ENGR 2001 Week 7 Applications

ENGR 2001: Week 7 Applications Notes

Today's Plan

• Review of Schaum's Chapter 3: Linear Systems
• Circuit Analysis

Linear Simultaneous Equations

• Application Example: Electric Circuits

Application – Electric Circuits

• Voltage Relationship: In a simple circuit, the voltage supplied ($V$) equals the current ($I$) times the resistance ($R$): $V = I imes R$
  • Battery: Voltage Source
  • Resistor
  • Current Flow: Positive and Negative terminals involved.

Kirchhoff's Laws

• KCL (Kirchhoff Current Law):
  • Definition: The algebraic sum of currents entering a node is zero.
  • Equivalent form: Sum of currents in = sum of currents out.
  • Node Definition:
  • A single node
  • A closed boundary (a super node).

Example 1

• KCL Application:

  • Given:
    $I<em>1 + I</em>3 + I<em>4 = I</em>2 + I_s$
    Where:
  • Currents involved: $I1$, $I2$, $I3$, $I4$, $I_s$.

• Symbols Used:

  • $I$: Current
  • $h$: Node placeholder for currents.

• Values Provided:

  • Example states currents as 12, 15, $I_s$.

• Conservation: Emphasizes the law of conservation of charge.

KVL (Kirchhoff Voltage Law)

• Definition: The algebraic sum of voltage drops around a loop (any closed loop) is zero.

Example 2

• Circuit Analysis:
  • Question posed: How many nodes are in the circuit?

Nodal Voltage Drop Calculation

• Voltage drop represented as follows: $V<em>{ab} + V</em>{bc} + V<em>{cd} + V</em>{de} + V_{ea} = 0$
  • Order Importance: Order of voltage nodes and their signs play a crucial role.

Systematic Methods for Circuit Analysis

• Two Approaches:

  1. Nodal Analysis
  2. Mesh Analysis

• Applicability: Both methods are applicable to DC and AC circuits.

Nodal Analysis

• General Approach:
  • Use voltages at nodes as variables:
  • Steps in Nodal Analysis:
  1. Identify “n” nodes in the circuit.
  2. Select a reference node (ground).
  3. Assign variables to the other $n-1$ nodes ($V1$, $V2$, …, $V_{n-1}$).
  4. Apply KCL to each of the $n-1$ nodes and form one equation per node for solving nodal voltages.
  5. Use Ohm’s Law ($V = I imes R$) to express relationships between current through resistors connected to the node and the node voltages.
  6. Form simultaneous equations.
  7. Solve the equations for unknown nodal voltages.
  8. Use solved voltages to find other needed quantities.

Example 3

• Identifying Nodes:

  • Identify three nodes (0, 1, and 2).
  • Select reference node (0) and mark it as ground (0 volts).
  • Complete assignment of nodal voltages as $V1$, $V2$.
  • Ground Selection: Mention that positioning ground can vary depending on convenience in complex problems.

• KCL Setup:

  • At Node 1:
  • Branches:
  1. $I_1$ entering
  2. $I2$, $I3$, $I_4$ leaving
  • Equation:
    $I<em>1 = I</em>2 + I<em>3 + I</em>4$

Consistent Current Directionality

• Determine directionality of currents based on circuit components:
  • For current sources, follow the given direction.
  • For resistors, keep a consistent approach (left to right, top to bottom).

KCL in Node 2

• Describe connections in Node 2:
  • Branches connect $I2$ and $I4$ entering Node 2, $I_5$ leaving Node 2.
  • Equation established as:
    $I<em>2 + I</em>4 = I_5$

Conclusion of Example 3: Setting up Nodal Equations

• Use Ohm’s Law to relate current with corresponding nodal voltages as derived from prior calculations in the KCL equations at both nodes.

Example 4: Find Nodal Voltages

• Step 1: KCL equations for three nodes will be established, including any dependent current sources.
  • Identify dependent sources and define their relationship, e.g., a current-controlled current source defined as $2I_x$.
• Step 2: Setup system of KCL equations with reference to prior nodes.
• Set equations for mesh analysis to derive around the selected nodes.

Mesh Analysis

• Define mesh currents as variables and follow similar steps laid out previously:
  1. Identify “n” meshes (independent loops).
  2. Assign mesh current variables: $I1, I2,
     I_n$.
  3. Direction of currents should be set consistently.

Formulation in Analysis

• Apply KVL to each mesh and form equations similarly as in nodal analysis.
• Emphasize relationship between voltage drops and mesh currents specifically pointing out that mesh currents differ from branch currents needing reconciling of values shared between meshes.

Example 5: Current I3 Analysis

• Define currents within meshes considering the dependency on KVL equations.

• Final equations to identify currents as required in analysis. Formulate with right signs and constants as per previous node definitions.

Summary of Homework Topics

• Review applications covered - Nodal Analysis and Mesh Analysis including defined principles of operation and example resolutions directly applied in practice problems
• Use MATLAB for simultaneous equations if noted—as demonstrated in example completion mentions.
  • Example MATLAB code provided for solving voltages in defined mesh analysis.
• Pictorial representations of circuits should be utilized in homework assignments as shown in conducted sessions.