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Single Loop Circuits * with a current source * with a voltage source * with multiple sources * voltage divider circu

Lecture 3. Single Loop Circuits & Superposition Method. Single Loop Circuits * with a current source * with a voltage source * with multiple sources * voltage divider circuits * Equivalent resistance Superposition method * Principle

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Single Loop Circuits * with a current source * with a voltage source * with multiple sources * voltage divider circu

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  1. Lecture 3. Single Loop Circuits & Superposition Method Single Loop Circuits * with a current source * with a voltage source * with multiple sources * voltage divider circuits * Equivalent resistance Superposition method * Principle * Procedures * How to apply

  2. I R1 R2 + – VS Rn Single Loop Circuits • The same current flows through each element of the circuit—the elements are in series. V1 V1 I R1 V2 V2 R2 IS V3 V3 Rn With an independent voltage source With an independent current source

  3. Single Loop Circuits – with a Current Source • What is I? V1 I • In terms of I, what is the voltage across each resistor? R1 V2 R2 IS V3 Rn …

  4. I R1 I + – + R1 R2 I R2 – + – VS + Rn I Rn – Single Loop Circuits – with a Voltage Source • In terms of I, what is the voltage across each resistor? • To solve for I, apply KVL around the loop. … IR1 + IR2 + … + IRn– VS = 0

  5. With Multiple Voltage Sources • The current i(t) is: • Resistors in series

  6. + + – – Voltage Division • Consider two resistors in series with a voltage v(t) across them: R1 v1(t) – v(t) + R2 v2(t) • If n resistors in series:

  7. Voltage Divider: A Practical Example Electrochemical Fabrication of Molecular Junction QuantumPoint Contact or Atomic-scale wire

  8. E - - + + Voltage Divider: An Example • Anode:Etching delocalized, but • Cathode:Deposition localized at sharpest point, • due to: • Self-focusing – directional growth • Decreasing Gap!

  9. Initially, Rgap >> Rext, Vgap ~ V0full speed deposition. • Finally, Rgap << Rext, Vgap ~ 0deposition terminates. Voltage Divider: An Example • The gap resistance is determined by Rext.

  10. 3 1 • Growth starts • after applying 1.5 V • Self-terminates after • forming a tunneling gap • Two electrodes with • 10 mm initial separation 2 Voltage Divider: An Example

  11. Stepwise increase in Conductance Ohmic behavior G (2e2/h) Time (sec.) Voltage Divider: An Example

  12. + I1 I2 I R1 R2 V – Example: Two Resistors in Parallel How do you find I1 and I2?

  13. + I1 I2 I R1 R2 V – Example: Two Resistors in Parallel • Apply KCL with Ohm’s Law

  14. Equivalent Resistance of Parallel Resistors • Two parallel resistors is often equivalent to a single resistor with resistance value of: • n-Resistors in parallel:

  15. What are I1 and I2 ? • This is the current divider formula • It tells us how to divide the current through parallel resistors

  16. + I1 I2 Is1 Is2 R1 R2 V – Circuits with More Than One Source How do we find I1 or I2?

  17. + I1 I2 Is1 Is2 R1 R2 V – What if More Than One Source? • Apply KCL at the Top Node

  18. Class Examples • Example: P1-33 (page 43). • Drill Problem P1-34 (page 43).

  19. Superposition Method – A More General Approach to Multiple Sources “In any linear circuit containing multiple independent sources, the current or voltage at any point in the circuit may be calculated as the algebraic sum of the individual contributions of each source acting alone.”

  20. How to Apply Superposition • To find the contribution due to an individual independent source, zero out the other independent sources in the circuit • Voltage source  short circuit • Current source  open circuit • Solve the resulting circuit using your favorite technique(s)

  21. 1kW 1kW 1kW 1kW 1kW 1kW + + + + – + – V1 + – + + – V2 V’out 1kW 1kW V’’out V1 Vout 1kW V2 – – – Superposition of Summing Circuit

  22. 1kW 1kW 1kW 1kW + + + – + – V1 + V2 V’out 1kW 1kW V’’out – – Superposition of Summing Circuit (cont’d) V’out = V1/3 V’’out = V2/3 Vout = V’out + V’’out = V1/3 + V2/3

  23. Superposition Procedure • For each independent voltage and current source (repeat the following): • Replace the other independent voltage sources with a short circuit (i.e., V = 0). • Replace the other independent current sources with an open circuit (i.e., I = 0). Note: Dependent sources are not changed! • Calculate the contribution of this particular voltage or current source to the desired output parameter. 2. Algebraically sum the individual contributions (current and/or voltage) from each independent source.

  24. Class Examples • Example 2-9 (page 70). • Drill Problem 2.7.

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