single loop circuits a single loop circuit is one which has only a single loop. the same current...

SINGLE LOOP CIRCUITSSINGLE LOOP CIRCUITS

• A single loop circuit is one which has only a single loop.

• The same current flows through each element of the circuit-the elements are in series.

• We will consider circuits consisting of voltage sources and resistors.

VOLTAGE DIVIDERVOLTAGE DIVIDER

Consider two resistors in series with a voltage v(t) across them:

R1

R2

-

v1(t)

+

+

-

v2(t)

+

-

v(t)21

11 )()(

RR

Rtvtv

21

22 )()(

RR

Rtvtv

IMPORTANT VOLTAGEDIVIDER EQUATIONS

+-

+-

+-

+ -

+-

+ -

FIRST GENERALIZATION: MULTIPLE SOURCES

i(t)

KVL

01542321 vvvvvvv RR

Collect all sources on one side

2154321 RR vvvvvvv

21 RReq vvv eqv

1R

2R

Voltage sources in series can be algebraically added to form an equivalent source.

We select the reference direction to move along the path.Voltage drops are subtracted from rises

1R

2R

1Rv

2Rv

1v

2v

3v

4v

5v

SECOND GENERALIZATION: MULTIPLE RESISTORS

APPLY KVLTO THIS LOOP

VOLTAGE DIVISION FOR MULTIPLE RESISTORS

iRv iRi

Multiple Sources/ResistorsMultiple Sources/Resistors

)(1 tv

)(2 tv

)(3 tv

1R

4R

2R

3R+

+

+

_

_

_

_+)(tveq eqR

These two circuits are equivalent where

)()()()( 321 tvtvtvtveq and 4321 RRRRReq

I R1 R2 V

+

-

I1 I2

Single Node Pair CircuitSingle Node Pair Circuit

How do we find I1 and I2?

Apply KCL at the Top NodeApply KCL at the Top Node

I1 + I2 = I

11 R

VI

22 R

VI

R1 R2 V

+

-

I1 I2I

Solve for Solve for VV

2121

11

RRV

R

V

R

VI

21

21

21

111

RR

RRI

RR

IV

Equivalent ResistanceEquivalent Resistance

If we wish to replace the two parallel resistors with a single resistor whose voltage-current relationship is the same, the equivalent resistor has a value of:

21

21

RR

RRReq

21

21

RR

RRIV

21

2

11 RR

RI

R

VI

Now to find Now to find II11

• This is the current divider formula.

• It tells us how to divide the current through parallel resistors.

What is the formula for What is the formula for II22??

Is2 VR1 R2

+

-

I1 I2

More Than One SourceMore Than One Source

How do we find I1 or I2?

Is1

Apply KCL at the Top NodeApply KCL at the Top Node

I1 + I2 = Is1 - Is2

212121

11

RRV

R

V

R

VII ss

21

2121 RR

RRIIV ss

Multiple Current SourcesMultiple Current Sources

• We find an equivalent current source by algebraically summing current sources.

• We find an equivalent resistance.

• We find V as equivalent I times equivalent R.

• We then find any necessary currents using Ohm’s law.

SERIES AND PARALLEL SERIES AND PARALLEL RESISTOR COMBINATIONSRESISTOR COMBINATIONS

• For analysis, series resistors can be replaced by an equivalent resistor.

• Parallel resistors can be replaced by an equivalent resistor/ impedance.

• Complicated networks of resistors can be replaced by a single equivalent resistor.

Equivalent ResistanceEquivalent Resistance

i(t)

+

-

v(t)

i(t)

+

-

v(t)Req

Req is equivalent to the resistor network on the

left in the sense that they have the same i-v characteristics.

Equivalent ResistanceEquivalent Resistance

• The rest of the circuit cannot tell whether the resistor network or the equivalent resistor is connected to it.

• The equivalent resistance cannot be used to find voltages or currents internal to the resistor network.

Series ResistanceSeries Resistance

R1

R3

R2 Req

Req = R1 + R2 + R3

Two elements are in series if the current that flowsthrough one must also flow through the other.

Parallel ResistanceParallel Resistance

Req

1/Req = 1/R1 + 1/R2 + 1/R3

R3R2R1

Two elements are in parallel if they areconnected between the same two nodes.

Circuits with Series and Parallel Circuits with Series and Parallel CombinationsCombinations

• The combination of series and parallel resistances can be used to find voltages and currents in circuits

• Simplification– Resistances are combined to create a simple

circuit (usually one source and one resistance), from which a voltage or current can be found. Start from the furthest branch from the source.

• Backtracking– Once the voltage or current is found, KCL and

KVL, Ohm’s Law, Voltage and Current Dividers are used to work back through the network to find voltages and currents.

FIRST WE PRACTICE COMBINING RESISTORS

6k||3k

(10K,2K)SERIES

SERIESk3

kkk 412||6

k12k3

k5

EXAMPLES COMBINATION SERIES-PARALLEL

k9

kkk 69||18

kkk 1066

AN EXAMPLE WITHOUT REDRAWING

kkk 612||12 kkk 26||3

)24(||6 kkk

RESISTORS ARE IN SERIES IF THEY CARRYEXACTLY THE SAME CURRENT

RESISTORS ARE IN PARALLEL IF THEY ARECONNECTED EXACTLY BETWEEN THE SAME TWONODES

AN “INVERSE SERIES PARALLEL COMBINATION”

AVAILABLE ARERESISTORS ONLY

WHEN600mV BE MUST

1.0

3AIVR

2.03

6.

A

VR REQUIRED 1.01.0R

AVAILABLE ARERESISTORS ONLY

WHEN600mV BE MUST

1.0

9AIVR

0667.09

6.

A

VR REQUIRED

SIMPLE CASE

NOT SO SIMPLE CASE

FIRST REDUCE IT TO A SINGLE LOOP CIRCUITk12kk 12||4

k6

kk 6||6

k

VI

12

121

)12(93

3

aV

SECOND: “BACKTRACK” USING KVL, KCL OHM’S

k

VI a

62 :SOHM'0321 III :KCL

3*3 IkVb :SOHM'

3I

…OTHER OPTIONS...

4

34

*4124

12

IkV

II

b

5

345

*3

0

IkV

III

C

:SOHM'

:KCL