lecture 4 Resistor Networks and Circuit Analysis

Section 1

• The lecture covers resistor networks, circuit analysis, and practical lab notes. 🧪

• Friday window: class runs from 9 to 12. ⏰ If you
  accepted, you
  re set; if not, a note will be sent indicating you have first priority for October 11.

• If you have a car, offer rides to classmates to help everyone engage and get to know each other (e.g., hiking trip, coordinate rides around 23). 🚗 This is framed as a social/learning strategy in the course.

• Photos shown are from last semester: sophomore Cascade students swimming, potential social activities (s'mores, grilling). 📸🍽 Coordination of food is encouraged for labs and fun, which builds class cohesion.

• A few logistical slides mention kit pickups and deadlines: a kit pickup extension to Friday 11:59 PM for Google Drive uploads, etc. 🎒

• The course uses Yasmin (a messaging tool) on the left side of Canvas to message the board for questions. 📱 The first four questions have been posted.

• Homework guidance emphasized:

  • Always include units (e.g., 10 V, 10 A, 10 W) and use correct prefixes. 📏

  • Use decimal notation with a reasonable precision (e.g., keep numbers like 0.0123 rather than a long tail; prefer 2.3
     10^3 or 2.3e3 when appropriate). 🔢

  • Show your work; don
    t just write the final answer or you
    ll lose points. Partial credit will be given for correct method. ✅

• Quick preview of today and upcoming topics: sections 1.4
  1.7 in the textbook cover solving quantities in resistor networks and cost curves; today
  s practical examples and analysis will tie into that. 📚

• A
  code for the clues
   activity labeled Metallica is due by 3 PM today; students may start but if they do, they
  ll be marked as fifteen minutes late. 🎸 Waiting until after to check slides is encouraged.

• If you get bored during the lecture, you can start the assignment now, but be aware of the late-minute policy. 😴

Section 2

• Resistors are elements that current flows through, and they dissipate energy as heat (power dissipation) rather to store it; electrons move through, producing friction-like losses. 🔥

• Example: a 9 V source across a 10
   resistor yields current $I =\frac{V}{R} =\frac{9}{10} = 0.9 A$ and power $P = VI = 9 \times 0.9 = 8.1 W$. Alternatively, $P = I^2R = (0.9)^2 \times 10 = 8.1 W$. ⚡

• Typical resistor power ratings: common fixed resistors are rated around 0.5 W (0.5 W = 1/2 W). Exceeding this rating causes overheating and damage (blackening, burning smell, etc.). ♨ A demonstration was used to emphasize this risk in the lab.

• A cautionary anecdote: misused lab components can burn or fail, underscoring the importance of staying within safe voltage/power ratings and not improvising dangerous configurations. ⚠

Section 3

• Ohm
  s law in three common forms:

  • $I =\frac{V}{R}$ 💡

  • $V = I R$

  • $P = V I = I^2 R =\frac{V^2}{R}$

• Power sign convention (absorption vs supply):

  • If a resistor absorbs power, the power is positive (P > 0) and the source supplies that power. ➕

  • If an element supplies power, the power is negative (P < 0). ➖

• Worked example (in lecture):

  • A 20 V source across a 4
     resistor yields $I =\frac{20}{4} = 5 A$ and $P = VI = 20 \times 5 = 100 W$. The resistor absorbs power (positive), the source supplies power (negative for the source, since it provides the energy). 🔋

• A second example discussed: a 12
   resistor with 1 A current yields $P = I^2 R = 1^2 \times 12 = 12 W$; the resistor dissipates 12 W. 🔌

Section 4

• Three equivalent ways to calculate power in a resistor:

  • $P = V I$

  • $P = I^2 R$

  • $P =\frac{V^2}{R}$

• Example: A circuit with a 12 V source and a 2
   resistor would have $I =\frac{V}{R} =\frac{12}{2} = 6 A$ and $P = VI = 72 W$ (representing the power dissipated by the resistor). 📈

• Remember: the current through a resistor in a particular circuit is determined by the rest of the network; the formulas above apply to the component once V and I are known. 🌐

Section 5

• A resistor is made from a material with a certain resistivity
  , which affects how easily current flows. ⚛

Resistance formula for a resistor: $R =\frac{\rho \, L}{A}$ where

 is the resistivity (

 m)

• L is the length of the resistive material (m)

• A is the cross-sectional area (m^2)

• Temperature dependence: Resistance is a function of temperature; the table in the text lists resistivity values at a specific temperature, and resistance can vary if temperature changes beyond a narrow range. 🌡

• Extreme resistance concepts:

  • Zero ohms: a short circuit (two points connected with effectively no resistance). Short circuits cause very high current and are dangerous. 💥

  • Infinite resistance: an open circuit (no complete path for current). In practice, open circuit means effectively no current flows; measuring resistance with an ohmmeter (which uses a small internal battery) will read a very large value if the circuit is open. ♾

• Real-world analogy: current takes the path of least resistance; a short circuit provides a path of near-zero resistance, bypassing other resistors. 🛤

Section 6

• Resistor color code mnemonic given in class: "Bad burgers ruin our yum guts, but vegetables go" (colors in order). 🌈 This helps you read the resistance value from color bands.

• Standard tolerances: most fixed resistors have a tolerance of
  5%.
  

• Example interpretation: A labeled resistance of 1000
   with
  5% tolerance would actually be in the range from 950
   to 1050
  . 🔍

• Important note: If you misread a resistor or misinterpret color bands, you can obtain incorrect values; always double-check before building a circuit. ✔

Section 7

• Nodes: A node is a connection point between two or more circuit elements. If two wires meet and there is no element between them, they are the same node. 📍

• Series connections:

  • Elements in series share a single exclusive connection point; no other element is connected to that junction.

  • In a true series chain, the same current flows through all components: IA = IB = I_C, etc. ⛓

• Parallel connections:

  • Elements in parallel share two common nodes (the same two nodes connect across each element).

  • In parallel, the voltage across each element is the same (V across all parallel branches is identical) but the currents can differ based on each branch
    s resistance. ↔

• The notion of exclusivity is emphasized: in series, the junction is exclusive (no other elements connect to that junction); in parallel, branches share the same two nodes. 🤝

• Node counting example concept:

  • In a diagram with multiple junctions, you count distinct nodes by grouping points that are connected without an intervening element.
    

  • If an element lies between two points, that creates a separate node; if no element lies between, those points are the same node. 🗺

• Practical takeaway: when solving circuits, look at the connection points (nodes) rather than focusing on wires; this helps identify which components are in series or in parallel. 🤔

Section 8

• Kirchhoff
  s Current Law (KCL): The algebraic sum of currents at a node is zero, i.e., currents entering equal currents leaving. ⚖

  • Example form used in lecture: IA + IC = IB + ID at a particular node, or equivalently IA + IC
     IB
     ID = 0.

• In a simple series chain, currents through all elements are equal (IA = IB = IC) because the current has only one path through the series chain. ➡

• In a parallel group, the currents may differ, but the voltages across all parallel branches are the same. ↕

• Direction conventions: currents can be assumed in a direction to begin solving; check results and flip directions if the math yields a negative current. 🔄

• Node-based equations are the primary tool for network analysis; KCL is the basis for setting up equations around nodes. ✏

Section 9

• Prototyping boards and boards:

  • A prototyping board (often called a perf board) is used to assemble circuits before committing to a printed circuit board (PCB). Connections are made by soldering wires or components to the board. 🛠

  • In production, circuits may go from a prototype board to a PCB with etched copper traces or a perf board with drilled holes. 🏭

• Fixed resistors vs variable resistors:

  • Fixed resistors have fixed values; variable resistors can change value and are used as rheostats (variable resistors) or potentiometers when used as a voltage divider. ⚙

  • Potentiometers (pots) are adjustable resistors with a sliding contact (the wiper) that allows varying resistance or voltage. 🖐

  • Example mechanism: The potentiometer typically has two end terminals and a wiper; measurement shows resistance varies from one end to the wiper as you turn the knob. 🎚

  • Typical small pots in kits are blue with a white adjustment screw (as described in the lecture). 🔵⚪

• Measurement and practical use: when you measure a resistance across a pot, you
  ll see the resistance vary as you move the wiper; the device is frequently used to adjust a voltage ratio in a circuit. 📊

Section 10

• The reduction of resistance in wire paths is often neglected in basic circuit models; wire resistance is typically small compared to component resistances but grows more significant in larger circuits. 🤏

• Conductance g is the reciprocal of resistance: $g = 1/R$. Unit-wise, conductance is measured in siemens (S); earlier term used was mhos (the inverse of ohm). ➰

• The ohmmeter uses a small internal battery (e.g., nine volts) to apply a test voltage and measure current to determine resistance. 🔋 If no current flows (open circuit), R appears infinite; if a short to ground occurs, R approaches zero. 📏

• Color code mnemonic: the class used "Bad burgers ruin our yum guts, but vegetables go" to memorize the color bands and their digit values (Black 0, Brown 1, Red 2, Orange 3, Yellow 4, Green 5, Blue 6, Violet 7, Gray 8, White 9). 🍔🥕

• Tolerances: most resistors have
  5% tolerance, meaning the actual resistance can vary within that band. ✨

Section 11

• Case study: component quality and supply chain decisions matter; even a small 10-cent capacitor can cause a board failure if it fails under stress, moisture ingress, or poor soldering/assembly practices. 💧 A vendor delivering cheaper capacitors with poorly sealed ends allowed moisture to enter and caused failure, mold growth, and conduction faults. 📉

• The moral: cost-cutting in components can have outsized consequences on reliability and the overall board cost due to failure modes and rework. 💸 Quality control and robust design matter in engineering practice. 👷

• The lesson for students: always consider reliability and safety, not just cost when selecting components; use proper testing, and design with protection against faults (e.g., fusing, proper ratings).🛡

• Discussion about industry realities: the balance between price and reliability; engineers must advocate for design choices that minimize risk, not just chase the lowest component price. ⚖

Section 12

• When facing complex circuits, start with a clear node analysis using KCL, then identify series and parallel groups to simplify the network step by step. 🧩

• Use the node-pair method to decide which elements are in series or parallel based on shared nodes and exclusivity of connections. 🤝

• If you encounter uncertain current direction, assume a direction; negative results indicate the actual direction is opposite to your assumption. ⬅➡

• In exercises, always report units and proper prefixes for all quantities; avoid ambiguous numbers (e.g., 10 vs 10.0) and show your reasoning to earn partial credit if needed. 💯

• Be mindful of the typical values and tolerances when selecting resistors
  tolerances can alter the effective performance of a circuit, especially in precision applications. 🎯

Section 13

• Given a 9 V source across a 10
   resistor:

  • Compute I and P using all three power forms. 💪

• If two resistors R1 and R2 are in parallel across a 12 V source, write the expression for the equivalent resistance R_eq. ↔

• A resistor with
   = 1.7
  10^-8
  
   m, length L = 2.0 cm, cross-sectional area A = 0.2 mm^2; calculate R. (Convert units as needed.) 📏

• Explain why a short circuit (R
   0) is dangerous and how it affects current in a circuit. ⚡

• Draw a simple schematic with a 12 V supply and two parallel resistors (R1 and R2) plus a series resistor R3; write the node equations for the circuit and determine the currents in each branch (qualitative rather than numeric). ✍

• Describe the function of a potentiometer in a circuit and how you would test a pot
  s resistance from one end to the wiper. 🎛

Section 14

• Node: A connection point where two or more circuit elements meet. If no element lies between two points, they are the same node. 📍

• Bus: Another term sometimes used for a node (in power systems contexts). 🚌

• Series connection: Elements share a single exclusive node; same current flows through each. ⛓

• Parallel connection: Elements share two common nodes; same voltage across each element, currents can differ. ↔

• Short circuit: Path of zero resistance causing excessive current; dangerous, avoid in design. 💥

• Open circuit: Infinite resistance along the path; no current flows. 🚫

• Potentiometer (pot): A three-terminal variable resistor used to vary a voltage divider. 🎛

• Rheostat: A variable resistor used to adjust current by changing resistance. ⚙

• Conductance (g): The inverse of resistance, $g = 1/R$; measured in siemens (S). ➰

• Ohm
  s law: Forms $I = V/R$, $V = IR$, $P = VI = I^2R = V^2/R$. 💡

• Power conventions: Positive P for absorption by a component; negative P indicates energy delivery by the component. ➕➖

Section 15

• Recognize and explain the three fundamental Ohm
  s law forms and the three equivalent expressions for power. 💡

• Distinguish between series and parallel configurations by node analysis and the rule about currents/voltages in each configuration. ↔⛓

• Apply node-based Kirchhoff
  s Current Law to write and solve circuit equations. ✍

• Understand resistor properties: $R = \rho L/A$, R in
  ,
   in
  
   m, R depends on temperature, and the cross-sectional area and length are design levers. ⚛

• Use color codes and tolerance to select proper components; know the common tolerance (
  5%) and how it affects circuit behavior. 🌈

• Be able to identify short/open circuits visually and conceptually, including their impact on current and readings on instruments like ohmmeters. 💥🚫

• Understand practical lab practices: how to read schematics, how to interpret a board layout (prototype vs PCB), and how to reason about equipment safety and reliability (e.g., component failures). 🧪👷