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

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

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

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

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

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

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

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

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

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

Superposition – example 1 Determine the current i in the circuit below

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

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

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

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

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  

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

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)

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

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

Example 1 – continued