Superposition

1
Superposition
2
Superposition

• 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.

3
The Summing Circuit
4
Superposition
1kW
1kW


Vout
1kW
-
5
More Superposition

• Vout V1/3
• Vout V2/3
• Vout Vout Vout V1/3 V2/3

6
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).

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Problem 4.53
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2mA Source
2kW
2kW
1kW
2mA
I0
9
4mA Source
4mA
2kW
2kW
1kW
I0
10
12V Source
12V
2kW

-
2kW
1kW
I0
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Results

• I0 -4/3 mA
• I0 0
• I0 -4 mA
• I0 -16/3 mA

12
Superposition in Frequency Domain Analysis

• Periodic signals can be represented using Fourier
  series.
• Each term in the Fourier series can be expressed
  as a cosine plus a sine, each with a given
  magnitude and phase.
• Superposition allows us to consider the response
  of a circuit to an input as the sum of responses
  to its Fourier series terms.

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Example

• A square wave signal (for example, a clock
  signal) in a metal interconnect on an integrated
  circuit.

vs(t)
3.3V
2.5ns
-2.5ns
14
Fourier Series
1.65V
15
Superposition





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Solve Each Circuit
v0(t) 1.65V
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Solve Each Circuit

100W

V0
2.1V?0?
-j40W
-
-
V0 0.78V? -68.2?
v0(t) 0.78Vcos(2.5?109t-68.2?)
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Solve Each Circuit

100W

V0
0.7V?0?
-j13.3W
-
-
V0 0.09V? 97.6?
v0(t) 0.09Vcos(7.5?109t97.6?)
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Solve Each Circuit

100W

V0
0.4V?0?
-j8W
-
-
V0 3.2?10-3V? -85.4?
v0(t) 3.2?10-3V cos(12.5?109t- 85.4 ?)
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Sum of Results