Liner Resistive networks for electrical engineers

Chapter Three Linear Resistive Networks
3.0 Introduction
3.1 Resistance
3.2 Basic Network Configurations
3.3 Superposition
3.4 Equivalent Circuits
3.5 Sources and Loads
3.6 Consistent Unit
3.7 Electrical Measurements
Contents of this Chapter:

• 3.0 Introduction
In this chapter we are going to learn basic methods for
circuit analysis.
1. Resistive Elements
Element Characteristic
A kind of relation between the voltage at the terminals and the
current through the element.
Resistive Characteristic:
The relation between the voltage at the terminals and the current
through the element is algebraic and can be presented by a curve in
v-i plane. So the resistive characteristic is often called v-i
characteristic. For example

• Resistive Element:
All elements, which have resistive characteristic, are
called resistive elements.

1. Linearity
Linearity contains two properties, additivity and multiplicity
•Additivity
For a function of x, f (x), suppose that when x=x1, the function
takes on the specific value f (x1), and when x=x2, the function takes
on a different value f (x2), then if x=x1+x2, the function takes on the
value
•Multiplicity
If x=x1, the function takes on the value f (x1), then when x= a×x1
(a is a constant), the function takes on the value
•Linear function:
It can be presented by a line going through the origin.
1 2 1 2
( ) ( ) ( )
f x x f x f x
  
1 1
( ) ( )
f a x a f x
  

3.1 Resistance
1. The Ideal Resistor
An element with constant positive slope v-i characteristic is called
ideal resistor. Its circuit symbol and characteristic are shown as
The left figure is accepted in our country. Ideal resistor is
labeled R.

2. Ohm's Law
The relation of the voltage between terminals of a resistor and the
current through the resistor is called Ohm's Law (relative reference
polarity)
or
v i R i v G
   
Where, R is the resistance of the resistor (its unit is ohm, W). G is
the conductance of the resistor (its unit is siemens, S). Obviously,
G=(1/R) or R=(1/G).
_
+ v
i
R

3. The linearity of resistor
Two voltage sources are connected in series with an ideal resistor, R.
According to KVL for the loop in Figure (a), we have
1 2 1 2
1 2 0
R R
v v v v
v v i R or i
R R R

      
In Figure (b), the two voltage sources connected separately to an
ideal resistor R. According to KVL, the current iR1 and iR2 are
given by
1 2
1 2
R R
v v
i i
R R
 
Comparing above two equations, 1 2
R R R
i i i
 
It is to say that the ideal resistor is linear. Therefore, the
ideal resistor is also called linear resistor.

4. Power in an Ideal Resistor
When we adopt the relative reference polarity, as the figure,
_ +
v
i
R
the power of an ideal resistor can
calculate as follows
2
2
( )
( )
R
p v i i R i i R
v v
v i v
R R
      
    
If non-relative reference polarity is used,
_
+ v
i
R
2
2
( )
( )
R
p v i i R i i R
v v
v i v
R R
         
      

5. Actual Resistors
The actual electrical resistors, which commonly made
from pressed carbon powder, metal films, or fine wires,
are manufactured to have specific value of resistance.
They behave very nearly like the ideal resistors in a
wide range. Therefore, we shall use the term "resistor"
to refer both to the actual and to the ideal resistors.

3.2 Basic Network Configurations
1. Simple Applications of Kirchhoff's Laws
Now we build networks out of ideal sources and resistors. The
simplest connections involve one source and one resistor, and are
shown as follows.
The left figure shows a voltage source connected to a resistor. By the
definition of the voltage source, the voltage across the resistor must
equal v0. By Ohm's law, therefore, 0
v
v
i
R R
 
The right figure shows the corresponding network involving a
current source. By definition of the current source, the current is
fixed at i = i0. Ohm's law then yields
0
v i R i R
   

2. Resistors in Series
• Concept of Equivalent Circuit:
If two sub-circuits or circuit elements have the same behaviors in
any network, they are equivalent. As in the following networks, N1
and N2 are connected to an arbitrary networks N respectively. If the
working conditions of N in two networks are the same, then we can
say that network N1 and N2 are equivalent each other.

In other words, if two sub-circuits or circuit elements have the
same v-i characteristic, they are equivalent.

• Equivalent for Resistors in Series :
When two resistors are connected in series as follows:
They form a sub-circuit. Because they are in
series, the same current i flows through
them.
Ohm's Law:
KVL:
1 1 2 2
1 2
v i R and v i R
v v v
   
 
Combine above two equations and yield the v-i
characteristic of this sub-circuit
1 2
( )
v R R i R i
    
Therefore, two resistors R1 and R2 in series
are equivalent to one resistor R=R1+R2

More generally, N resistors R1, R2, …., RN are connected in series to
form a sub-circuit, this sub-circuit is equivalent to a single resistor R,
which is called equivalent resistor:

• Voltage divider relation
Recall the Ohm's law in the equivalent circuit to determine the
current
Apply the current to the resistors in series and recall Ohm's
law again
Therefore we yield the voltage divider relation of resistors in series
1 2 N
v v
i
R R R R
 
  
1
1 1
1 2
;
N
R
v i R v
R R R
   
  
1 2
1,2,
k k
k
N
R R
v v v for k N
R R R R
    
  

3. Resistors in Parallel
• Equivalent for Resistors in Parallel
When two resistors are connected in parallel as follows
they form a sub-circuit.
Because the two resistors are
in parallel, they have the same
voltage cross v.
Ohm's Law:
1 1
1
2 2
2
1 2
:
v
i G v
R
v
i G v
R
KCL i i i
  
  
  1 2
1 2
1 1
( ) ( )
i v G G v
R R
     

Comparing the Ohm's law, it is found that two resistors R1 and
R2 in parallel are equivalent to one resistor
1 2
1
1 2
1 2 2
1 1 1
,
R R
or G G G
R R R
R
R
R


  


More generally, N resistors R1, R2, …., RN are connected in
parallel to form a sub-circuit, this sub-circuit is equivalent to a
single resistor R, which is called equivalent resistor.

• Current divider relation
Recall the Ohm's law in the equivalent circuit to determine the
voltage across
1 2
1 1
N
v i R i i
G G G G
     
  
Apply the current to the resistors in parallel and recall Ohm's law
again
1
1 1
1 1 2
,
N
v G
i G v i
R G G G
    
  
Therefore we yield the current divider relation of resistors in
parallel
1 2
for 1, 2,
k
k k
k N
G
v
i G v i k N
R G G G
     
  
Especially, when N=2, we have
2 1
1 2
1 2 1 2
and
R R
i i i i
R R R R
   
 

4. A Network Example
0 2
3 1 3
3 1 2 1 2
6
0.3 0.2
|| 20
v V R
i A i i A
R R R R R
     
 W 

3.3 Superposition迭加定理
An Example
KCL: 1 0 2 0
i i i
  
0 1 2
v v v
 
KVL:
Ohm’s Law:
1 1 1 2 2 2
v i R v i R
   
Combine these three basic relations
0 1 1 2 2 1 1 0 1 2
( )
v i R i R i R i i R
        
0 0
2 1
1 0 2 0
1 2 1 2 1 2 1 2
v v
R R
i i i i
R R R R R R R R
   
   

Each current is the sum of two terms, and each term is exactly
proportional to one and only one independent source. This result
illustrates an extension of the concept of linearity to cases with
more than one independent source.
If i0 happened to be zero, i1 and i2 would be exactly proportional to
v0 (linearity), and if v0 were zero, i1 and i2 would be exactly
proportional to i0 (linearity).
We say then that the solutions for i1 and i2 are superposition of
linear responses, one response from each independent source.
If the network had three independent sources, each response would
consist of three terms, one term proportional to each independent
source. This suggests a way of breaking up problems containing
more than one independent source into several smaller problems.
This method is called superposition.
0 0
2 1
1 0 2 0
1 2 1 2 1 2 1 2
v v
R R
i i i i
R R R R R R R R
   
   
Determine the response to each
independent source, one at a time,
assuming that all other independent
sources are zero. Sum the results
to get the total response.

• Suppression of Independent Sources
To use the superposition method, we must know what happens to
a network when one sets an independent source to zero.
Suppressing an independent voltage source: Let the terminal
voltage of the voltage source be zero! In other words, substitute
a short circuit for the voltage source.
Suppressing an independent current source: Let the current
through the current source be zero! In other words, substitute a open
circuit for the current source.
Cautionary Note: The superposition can be used only in linear
networks.

Combining the individual response to find the total response
1 2
2 1
3 31 32
2 3 1 2 3 1 3 2 1 3
|| ||
s s
v v
R R
i i i
R R R R R R R R R R
     
   
1
2
31
2 3 1 2 3
||
s
v
R
i
R R R R R
 
 
11
i
31
i
21
i
11
v
31
v
21
v
11
i
31
i
21
i
11
v
31
v
21
v
2
1
32
1 3 2 1 3
||
s
v
R
i
R R R R R
 
 
32
i
12
i 22
i
12
v
32
v
22
v
Example: Find the current flowing through resistor R3 using
superposition.

3.4 Equivalent Circuits
Equivalent circuit: If two sub-circuits or circuit elements have the
same behaviors in any network, they are equivalent. In other words,
if two sub-circuits or circuit elements have the same v-i
characteristic, they are equivalent.
1. Terminal Characteristics
The relation between the terminal voltage of a circuit component
and current through it is called its terminal characteristic. For a
linear two-terminal element or sub-circuit the terminal
characteristic must be linear and can be represented by a straight
line in v-i plane. As we know, a straight line can be determined by
two parameter, intercept at v-axis or at i-axis and the slope.

The terminal characteristic of two-terminal network N can
also be represented in equation:
OC SC
v
v v i R i i
R
    
Where, vOC is the open-circuit (terminal) voltage, which is the
voltage cross N when i=0 (open circuit). iSC is the short-circuit
(terminal) current, which is the current through N when v=0
(short circuit). R is the internal resistance of N.

Recall the characteristic of ideal independent sources and resistance,
we yield two equivalents to a linear two-terminal network:
OC OC
OC SC SC
SC
v v
v i R i R
R i
   
Obviously, there are relations between the three variables:

2. The terminal characteristics of circuit elements
Resistor
0; 0
OC SC
v
v i R i v i
R
     

Ideal voltage source
0 0; 0
OC
v v v v R
   

Ideal current source
0 0;
SC
i i i i R
     

3. Thévenin and Norton Equivalents 戴维南和诺顿等效
Linear resistive network: If a network consists of linear resistive
elements, e.g. resistors, ideal voltage sources and ideal current
sources, it is called linear resistive network.
Consider a two-terminal linear resistive network N, it has a linear
characteristic. Now we construct a circuit or network with N and
a ideal voltage source as follows

According to the superposition, the response i must be
 
T k sk m sm
i G v i g v

     
 
where G, ak and gm are constants determined from the simple
superposition networks with all but one independent source
suppressed. When v = 0 (short-circuit condition), we know the
terminal current i must equal -iSC, which depends only on the
elements inside the network N. Therefore, the above equation can
be converted to
T SC
i G v i
  
GT must have the dimension of a conductance, GT is equivalent to
l/RT, where RT is the effective, or equivalent resistance for the
network. Furthermore, the above equation may be rewritten as
where
T OC OC T SC
v R i v v R i
    

According to our above discussion, an arbitrary linear resistive
network (consists of resistors, ideal voltage sources and current
sources) can be replaced by a simple equivalent network, Thévenin
equivalent network or Norton equivalent network.
It is important to note the relation between reference direction of
equivalent current source in Norton equivalent network and reference
polarity of equivalent voltage source in Thévenin equivalent network.

4. To determine the Thévenin or Norton equivalent network
For any given network, there are two ways to determine Thévenin
or Norton equivalent network: two-step method and one-step
method .
Two-step method
step1 Find vOC or iSC using standard network analysis
methods.
Step2 Calculate the equivalent resistance RT
(a) suppressing all ideal sources inside the
network, and using the series and parallel
equivalent.
(b) using the formula OC
T
SC
v
R
i

One-step method
Connect a general ideal voltage source (or current) to the terminal
of the network. then determine the terminal current i (or voltage v)
using a standard network analysis methods, which is a relation
between terminal voltage and current, named v-i characteristic.
The equivalent circuit will be given by the v-i characteristic.
Both methods are illustrated below through examples

Example 1. Find the equivalent for the following
two-terminal network.
Solution 1 (two-step method):
Step 1 Determine open-circuit voltage
Recall the voltage divider relation of
resistors in series 2
1
1 2
OC
R
v v
R R


Determine short-circuit current. According
to Ohm's law 1
1
SC
v
i
R

Step 2 Calculate the equivalent resistance
1 2
1 2
OC
T
SC
v R R
R
i R R
 


Solution 2 (two-step method):
Step 1 Determine open-circuit voltage
Recall the voltage divider relation of
resistors in series
2
1
1 2
OC
R
v v
R R


Step 2 Determine the equivalent resistance
Suppress all ideal sources inside the network and recall the
equivalent for resistors in parallel, the above circuit equivalent to
one resistor (i.e. equivalent resistor)
1 2
1 2
1 2
||
T
R R
R R R
R R
 


Solution 3 (one-step method):
Connect a general voltage source to the terminal
Determine the terminal
current i by KVL, KCL
and Ohm's law
2 1 1
1 1 1 2 2 2
1 2
,
,
R R
R R R R
R R
v v v v v
v R i v R i
i i i
  
 
 
Combining the above
equations, we have
1 1
2 1 1 2 1
1 1
SC
T
v v v v v
i v i
R R R R R R
 

       
 
 
Then we yield the a Norton equivalent.

Example 2. Find the Norton or Thévenin equivalent for the
following two-terminal network.

Solution 1 (two-step method)
Step 1 Determine the open-circuit voltage Recall the superposition
1 2 2
1 2 0 0
1 2 1 2
OC OC OC
R R R
v v v i v
R R R R
   
 

Step 2 Determine the equivalent resistance
Suppress all ideal sources inside the network:
1 2
1 2
T
R R
R
R R



Finally we yield the equivalent circuit

Solution 2 (one-step method)
Connect a general current source to the terminal.
 
1 2 2 2 1 2
0 0 1 0 0
1 2 1 2 1 2 1 2
( ) OC T
R R R R R R
v i i v R i v i v R i
R R R R R R R R
       
   

Then the Thévenin equivalent circuit is

3.5 Sources and Loads
1. Voltage and Current Transfer
Source: the deliver of electrical power in network and its power<0
Load: the receiver of electrical power in network and its power>0
Generally, a source network can be represented in the Thévenin
or Norton equivalent, and a load network can be represented in a
resistance. As the following figure shows.

Voltage transfer: Relation between the voltage across the load v
and the open-circuit source voltage vs
Current transfer: Relation between the current through the load
i and the short-circuit source current is
If RL is larger than RS, the voltage transfer ratio would larger. If RL
is smaller than RS, the current transfer ratio would larger.
In summary, source network with low resistance compared with
the load behave like voltage sources; source with higher
resistance compared with the load behave like current sources.
L
S S L
v R
v R R


S
S S L
R
i
i R R



2. Power Transfer
The power dissipated in load (transferred from the source) is
called power transfer.
Obviously, if the source is given the power transfer varies with the
load. Therefore there is maximal power transfer theory.
2
2
( )
S
L L
L S S
S L S L S L
v
R R
P v i v v
R R R R R R
    
  
If the source network is fixed, with both vs and RS given, when and
only when the load resistance RL is equal to the source equivalent
resistance RS, the load received maximum power from the source.
3. Maximal power transfer theory
In mathematic language, If and only if L S
R R

The source transfer maximal power to the load,
2
1
4
S
L
S
v
P
R
 

3.6 Consistent Units
Quantity Consistent Units
Voltage V V V V V
Current A mA mA mA A
Resistance W kW kW kW MW
Conductance S mS mS mS S
Capacitance F mF nF pF mF
Inductance H H mH H -
Time s ms ms ns s
Frequency Hz kHz MHz GHz Hz

3.7 Electrical Measurements
•Ammeter: 安培计或电流表
Ammeter is an instrument for measuring the current.
An ammeter consists of a coil of wire suspended in a magnetic field.
When current is passed through the coil, the magnetic field exerts a
force on the coil that causes it to rotate. The amount of rotation
depends on the amount of current in the coil. A meter needle
attached to the coil enables the position of the coil to be observed.
Ideally, an ammeter should have zero resistance. In practice, the
coil resistance is never zero. Thus the simplest circuit model for an
ammeter consists of an ideal ammeter in series with suitably chosen
coil resistance.
In order to measure the current through a
Branch, an ammeter must be interposed in
series with the branch. In practical
measurement, the coil resistance should be
very small (generally, <1mW). Otherwise,
the result of measurement would be not
accurate.

•Voltmeter: 伏特计或电压表
Voltmeter is an instrument for measuring the voltage.
For measuring the voltage cross a Branch, a Voltmeter must be
connected in parallel with the branch. In order that the ideal
voltmeter not change any of the voltages in the network to which it
is connected, the ideal voltmeter should have an infinite resistance.
In practice, all voltmeters draw a small amount of current. The
simplest circuit model for an actual voltmeter, therefore, consists of
an ideal voltmeter in parallel with a suitably chosen shunt
resistance(漏电电阻).
In practical measurement, the shunt resistance
should be very large (generally, >1MW). Otherwise,
the result of measurement would be not accurate.

• Potentiometer 电位器
A potentiometer is a resistor constructed with a third contact whose
position of contact can be varied. This third contact is called the
wiper (滑动端W). As the wiper position is moved from terminal 1
to terminal 2, the resistance between the wiper and terminal 1
increases from zero to R, while the resistance between the wiper
and terminal 2 decreases from R to zero.
The voltmeter and ammeter discussed above are called direct-
reading instruments because the electrical quantity of interest is
converted directly into an observable meter motion. However,
many electrical measurements use indirect or comparison methods.
A network element of great value in such measurements is the
potentiometer.
As the wiper position is moved, both the Thévenin voltage and the
Thévenin resistance of the variable voltage divider are changed.

Exercises of this chapter
E3.3, E3.4, E3.5, E3.7, E3.8, E3.10

Liner Resistive networks for electrical engineers

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