Parallel and Series Resistors: Analysis and Equivalent Circuits, Slides of Electrical Circuit Analysis

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Series/Parallel,

Dividers,

Nodes & Meshes

Series Parallel

• Up To Now We Have Studied Circuits That Can Be

Analyzed With One Application Of KVL Or KCL

• We will see That In Some Situations It Is

Advantageous To Combine Resistors To Simplify

The Analysis Of A Circuit

• Now We Examine Some More Complex Circuits

Where We Can Simplify The Analysis Using

Techniques:

• Combining Resistors
• Ohm’s Law

Conductance Equivalents

• ReCall: G = 1/ R
• For SERIES

Connection

• For PARALLEL

Connection

S N

S N

G G G G G

R R R R R

1 1 1 1 1

1 2 3

1 2 3

= + + + +

⇓

= + + + +





P N

P N

G G G G G

R R R R R

= + + + +

⇓

= + + + +





1 2 3

1 2 3

1 1 1 1 1

G (^) S = 1.479 S

G (^) P = 15 S

Combine

Resistors

Example:

Find R

AB

6k||3k = 2k

(10K,2K)SERIES

SERIES

3 k

6 k || 12 k = 4 k

12 k

5 k Ω

(4K,2K)SERIES

6 k || 6 k = 3 k

(3K,9K)SERIES

4 k || 12 k = 3 k

Example w/o Redrawing

 Step-2: 12k 12k = 6k

 Step-3: 3k 6k = 2k

 Step-1: 4k↔8k = 12k

 Step-4: 6k (4k↔2k) = 3k = RAB

12 k || 12 k = 6 k

3 k || 6 k = 2 k

6 k ||( 4 k + 2 k )

12 k

Series-Parallel Resistor Circuits

• Combing Components Can Reduce The

Complexity Of A Circuit And Render It Suitable

For Analysis Using The Basic Tools Developed

So Far

• Combining Resistors In SERIES Eliminates One

NODE From The Circuit

• Combining Resistors In PARALLEL Eliminates One

LOOP From The Circuit

Example – Ladder Network

• Find All I’s & V’s in Ladder Network
  • 1

st

: S-P Reduction

12 k

4 k || 12 k

6 k

(^36) k || 6 k

2

3

6

V k I

k

V I

b

a

= Ω ×

Ω

=

I 3

• 2

nd

: S-P Reduction

• Also by Ohm’s Law

Ladder Network cont.

• Final Reduction; Find Calculation Starting

Points

V ( k )( mA ) V

mA

k k

V

I

a^31.^03

1

V k I V V

mA I I I I mA k

V

k

V I

b b

a

3 1. 5

1. 5 0. 5 6

3

6

3

2 1 2 3 3

= Ω× ⇒ =

= = + ⇒ = Ω

= Ω

=

• Now “Back Substitute” Using KVL, KCL,

and Ohm’s Law

• e.g.; From Before

Voltage Divider cont.

• Now Sub i(t) Into Ohm’s

Law to Arrive at The

Voltage Divider Eqns

( ) ( )













= 











=

1 2

2

1 2

1 1 2

and

R R

v t v R

R R

v t v R R R

v ( ) t

R

vt v R R

vt R vR R = 

  



= = 

  



= ⇒ = 2

2 2

1 0

0 and 0

(^0 )

0 0

and 0 0

0 1 1

2 1 1 2 = 

  



= = 

  



= ⇒ = R

vt vt v R

vt R vR R R





 Quick Chk → In Turn, Set R

, R

to 0

KVL ON THIS LOOP

V-Divider Summary

• Governing Equations

( )

1 2

1 1

v t R R

R v (^) R

= ( )

1 2

2 2

v t R R

R v (^) R

=

• The Larger the R , The Larger the V-drop

 Example

• Gain/Volume Control
  • R 1 is a Variable

Resistor Called a

Potentiometer, or

“Pot” for Short

Practical Example  Power Line

Also

8.25% of Pwr Generated is Lost to Line Resistance!

• How to Reduce Losses?

 Power Dissipated by the Line is a LOSS

 Using Voltage Divider

( ) 367 kV

400 kV 183.5 16.5 Ω

183.5 Ω load

=

 

  



V = ( (^2) kA) ( 183. 5 ) (^734) MW

2 load

load

2 load

= ⋅ Ω =

= ⇒

P

P I R

( 2 kA) ( 16. 5 ) 66 MW

2 line

line

2 line-LOSS src load

= ⋅ Ω =

= − = ⇒

P

P P P I R

Equivalent Circuit

• The Equivalent Circuit Concept Can Simplify

The Analysis Of Circuits

• For Example, Consider A Simple

Voltage Divider

R 1

R 2

v S

i

R 1 R 2

v i S

=

v (^) S - R 1^ + R 2

i

SERIES Resistors → 1

R R 2

R 1 + R 2

• As Far As The

Current Is

Concerned Both

Circuits Are

Equivalent

 The One On The Right Has Only One Resistor

COMPONENT SIDE

CONNECTOR SIDE

ILLUSTRATING THE DIFFERENCE BETWEEN PHYSICAL LAYOUT AND ELECTRICAL CONNECTIONS

PHYSICAL NODE

PHYSICAL NODE

SECTION OF 14.4 KB VOICE/DATA MODEM

CORRESPONDING POINTS

Generalization  Multiple v-Sources

• Voltage Sources In Series

Can Be Algebraically

Added To Form An

Equivalent Source

• We Select The Reference

Direction To Move Along

The Path

i(t)

R 1

R 2

• vR 1 −

−

vR 2

−

v 1

• v 2 −

−

v 3

− v 4 +

− v 5

0 1 2 3 2 4 5 1

v + v − v + v + v + v − v = R R

• Voltage Rises Are Subtracted From Drops

 Apply KVL