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Understanding Circuit Reduction: Series and Parallel Elements

Explore how to reduce circuit elements in series and parallel to simplify circuits and solve related problems efficiently. Learn techniques through examples and understand the concepts of voltage division and current division in a clear and concise manner.

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Understanding Circuit Reduction: Series and Parallel Elements

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  1. Lecture 5 Review: Series, parallel circuit elements Circuit reduction Related educational materials: Chapter 2.1, 2.2, 2.3

  2. Review: series and parallel circuit elements • Elements in series if they have the same current • Elements in parallel have the same voltage

  3. Circuit reduction • Some circuit problems can be simplified by combining elements to reduce the number of elements • Reducing the number of elements reduces the number of unknowns and thus the number of equations which must be written to determine these unknowns

  4. Series circuit elements – example 1 • Apply KCL at any node  all elements have the same current • All of the above circuit elements are in series

  5. Series element circuit reduction – example 1 • KVL around the loop: -V1 + i·R1 + V2 + i·R2 + i·R3 – V3 + i·R4 = 0 (-V1 + V2– V3) + i(R1 + R2 + R3 + R4) = 0

  6. Series circuit reduction • Notes: • Voltage sources in series add directly to form an equivalent voltage source • Resistances in series add directly to form an equivalent resistance

  7. Series circuit reduction – Example 2 • Determine the power delivered by the 20V source

  8. Voltage Division • Series combination of N resistors:

  9. Voltage Divider Formula • Ratio of VK to the total voltage is the same as the ratio of RK to the total series resistance

  10. Voltage Dividers – special case • Voltage source in series with two resistors:

  11. Voltage division – example 1 • Determine the power dissipated by the 2 resistor

  12. Voltage division – example 2 • Determine the voltage V1 in the circuit below.

  13. Parallel circuit elements – example 1 • Apply KVL around any loop  all elements have the same voltage • All of the above circuit elements are in parallel

  14. Parallel element circuit reduction – example 1 • KCL at upper node:

  15. Parallel circuit reduction • Notes: • Current sources in parallel add directly to form an equivalent current source • Definition: Conductance is the inverse of resistance • Units are siemens or mhos (abbreviated S or ) • Conductances in parallel add directly to form an equivalent conductance

  16. Go back to previous example, look at it in terms of conductances

  17. Parallel element circuit example 1 – revisted

  18. Parallel circuit reduction – Example 2 • Determine the power delivered by the 2A source

  19. Current Division • Parallel combination of N resistors:

  20. Current Divider Formula • Ratio of iK to the total current is the same as the ratio of GK to the total parallel conductance

  21. Current Divider – special case • Current source in parallel with two resistors

  22. Current division – example 1 • Determine the current in the 2 resistor

  23. Current division – example 2 • Determine the value of R which makes i = 2mA

  24. Circuit Reduction • Series and parallel combinations of circuit elements can be combined into a “equivalent” elements • The resulting simplified circuit can often be analyzed more easily than the original circuit

  25. Circuit Reduction – example 1 • Determine the current in the 2 resistor. (Note: we wrote the governing equations for this example in lecture 3.)

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