Circuit Theory Study Notes

Circuit Analysis – Voltage Sources in Series

• Overview of Two Voltage Sources in Series

  • Explanation of two voltage sources in series pushing against each other.

  • Result: Their effects partially cancel to ground.

  • How to determine if they add or subtract: Depend on their signs; the larger one dictates current direction.

• Practical Implications

  • Rarely recommended to build circuits this way in practice, yet knowledge is important for future studies.

  • Situational modeling may require components to be treated as fixed voltage sources opposing one another.

• Kirchhoff’s Voltage Law (KVL) Application

  • Analyze a circuit with voltage sources (V1, V2, V3) and their effects:

  - Starting Point: Bottom left, traveling through each source.

  1. Travel through V1:

     • Decreasing potential (from higher to lower potential) → Negative sign.

  2. Travel through V2:

     • Increasing potential (from negative to positive) → Positive sign.

  3. Travel through V3:

     • Increasing potential (from negative to positive) → Positive sign.

  4. Measure the potential across voltage source Vx from + to -.

     • Decreasing potential → Negative sign for Vx.

• KVL Equation Formulation

  • Sum of voltage changes = 0:
    -V1 + V2 + V3 - Vx = 0

  • Rearranging gives:
    Vx = -V1 + V2 + V3

  • Example Calculation:

  • Plugging numbers:

    • Substitute V1 = -4V, V2 = 2V, V3 = 8V

  • Result:
    V_x = 6V

Voltage Measurement Polarity

• Measuring Voltage

  • Change measurement poles:

  • Positive lead on top, negative on bottom → yields Positive reading.

  • Switch leads → yields Negative reading.

  • Example with Circuit Simulation:

  • Two resistors and a measurement experiment using a multimeter to demonstrate voltage readings based on connection order.

  • With R1: Expecting readings to match polarity as expected from the passive sign convention.

• Importance of Measuring Method

  • Consistent methodology needed to ensure correctness of voltage readings.

  • Plugging leads in reverse results in opposite signs.

Current in Series Circuits

• Understanding Series Circuit Behavior

  • Example Circuit: Two sources (10V and 5V) with 1k and 2.7k resistors.

  • Current direction: Determined to be Clockwise by analysis.

• Voltage Analysis and KVL Application

  • Assume a starting point and travel through each voltage source, while tracking potential changes:

  • V1 - I imes R1 - V2 - I imes R2 = 0

  • Substitute the known values for V1 and V2 to solve for the current I.

  • Resulting Current Formula:
    I = rac{V1 - V2}{R1 + R2}

  • Validate current calculations against physical intuition.

Kirchhoff’s Current Law (KCL)

• Explanation of KCL

  • Total current entering a junction = Total current leaving the junction.

• Example Application of KCL

  • A junction with currents labeled: I1, I2 (entering) and I3, I4 (leaving). Formulate KCL equation:
    I1 + I2 = I3 + I4

• Understanding Current Flow

  • The significance of mislabeling current directions leading to incorrect interpretations and a negative result indicating flow opposite to assumption.

Parallel Circuits Overview

• Structure of Parallel Circuits

  • Resistors connected in parallel share terminal connections; voltage remains constant across components while current splits.

• Key Characteristics

  • Same voltage across all parallel components (V = V1 = V2 = V3)

  • Currents add up to the total source current.

  • Formula: Current provided by source (I_total) = Sum of individual branch currents.

• Example & Calculation with KCL

  • Given three resistors connected in parallel:

  • Use current values to find total current vs individual branch current. Example:
    Itotal = I1 + I2 + I3

• Total Resistance in Parallel Circuits

  • Total resistance of multiple parallel resistors is calculated with:
    rac{1}{RT} = rac{1}{R1} + rac{1}{R2} + rac{1}{R3}

  • Ensure to take the reciprocal of the sum to find RT.

Current Divider Rule (CDR)

• Introduction to CDR as a tool for analyzing current distribution in parallel circuits.

  • CDR Formula:
    Ix = I{source} imes rac{RT}{Rx}

  • Applicable when voltage is unknown or to find how current divides across parallel paths.

• Special Case of Two Parallel Resistors

  • For two resistors R1, R2 connected in parallel — shortcuts to simplify calculations:
    I1 = I{source} imes rac{R2}{R1 + R2} I2 = I{source} imes rac{R1}{R1 + R2}

• Practical Application

  • When solving KCL in a circuit junction, observe that each resistor's current should lead to confirmations with overall circuit behavior and KCL.

  • Validate current division against KCL to ensure results align with expected theory.