Understanding Linear Systems and Thevenin Theorem in Circuit Analysis

1. Lecture 10 Signals and systems Linear systems and superposition Thévenin and Norton’s Theorems Related educational modules: Sections 1.7.0 through 1.7.4

2. Review: System representation of circuits • In lecture 1, we claimed that it is often convenient to use a systems-level analysis: • We can define inputs and outputs for a circuit and represent the circuit as a system • The inputs and outputs are, in general, functions of time called signals

3. What’s the difference? • Previously, our circuit analysis has been for a specific input value • Example: Determine the current i

4. System-level approach • Let the voltage source be the “input” and the current the “output” • Represent as system: • Output can be determined for any value of Vin

5. Linear Systems • In lecture 1, we noted that linear systems had linear relations between dependent variables • A more rigorous definition:

6. Linear system example • Dependent sources are readily analyzed as linear systems:

7. System representation of circuit – example 1 • Determine the input-output relation for the circuit (Vin is the input, VX is the output)

8. Example 1 – continued • Determine VX if Vin is: (a) 14V (b) 5cos(3t) – 12e-2t

9. Superposition • Special case of linear circuit response: • If a linear circuit has multiple inputs (sources), we can determine the response to each input individually and sum the responses

10. Superposition – continued • Application of superposition to circuit analysis: • Determine the output response to each source • Kill all other sources (short voltage sources, open-circuit current sources) • Analyze resulting circuit to determine response to the one remaining source • Repeat for each source • Sum contributions from all sources

11. Superposition – example 1 • Determine the current i in the circuit below

12. Two-terminal networks • It is sometimes convenient to represent our circuits as two-terminal networks • Allows us to isolate different portions of the circuit • These portions can then be analyzed or designed somewhat independently • Consistent with our systems-level view of circuit analysis • The two-terminal networks characterized by the voltage-current relationship across the terminals • Voltage/current are the input/output of the system

13. Two-terminal networks – examples • Resistor: • System representations: • Voltage-current relation:

14. Two-terminal network examples – continued • Resistive network: • Resistor + Source:

15. Thévenin and Norton’s Theorems • General idea: • We want to replace a complicated circuit with a simple one, such that the load cannot tell the difference • Becomes easier to perform & evaluate load design

16. Thévenin and Norton’s Theorems – continued • We will replace circuit “A” of the previous slide with a simple circuit with the same voltage-current characteristics • Requirements: • Circuit A is linear • Circuit A has no dependent sources controlled by circuit B • Circuit B has no dependent sources controlled by circuit A

17. Thévenin’s Theorem • Thévenin’s Theorem replaces the linear circuit with a voltage source in series with a resistance • Procedure:

18. Thévenin’s Theorem – continued • Notes: • This is a general voltage-current relation for a linear, two-terminal network • Voc is the terminal voltage if i = 0 • (the open-circuit voltage) • RTH is the equivalent resistance seen at the terminals (the Thévenin resistance)

19. Creating the Thévenin equivalent circuit • Identify and isolate the circuit and terminals for which the Thévenin equivalent circuit is desired • Kill the independent sources in circuit and determine the equivalent resistance RTH of the circuit • Re-activate the sources and determine the open-circuit voltage VOC across the circuit terminals • Place the Thévenin equivalent circuit into the original overall circuit and perform the desired analysis

20. Thévenin’s Theorem – example 1 • Replace everything except the 1A source with its Thévenin equivalent and use the result to find v1

21. Example 1 – continued