Summary
Highlights
Key circuit components are defined: a branch is a single element, a node is a connection point between two or more branches, and a loop is any closed path. A general formula for branches and nodes (B = L + N - 1) is mentioned. An example circuit is analyzed to identify the number of branches (4), nodes (3), and elements in series (5-ohm resistor with voltage source) and parallel (current source with 6-ohm resistor).
The lecture begins by introducing the topic of basic laws in electrical circuits, emphasizing that 'it doesn't matter how slow you go, as long as you never stop pushing on.' The learning outcomes for the module include determining variable values, applying Ohm's and Kirchhoff's laws, understanding series and parallel resistors, voltage and current division, and Y-delta transformations.
Ohm's Law is introduced, defining resistance as a material's ability to resist current flow (R = ρL/A). The resistivity (ρ) of common materials like silver, copper, carbon, and glass is discussed. Ohm's Law (V = IR) states that voltage across a resistor is directly proportional to the current. The concepts of short (R=0) and open circuits (R=∞) are also explained.
Different types of resistors are presented, including fixed resistors like wire-wound (high power) and carbon-film (common, with color codes for nominal value, multiplier, and tolerance), and variable resistors (potentiometers and rheostats). Conductance (G = 1/R = I/V) is introduced as the reciprocal of resistance, measured in Siemens (S).
Several problems are solved using Ohm's Law. Examples include calculating resistance for an electric iron (R = 60 ohms), finding internal resistance of a battery (R = 0.169 ohms), determining current, conductance, and power for given circuits, and calculating current and power for a voltage source with a sinusoidal function (I = 4sin(πt) mA, P = 80sin²(πt) mW). A problem involving power and voltage to find current and resistance is also covered (I = 2cos(πt) mA, R = 5 ohms).
Series circuits are characterized by different voltages across elements, where the sum of individual voltages equals the source voltage, and current is the same throughout. Parallel circuits have the same voltage across each element, while current may vary and is divided among branches. An analogy with home wiring is used to explain the benefits of parallel connections. A problem involving three parallel resistors is solved to find the total current given the current in one branch (It = 5A).
Kirchhoff's Laws are introduced: Kirchhoff's Current Law (KCL) states that the algebraic sum of currents entering a node is zero, and Kirchhoff's Voltage Law (KVL) states that the algebraic sum of all voltages around a closed path is zero. KVL is applied to find unknown voltages in a multi-loop circuit (V1=35V, V2=5V, V3=30V), and KCL is used to find unknown currents in a nodal circuit (I1=12A, I2=-10A, I3=5A).
Problems involving dependent sources are demonstrated. In one example, KVL is used for a circuit with a dependent voltage source (2Vo), where Vo is the voltage across a 6-ohm resistor. The current 'i' is found to be -8A and Vo to be 48V. Another example uses KCL for a circuit with a dependent current source (0.5Io), finding Io = 6A and Vo = 24V.
The equivalent resistance of series resistors is the sum of individual resistances (Req = R1 + R2 + ... + Rn). The voltage divider formula (Vn = R_n / (ΣR) * V_source) is introduced. An example problem calculates V1 (8V) and V2 (12V) in a series circuit connecting a 20V source to 2-ohm and 3-ohm resistors.
The equivalent resistance for parallel resistors is 1/Req = 1/R1 + 1/R2 + ... + 1/Rn. For two parallel resistors, Req = (R1 * R2) / (R1 + R2). The current divider formula is provided: for I1 in a two-resistor parallel branch, I1 = R2 / (R1 + R2) * I_total. Calculations for conductance in both parallel and series configurations are also explained. A complex circuit involving series and parallel combinations is simplified to find the equivalent resistance (14.4 ohms).
The Y-Delta transformation is introduced as a method to simplify complex circuits. For Delta-to-Y conversion, each Y-resistor is the product of two adjacent delta branches divided by the sum of all three delta resistors (e.g., R1 = (Rb * Rc) / (Ra + Rb + Rc)). For Y-to-Delta conversion, each delta resistor is the sum of all possible products of Y-resistors taken two at a time divided by the opposite Y-resistor. Balanced networks (where all resistors are equal) lead to simplified conversion formulas (Ry = R_delta / 3 or R_delta = 3Ry). Examples demonstrate converting a delta network to an equivalent Y network (R1=5Ω, R2=3Ω, R3=7.5Ω) and a Y network to an equivalent delta network (Ra=140Ω, Rb=70Ω, Rc=35Ω). Lastly, a complex circuit with two delta configurations is solved by converting one delta to a Y network and then simplifying the series and parallel combinations to find the equivalent resistance Rab (142.32 ohms).