ENGR-241 Week 1 Notes (Chapter 1: SI Units, Circuit Variables, Power & Energy)

The International System of Units (SI)

• Memorize the boxed prefixes; they are used constantly in circuit analysis.
• Common prefixes (SI):
  • $10^3 = \text{k}$ (kilo), $10^6 = \text{M}$ (mega), $10^9 = \text{G}$ (giga), etc.
  • $10^{-3} = \text{m}$ (milli), $10^{-6} = \mu$ (micro), $10^{-9} = \text{n}$ (nano), $10^{-12} = \text{p}$ (pico), etc.
• These prefixes allow us to express large and small quantities succinctly in volts, amperes, ohms, farads, henries, watts, etc.

Circuit Analysis: An Overview

• Purpose: Introduce how circuit theory fits into broader engineering practice.
• Key concepts likely covered here include modeling circuits with elements, sources, and interconnections, and using voltage/current variables to analyze systems.

Voltage and Current

– Part 1

• Voltage (v): energy per unit charge created when + and - charges are separated.
• Defining equation for voltage: $v = \frac{d w}{d q}$
  • where
  • $v$ is voltage in volts [V],
  • $w$ is energy in joules [J],
  • $q$ is charge in coulombs [C].
• Relationship indicates that voltage is the differential change of energy with respect to charge.

Voltage and Current

– Part 2

• Current (i): the time rate of change of charge flow.
• Defining equation for current: $i = \frac{d q}{d t}$
  • where
  • $i$ is current in amperes [A],
  • $q$ is charge in coulombs [C],
  • $t$ is time in seconds [s].
• Intuition: current measures how quickly charge passes a point in a circuit.

The Ideal Basic Circuit Element

• Definition of a few terms:
  • Ideal: the element can be described solely by the relationship between its voltage and current.
  • Basic: the element cannot be subdivided into simpler elements.
  • Circuit Element: an entity with two terminals that connects to other elements to form a circuit.
• This forms the foundational abstraction used to analyze circuits (resistors, capacitors, inductors, sources are all cross-cutting concepts under this umbrella).

Power and Energy

– Part 1

• Power (p): the time rate of change of energy.
• Defining equation: $p = \frac{d w}{d t}$
  • where
  • $p$ is power in watts [W],
  • $w$ is energy in joules [J],
  • $t$ is time in seconds [s].
• Relationship to energy: energy is the integral of power over time, i.e., $w = \int p\,dt$
• Practical implication: knowledge of either power as a function of time or energy over a period allows determination of the other.

Power and Energy

– Part 2

• Fundamental identity: $p = v i$

  • where
  • $v$ is the voltage across the element,
  • $i$ is the current through the element.

• This can be equivalently written in differential form using energy: $p = \frac{d w}{d t}$

  and using the chain rule: $p = v i$ (with v and i defined consistently).

• In many cases it is useful to remember the relationship in a compact form:

  • $p = v i$

Power and Energy

– Part 3

• Sign conventions are essential for correctly determining whether an element is delivering or absorbing power.
• The passive sign convention states:
  • If the reference direction for the current in an element is the same as the reference voltage drop across the element, use a positive sign in expressions relating voltage to current.
  • If not, use a negative sign.
• In practice: the common working form is
  • $p = \pm v i$
• The sign is chosen so that positive power means absorbing (dissipating) power, and negative power means delivering (generating) power.

Power and Energy

– Part 4

• The practical version of the passive sign convention:
  • If the current arrow points toward the + terminal of the voltage, use a + sign in the expression for power.
  • Otherwise, use a − sign.
• For the ideal basic circuit element shown (in typical introductory circuits), the power expression becomes
  • $p = + v i = v i$
• Summary rule: use the sign convention that makes absorbed power positive and delivered power negative, consistent with the orientation of current and voltage.

Power and Energy

– Part 5

• Examples of interpreting the algebraic sign of power (typical outcomes):
  • If the algebraic power value is negative, the element is delivering (supplying) power to the circuit.
  • If the algebraic power value is positive, the element is absorbing (dissipating) power from the circuit.
• Illustrative statements:
  • p is negative -> element supplies power.
  • p is positive -> element absorbs power.

Power and Energy

– Part 6

• Meaning of positive vs negative power:
  • Positive power means the element is absorbing (dissipating) energy from the circuit.
  • Negative power means the element is generating (delivering) energy to the circuit.
• This interpretation helps verify circuit analyses by checking energy balance.

Power and Energy

– Part 7

• Problem intuition examples (conceptual):
  • For a given circuit element, determine whether the element is supplying or absorbing by inspecting the direction of current with respect to the voltage polarity.
  • This is a quick sanity check in problem solving to ensure consistency with the passive sign convention.

Problem 1 (Chapter 1)

– Charge delivered to terminal 1

• Setup: total charge entering terminal 1 is computed from current-time history: $q(\text{terminal 1}) = \int_{0}^{\infty} i(t) \, dt$
• Result: the total charge entering terminal 1 is $q = 0.004\ \text{C}$
• Equivalent values of 0.004 C:
  • $4\ \text{mC}$
  • $4000\ \mu\text{C}$
• Non-equivalent values (for reference):
  • $4\ \text{kC}$ would be $4000\ \text{C}$ (not equal)
  • $40{,}000\ \text{pC}$ would be $4\times 10^{-8}\ \text{C}$ (not equal)
  • $0.04\ \text{C}$ (not equal)
• Takeaway: convert to common units to verify equivalence with known prefixes.

Problem 2

– Power in a circuit element

• Given a circuit with a voltage source and current, determine power and whether it is generated or absorbed.
• Example result using the passive sign convention:
  • If the current and voltage have the orientation such that the element is delivering energy to the rest of the circuit, then
  • $p = - v i$ and numerically this equals -80 W for the cited values.
  • If the orientation is such that the element is absorbing energy, then
  • $p = + v i$ and numerically this equals +80 W for the cited values.
• Practical takeaway: for a given element with voltage v and current i, compute p = v i and assign the sign according to the passive sign convention to determine generation vs absorption.

Problem 2

– Solution (conceptual approach)

• To find the total energy delivered to a circuit element, use the energy integral over the time interval of interest:
  • $w = \int<em>{t</em>0}^{t<em>1} p(t) \, dt = \int</em>{t<em>0}^{t</em>1} v(t) i(t) \, dt$
• If explicit time-varying forms of v(t) and i(t) are given, substitute and evaluate the integral.
• The example emphasizes that energy can be computed from the integral of power and that units and signs must be consistent.

Problem 3

– Power in a multi-element circuit

• Task: find the power associated with each circuit element a, b, c, d, e, f given voltages and currents.
• Procedure:
  • For each element, determine p = v i with the sign determined by the element’s current direction relative to the voltage polarity.
  • Classify each as supplying or absorbing based on the sign of p.
• Example (illustrative values):
  • a: p = -56 W (supply)
  • b: p = -14 W (supply)
  • c: p = 150 W (absorb)
  • d: p = -50 W (supply)
  • e: p = -18 W (supply)
  • f: p = -12 W (supply)
• Power balance principle (most important takeaway):
  • The total power supplied by all sources equals the total power absorbed by all elements:
  • $\sum p<em>{\text{supplied}} = \sum p</em>{\text{absorbed}}$
  • Therefore, the circuit is in power balance with no net power generation or dissipation.

Problem 3

– Solution (Part 2) – Power balance

• Statement: The power in a circuit always balances; the total power supplied equals the total power dissipated.
• Importance: This balance serves as a consistency check when using circuit analysis methods to find voltage and current for every element.

Assignment and Course Logistics

• Homework due date: Thursday, 08/28/2025, at the start of class.
• Assigned problems (Nilsson & Riedel):
  • Prob. 1.14
  • Prob. 1.18
  • Prob. 1.20
  • Prob. 1.33
  • Prob. 1.35
• Textbook: Nilsson & Riedel, Electric Circuits, 12th ed., Pearson. ISBN-13: 978-0-13-37648375
• Textbook and course details are from ENGR-241, Fall 2025, Week 1.

Agenda (Week 1, Chapter 1 scope)

• The International System of Units (SI)
• Circuit Analysis: An Overview
• Voltage and Current
• The Ideal Basic Circuit Element
• Power and Energy
• Problems

Foundational References and Context

• Textbook source: J. W. Nilsson and S. A. Riedel, Electric Circuits, 12th ed. (Pearson)
• This week’s material lays the groundwork for circuit variables, passive sign convention, and energy/power relationships that are used throughout ENGR-241.
• Real-world relevance: Understanding SI units and sign conventions is essential for properly modeling and analyzing electrical circuits in engineering practice.

Key Takeaways for Exam Preparation

• Master the definitions:
  • Voltage: $v = \frac{d w}{d q}$
  • Current: $i = \frac{d q}{d t}$
  • Power: $p = \frac{d w}{d t} = v i$
• Know the passive sign convention and how to apply sign to determine whether an element is delivering or absorbing power.
• Remember the power balance principle: total power supplied equals total power absorbed in any circuit.
• Be comfortable converting between units using SI prefixes (k, M, m, μ, n, p, etc.).
• For problem-solving: use integrals for energy, and use p = v i with correct sign to evaluate power for each element.