Download Resistors in Parallel

Resistors in Parallel
• Resistors connected at a single node pair
• Voltage across each resistor is the same
EGR 101 Introduction to Engineering I
1
Caution on Parallel Connection
• Here, R1 and R3 are not in parallel
• Resistor R1 is in parallel with the series
combination of R2 and R3
EGR 101 Introduction to Engineering I
2
Apply KCL and Ohm’s Law
i1
i2
i3
i4
iS  i1  i2  i3  i4
vS  i1R1  i2 R2  i3 R3  i4 R4
EGR 101 Introduction to Engineering I
3
Solving for each current
vS
i1 
R1
vS
i2 
R2
vS
i3 
R3
vS
i4 
R4
EGR 101 Introduction to Engineering I
4
Substitute back
iS  i1  i2  i3  i4
1 1
vS vS vS vS
1
1 
iS  
 
 vS  
  
R1 R2 R3 R4
 R1 R2 R3 R4 
iS
1
1 1
1
1

 
 
vS Req R1 R2 R3 R4
EGR 101 Introduction to Engineering I
5
Equivalent Circuit
k
1
1
1 1
1
     ..... 
Req i 1 Ri R1 R2
Rk
k
Geq   Gi  G1  G2  .....  Gk
i 1
EGR 101 Introduction to Engineering I
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Special case for only 2 resistors
1
1
1 R2  R1
 

Req R1 R2
R1 R2
R1 R2
Req 
R1  R2
Product
Sum
EGR 101 Introduction to Engineering I
7
Summary
• Resistors in parallel have the same
voltage
• The resistors can be replaced by an
“equivalent” resistance whose
conductance is the sum of the individual
conductances
• The “equivalent” resistor is smaller than
the smallest of the individual resistors
EGR 101 Introduction to Engineering I
8
Classroom Problem #2
Determine the equivalent resistance seen by the current
source.
R2
3k
R4
5k
R3
10k
R5
7k
R1
4k
I
3mA
EGR 101 Introduction to Engineering I
9