Objective of LectureExplain mathematically how a voltage that is applied

to resistors in series is distributed among the resistors.Chapter 2.5 in Fundamentals of Electric CircuitsChapter 5.7 Electric Circuit Fundamentals

Explain mathematically how a current that enters the a node shared by resistors in parallel is distributed among the resistors.Chapter 2.6 in Fundamentals of Electric CircuitsChapter 6.7 in Electric Circuit Fundamentals

Work through examples include a series-parallel resistor network (Example 4).Chapter 7.2 in Fundamentals of Electric Circuits

Voltage DividersResistors in series share the same current

Vin

Voltage DividersResistors in series share the same current

From Kirchoff’s Voltage Law and Ohm’s Law :

22

11

210

IRV

IRV

VVVin

+

V1-

+V2_

Vin

Voltage DividersResistors in series share the same current

From Kirchoff’s Voltage Law and Ohm’s Law :

in

in

in

VRRRV

VRRRV

RRVV

IRV

IRV

VVV

0

2122

2111

2112

22

11

21

+

V1-

+V2_

Vin

Voltage DivisionThe voltage associated with one resistor Rn in a chain of multiple resistors in series is:

or

where Vtotal is the total of the voltages applied across the resistors.

totaleq

nn V

R

RV

totalS

ss

nn V

R

RV

1

Voltage DivisionThe percentage of the total voltage

associated with a particular resistor is equal to the percentage that that resistor contributed to the equivalent resistance, Req.The largest value resistor has the largest

voltage.

Example 1Find the V1, the voltage

across R1, and V2, the voltage across R2.

+

V1-

+V2_

Example 1Voltage across R1 is:

Voltage across R2 is:

Check: V1 + V2 should equal Vtotal

+

V1-

+V2_

tVV

tVkkkV

VRRRV

tVV

tVkkkV

VRRRV

total

total

377sin4.11

377sin20 434

377sin57.8

377sin20 433

2

2

2122

1

1

2111

8.57 sin(377t)V + 11.4 sin(377t) = 20 sin(377t) V

+V1 -

+V2 -

+V3 -

Example 2Find the voltages listed in the

circuit to the right.

Check: V1 + V2 + V3 = 1V

VV

VV

VV

VV

VV

VV

R

R

eq

eq

143.0

1 700/100

571.0

1 700/400

286.0

1 700/200

700

100400200

3

3

2

2

1

1

+V1 -

+V2 -

+V3 -

Symbol for Parallel ResistorsTo make writing equations simpler, we use a symbol to indicate that a certain set of resistors are in parallel.

Here, we would write

R1║R2║R3

to show that R1 is in parallel with R2 and R3. This also means that we should use the equation for equivalent resistance if this symbol is included in a mathematical equation.

Current DivisionAll resistors in parallel share the same voltage

+

Vin

_

Current DivisionAll resistors in parallel share the same voltage

From Kirchoff’s Current Law and Ohm’s Law :

33

22

11

3210

RIV

RIV

RIV

IIII

in

in

in

in

+

Vin

_

Current DivisionAll resistors in parallel share the same voltage

+

Vin

_

in

in

in

IRRR

RRI

IRRR

RRI

IRRR

RRI

213

213

312

312

321

321

Current DivisionAlternatively, you can reduce the number of resistors in parallel from 3 to 2 using an equivalent resistor.

If you want to solve for current I1, then find an equivalent resistor for R2 in parallel with R3.

+

Vin

_

Current Division

+

Vin

_

ineq

eqeq I

RR

RI

RR

RRRRR

11

32

3232 and where

Current DivisionThe current associated with one resistor R1 in parallel with one other resistor is:

The current associated with one resistor Rm in parallel with two or more resistors is:

totalm

eqm I

R

RI

totalI

RR

RI

21

21

where Itotal is the total of the currents entering the node shared by the resistors in parallel.

Current DivisionThe largest value resistor has the smallest

amount of current flowing through it.

Example 3Find currents I1, I2, and I3 in the circuit to the

right.

Example 3 (con’t)

Check: I1 + I2 + I3 = Iin

AI

AI

AI

AI

AI

AI

Req

727.0

4 600/109

09.1

4 400/109

18.2

4 200/109

109600140012001

3

3

2

2

1

1

1

Example 4The circuit to the

right has a series and parallel combination of resistors plus two voltage sources.Find V1 and VpFind I1, I2, and I3

+

V1

_

+

Vp

_

I1

I2 I3

Example 4 (con’t)First, calculate

the total voltage applied to the network of resistors. This is the addition

of two voltage sources in series.

+

V1

_

+

Vp

_

I1

I2 I3

+

Vtotal

_

)20sin(5.01 tVVVtotal

Example 4 (con’t)Second, calculate

the equivalent resistor that can be used to replace the parallel combination of R2 and R3.

+

V1

_

+

Vp

_

I1

+

Vtotal

_

80100400

100400

1

1

32

321

eq

eq

eq

R

R

RR

RRR

Example 4 (con’t)To calculate the

value for I1, replace the series combination of R1 and Req1 with another equivalent resistor.

I1

+

Vtotal

_

280

80200

2

2

112

eq

eq

eqeq

R

R

RRR

Example 4 (con’t)I1

+

Vtotal

_

)20sin(79.157.3280

)20sin(5.0

280

1280

)20sin(5.01

1

1

1

21

tmAmAI

tVVI

tVVI

R

VI

eq

total

Example 4 (con’t)To calculate V1,

use one of the previous simplified circuits where R1 is in series with Req1.

+

V1

_

+

Vp

_

I1

+

Vtotal

_

)20sin(357.0714.0

or

1

111

1

11

tVVV

IRV

VRR

RV total

eq

Example 4 (con’t)To calculate Vp:

Note: rounding errors can occur. It is best to carry the calculations out to 5 or 6 significant figures and then reduce this to 3 significant figures when writing the final answer.

+

V1

_

+

Vp

_

I1

+

Vtotal

_

)20sin(143.0287.0

or

or

1

11

11

1

tVVV

VVV

IRV

VRR

RV

p

totalp

eqp

totaleq

eqp

Example 4 (con’t)Finally, use the

original circuit to find I2 and I3. +

V1

_

+

Vp

_

I1

I2 I3

)20sin(357.0714.0

or

2

12

12

132

32

tmAmAI

IR

RI

IRR

RI

eq

Example 4 (con’t)Lastly, the

calculation for I3.+

V1

_

+

Vp

_

I1

I2 I3

)20sin(43.186.2

or

or

3

213

13

13

132

23

tmAmAI

III

IR

RI

IRR

RI

eq

SummaryThe equations used

to calculate the voltage across a specific resistor Rn in a set of resistors in series are:

The equations used to calculate the current flowing through a specific resistor Rm in a set of resistors in parallel are:

totaln

eqn

totaleq

nn

VG

GV

VR

RV

total

total

I

I

eq

mm

m

eqm

G

GI

R

RI