = I R</em>2, \ v<em>3 = I R</em>3$
  • Therefore: $v<em>a - v</em>b = I\,(R<em>1 + R</em>2 + R_3)$

2.4 Series Resistors and Voltage Division

• Series definition: Two or more elements are in series if they are cascaded or connected sequentially and consequently carry the same current.
• The equivalent resistance for resistors in series is the sum of the individual resistances:$R<em>{eq} = R</em>1 + R<em>2 + \cdots + R</em>n$
• The voltage divider (introduction on the slide): can be expressed as a relation between the input and output voltages across series resistors. (Transcript shows the phrase “The voltage divider can be expressed as” but does not display the explicit formula.)
• Example (Resistors in Series in the transcript): A circuit shows multiple resistors in series with a 10 V source. The total series resistance is the sum of the individual resistances:
  • $R = R<em>1 + R</em>2 + \cdots + R_n$ where the numerical example lists values (5, 35, 25, 10, 5, 50, 15) kΩ.
  • Computed total: $R = (5 + 35 + 25 + 10 + 5 + 50 + 15)\ \text{k}\Omega = 145\ \text{k}\Omega$
• Example 3 (Transcript): “10V and 5W are in series” (note: this line indicates a series connection example; the exact interpretation of 5W in this context is unclear from the transcript).
• The Voltage Divider concept is associated with how the input voltage splits across series resistors, though explicit formula is not shown in the transcript.

2.5 Parallel Resistors and Current Division

• Parallel definition: Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them.
• The equivalent resistance for N resistors in parallel (transcript implies a general formula but does not display it):$R<em>{eq} = \left(\sum</em>{k=1}^{N} \frac{1}{R_k}\right)^{-1}$
• The total current in a parallel network is the sum of the branch currents; currents divide inversely proportional to resistance.
• Current divider formula (transcript):
  • For two parallel resistors R1 and R2 with total current i_total:
  • $i<em>1 = i</em>{TOTAL} \frac{R<em>2}{R</em>1 + R_2}$
  • $i<em>2 = i</em>{TOTAL} \frac{R<em>1}{R</em>1 + R_2}$
  • Note in the transcript: this current division relation differs slightly from the voltage division formula used for series resistors.
• Example 4 (transcript): “2W, 3W and 2A are in parallel” (illustrates a parallel network; specific calculations are not detailed in the transcript).

2.6 Wye-Delta Transformations

• Wye-Delta transformations are transformation techniques between the Delta (Δ) and Star (Y) networks:
  • Delta (Δ) → Star (Y)
  • Star (Y) → Delta (Δ)
• These transformations are used to simplify complex resistor networks by converting between equivalent Δ and Y configurations.

Notes and observations from the transcript

• Several items are presented as definitions and conceptual explanations (Ohm's Law, open/short circuits, node/branch/loop, KCL/KVL) with some equations shown and others implied or missing in the slides.
• Some slides explicitly show numerical examples (e.g., counts of branches/nodes/loops, series resistance sums, specific voltage/current relations) but do not always display all intermediate formulas; where appropriate, the standard related formulas are included here as clarifications.
• In several places, the transcript indicates a formula or expression but does not print the equation (e.g., the voltage divider in 2.4, and the parallel resistance formula in 2.5). Where helpful for study, conventional forms have been included in LaTeX notation to complete the concept.
• The overall structure follows a typical introductory circuit analysis sequence: Ohm’s Law, network topology concepts, Kirchhoff’s laws, series/parallel resistor analysis, and basic network transformations (Wye-Delta).