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Lecture 10
•Signals and systems
•Linear systems and superposition
•Thévenin and Norton’s Theorems
•Related educational materials:
–Chapter 4.1 - 4.5
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:
ax1(t)
+
S
bx2(t)
+
Linear
System
ay1(t) + by2(t)
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)
1W
2W
i1
3Vx
3W
4W
Vin +
-
i2
+
VX
-
5W
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:
• Voltage-current
relation:
• System representations:
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, twoterminal 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
1. Identify and isolate the circuit and terminals for which
the Thévenin equivalent circuit is desired
2. Kill the independent sources in circuit and determine
the equivalent resistance RTH of the circuit
3. Re-activate the sources and determine the open-circuit
voltage VOC across the circuit terminals
4. 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