series and parallel resistors

Objective of LectureExplain 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

210 VVVin

Series Resistors (con’t)Use KVL

Use Ohm’s Law

22

11

IRV

IRV

210 VVVin

Series Resistors (con’t)Use KVL

Use Ohm’s Law

Substitute into KVL equation

)(

0

2121

21

RRIIRIRV

IRIRV

in

in

22

11

IRV

IRV

210 VVVin

Equivalent Resistance: Series Connections

Req is equal to the sum of the resistors in series.

In this case: Req = R1 + R2

General Equations: Series ResistorsIf S resistors are in series, then

where Vin may be the applied

voltage or the total voltage dropped across all of the resistors in series.

S

sseq

S

ssin

RR

RIV

1

1

Parallel Resistors

Parallel Resistor (con’t)Use KCL

210 III in

Parallel Resistor (con’t)Use KCL

Use Ohm’s Law210 III in

22

11

RIV

RIV

R

R

Parallel Resistor (con’t)Use KCL

Use Ohm’s Law

Substitute into KCL equation

210 III in

22

11

RIV

RIV

R

R

2121

21

21

/

11

0

RRRRVI

RRVI

RVRVI

Rin

Rin

RRin

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 ResistorsIf P resistors are in parallel, then

where Iin may be the total current flowing into and out of the nodes shared by the parallel resistors. 1

1

1

1

P

p peq

P

p p

Rin

RR

R

VI

If you used G instead of RIn 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 RIn 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?

SummaryThe equivalent resistance and conductance of

resistors in series are:

where S is the total number of resistors in series.

The equivalent resistance and conductance of resistors in parallel are:

where P is the total number of resistors in parallel.

P

ppeq

P

p peq GG

RR

1

-1

1

1

1

11

1

S

s seq

S

sseq G

GRR