Electronic Devices and Circuit - Superposition Theorem, Study notes of Analysis and Design of Digital Integrated Circuits

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Superposition Theorem

Parallel Resistors and the Current

Divider Rule



Definition: Two or more circuit elements are said

to be in parallel if the identical voltage appears

across each of the elements.



Figure illustrates the notion of parallel resistors

connected to an ideal current source. Kirchhoff’s

current law requires that the sum of the currents

into, say, the top node of the circuit be zero:

Parallel Resistors and the Current

Divider Rule

 But by virtue of Ohm’s law we may express each current

as follows:

 (^) Since, by definition, the same voltage, v , appears across

each element. Kirchhoff’s current law may then be restated

as follows:

Or

Where

Divider Rule Parallel Resistors and

the Current

 As illustrated in Figure, one can generalize this result to an

arbitrary number of resistors connected in parallel by

stating that N resistors in parallel act as a single equivalent

resistance, R EQ, given by the expression

Or

Divider Rule Parallel Resistors and

the Current

 One can easily see that the current in a parallel circuit

divides in inverse proportion to the resistances of the

individual parallel elements. The general expression for

 (^) The current divider for a circuit with N parallel resistors is

the following:

Example: Current Divider

 (^) Determine the current i 1 in the circuit of Figure

 (^) Given Data: R

1 =^10 ;^ R 2 =^2 ^ ;^ R 3 =^20 ^ ;^ IS =^4

A.

 (^) Application of the current divider rule yields:

Linearity and

Superposition

Linearity

Consider a system (which may consist of a single network

element) represented by a block, as shown in figure &

observe that the system has an input designated by e (for

excitation) and an output designated by r (for response).

The system is considered to be linear if it satisfies the

homogeneity and superposition conditions.

System

A simple system

e r

Linearity and

Superposition

 (^) The homogeneity condition : If an arbitrary input to

the system, e , causes a response, r , then if ce is the

input, the output is cr where c is some arbitrary

constant.

 (^) The superposition condition: If the input to the

system, e 1, causes a response, r 1, and if an input to

the system, e 2, causes a response, r 2, then a

response, r 1 + r 2, will occur when the input is e 1 +

e 2.

 (^) If neither the homogeneity condition nor the

superposition condition is satisfied, the system is said

to be nonlinear****.

The Superposition

Theorem

 The superposition theorem specifies that,

in a linear circuit containing several

independent sources,the current or voltage

of a circuit element equals the algebraic

sum of the component voltages or currents

produced by the independent sources

acting alone.

A turned-off voltage source = a short circuit

A turned-off current source = an open circuit

The Superposition

Theorem



An elementary illustration of the

concept may easily be obtained by

simply considering a circuit with

two sources connected in series, as

shown in figure

The net current through R is the sum of the

individual source currents: i = i B 1 + i B 2

The Superposition

Theorem



The principle of superposition can easily be

applied to circuits containing multiple

sources and is sometimes an effective

solution technique.

 (^) In order to set a voltage source equal to zero, we

replace it with a short circuit.

Zeroing voltage source

The Superposition

Theorem

In order to set a current source equal to zero, we r

with an open circuit.

Zeroing current source

Example 1: Superposition

Theorem

 Using the superposition theorem we first of all

replace VS2 with a short circuit, giving the circuit to

the right. The current in each branch is calculated

using the basic laws of KCL, KVL and Ohm’s Law.

3k 

3k  6k 

Load

S

Total

S

R R R

V

R

V

I

1 3

1 1 1

I 2. 4 mA

5 10

12

(^1 ) 





Example 1: Superposition

Theorem

3

1

3

R R

R

I I

Load

Load 

 

I mA Load

I I I mA Load

2. 4 1. 6 0. 8 3 1

    

3k 

3k  6k 