Download Series Resistors

Objective of Lecture
 Explain mathematically how resistors in series are
combined and their equivalent resistance.
 Chapter 2.5
 Explain mathematically how resistors in parallel are
combined and their equivalent resistance.
 Chapter 2.6
 Rewrite the equations for conductances.
Series Resistors
Series Resistors (con’t)
 Use KVL
0  Vin  V1  V2
Series Resistors (con’t)
 Use KVL
0  Vin  V1  V2
 Use Ohm’s Law
V1  IR1
V2  IR2
Series Resistors (con’t)
 Use KVL
0  Vin  V1  V2
 Use Ohm’s Law
V1  IR1
V2  IR2
 Substitute into KVL equation
0  Vin  IR1  IR2
Vin  IR1  IR2  I ( R1  R2 )
Equivalent Resistance:
Series Connections
Req is equal to the sum of
the resistors in series.
In this case:
Req = R1 + R2
General Equations: Series
Resistors
 If S resistors are in series, then
S

Vin  I  Rs 
 s 1 
S
Req   Rs
s 1
where Vin may be the applied
voltage or the total voltage
dropped across all of the
resistors in series.
Parallel Resistors
Parallel Resistor (con’t)
 Use KCL
0  I in  I1  I 2
Parallel Resistor (con’t)
 Use KCL
0  I in  I1  I 2
 Use Ohm’s Law
VR  I1 R1
VR  I 2 R2
Parallel Resistor (con’t)
 Use KCL
0  I in  I1  I 2
 Use Ohm’s Law
VR  I1 R1
VR  I 2 R2
 Substitute into KCL
equation
0  I in  VR R1   VR R2 
I in  VR 1 R1   1 R2 
I in  VR R1 R2 / R1  R2 
Equivalent Resistance:
Parallel Connections
1/Req is equal to the sum
of the inverse of each of
the resistors in parallel.
In this case:
1/Req = 1/R1 + 1/R2
Simplifying
(only for 2 resistors in parallel)
Req = R1R2 /(R1 + R2)
General Equations:
Parallel Resistors
 If P resistors are in parallel, then
 P VR 
I in   
 p 1 R p 
P 1 
Req   
 p 1 R p 
where Iin may be the total
current flowing into and out of
the nodes shared by the
parallel resistors.
1
If you used G instead of R
 In series:
The reciprocal of the
equivalent conductance is
equal to the sum of the
reciprocal of each of the
conductors in series
In this example
1/Geq = 1/G1 + 1/G2
Simplifying
(only for 2 conductors in series)
Geq = G1G2 /(G1 + G2)
If you used G instead of R
 In parallel:
The equivalent
conductance is equal to the
sum of all of the
conductors in parallel
In this example:
Geq = G1 + G2
Electronic Response:
For the same value resistors
As you increase the number of resistors in series
Does Req increases or decreases?
As you increase the number of resistors in parallel
Does Req increases or decreases?
Summary
 The equivalent resistance and conductance of resistors
in series are:

1
Geq   
 s 1 Gs 
S
S
Req   Rs
s 1
1
where S is the total number of resistors in series.
 The equivalent resistance and conductance of resistors
in parallel are:
P 1 
Req   
 p 1 R p 
-1
P
Geq   G p
where P is the total number of resistors in parallel.
p 1