Quiz 1a) Find the currents i1 and i2 in the circuit in the figure. b) Find the voltage vo.c) Verify that the total power developed equals the total power

dissipated.

Quiz 2 The current ia in the circuit shown in the figure is 2 mA. Find (a) io

(b) ig

(c) Verify that the total power developed equals the total power dissipated.

Lecture - 3

Resistance equivalent circuits

Outline

• Resistors in series

• Resistors in parallel

• The voltage divider and the current divider circuits

• Voltage division and current division

• Measuring voltage and current

• The ∆-to-Y transformation.

Resistors in series• Series-connected circuit elements carry the same current.• Series resistors can be combined to obtain a single equivalent

resistance according to the equation:

• Note that the resistance of the equivalent resistor is always larger than that of the largest resistor in the series connection.

k

k

iieq RRRRR

......211

Resistors in parallel• Parallel-connected circuit elements have the same voltage

across their terminals.• Parallel resistors can be combined to obtain a single

equivalent resistance according to the equation:

• Note that the resistance of the equivalent resistor is always smaller than the resistance of the smallest resistor in the parallel connection.

k

k

i ieq RRRRR1........1111

211

Two resistors in parallel• When just two resistors are in parallel, the equation for

equivalent resistance can be simplified to give:

21

21

RRRRReq

Example 1Find is, i1, and i2 in the circuit shown in the figure.

-----------------------------------

and Avi

Vv

ARVi

eq

ss

41872

18

72)6)(12(

1210

120

11

1

Avi 8972

91

2

The voltage divider circuit• When voltage is divided between series resistors the voltage

across each resistor can be found according to the equations:

• These equations show that v1 and v2 are fractions of vs. • Each fraction is the ratio of the resistance across which the

divided voltage is defined to the sum of the two resistances.• Because this ratio is always less than 1, the divided voltages

v1 and v2 are always less than the source voltage vs.

s

s

vRR

Rv

vRR

Rv

21

22

21

11

Example 2The resistors used in the voltage-divider circuit shown in the figure have a tolerance of ±10%. Find the maximum and minimum value of vo?

-----------------------------The maximum value of vo occurs when R2 is 10% high and R1 is 10% low, and the minimum value of vo occurs when R2 is 10% low and R1 is 10% high.

Vv

Vv

o

o

6.765.2790)90)(100((min)

02.835.22110)110)(100((max)

The current divider circuit• When current is divided between parallel resistor the current

through each resistor can be found according to the equations:

• These equations show that the current divides between two resistors in parallel such that the current in one resistor equals the current entering the parallel pair multiplied by the other resistance and divided by the sum of the resistors.

s

s

iRR

Ri

iRR

Ri

21

12

21

21

Example 3Find the power dissipated in the 6 Ω resistor shown in the figure.

-----------------------------------------------First, we must find the current in the resistor by simplifying the circuit with series-parallel reductions. Then, we find the current io by using the formula for current division:

and the current in the 6 Ω resistor is:

and the power dissipated in the 6 Ω resistor is:p = (3.2)2(6) = 61.44W.

Aio 8)10(416

16

Ai 2.3)8(64

46

Voltage division • The voltage drop vj across a single resistor Rj from a collection

of series-connected resistors is proportional to the total voltage drop v across the set of series connected resistors.

• The constant of proportionality is the ratio of the single resistance to the equivalent resistance of the series connected set of resistors, or Rj/Req.

vRR

veq

jj

Current division• The current ij through a single resistor Rj from a collection of

parallel-connected resistors is proportional to the total current i supplied to the set of parallel connected resistors.

• The constant of proportionality is the ratio of the equivalent resistance of the parallel-connected set of resistors to the single resistance, or Req/Rj.

• The constant of proportionality in the current division equation is the inverse of the constant of proportionality in the voltage division equation.

iRjR

i eqj

Example 4 Use current division to find the current io and use voltage division to find the voltage vo for the circuit in the figure.

----------------------------we can find the equivalent resistance of the four parallel branches containing resistors:

and io and v:

and vo

6eqR

AAio 2)8(246

Vv 48)2)(24(

VVvo 18)48(8030

Measuring voltage & current• A voltmeter is an instrument designed to measure voltage; it is

placed in parallel with the element whose voltage is being measured.• An ideal voltmeter has infinite internal resistance and thus does not

alter the voltage being measured.• An ammeter is an instrument designed to measure current; it is

placed in series with the circuit element whose current is being measured.

• An ideal ammeter has zero internal resistance and thus does not alter the current being measured.

Digital vs. analog meters • An ideal ammeter or voltmeter has no effect on the circuit

variable it is designed to measure.

• Digital meters and analog meters have internal resistance, which influences the value of the circuit variable being measured.

• Digital meters offer several advantages over analog meters:– They introduce less resistance into the circuit to which they

are connected, – They are easier to connect, – The precision of the measurement is greater due to the nature

of the readout mechanism.

The ∆-to-Y transformation

cba

ba

cba

ac

cba

cb

RRRRRR

RRRRRR

RRRRRR

3

2

1

Ra Rb

Rc

R3

R2 R1

The Y-to-∆ transformation

3

133221

2

133221

1

133221

RRRRRRRR

RRRRRRRR

RRRRRRRR

c

b

a

Ra Rb

Rc

R3

R2 R1

Example 5Find the current and power supplied by the 40 V source in the circuit shown in the figure.

-------------------------------By replacing the upper ∆. We then compute the three Y resistances as:

10250

25*100

5.12250

25*125

50250

25*100

2

1

3

R

R

R

the equivalent resistance is

and the current in the circuit is:

i= v/R = 40/ 80= 0.5A

and the power is :

p= v*i = 40*0.5 = 20W

Example 5

80100

50*5055eqR

Summary• Series resistors can be combined to obtain a single equivalent

resistance according to the equation:

• Parallel resistors can be combined to obtain a single equivalent resistance according to the equation:

• When just two resistors are in parallel, the equation for equivalent resistance can be simplified to give:

k

k

iieq RRRRR

......211

k

k

i ieq RRRRR1........1111

211

21

21

RRRRReq

Summary• When voltage is divided between series resistors the voltage across

each resistor can be found according to the equations:

• When current is divided between parallel resistor the current through each resistor can be found according to the equations:

• Voltage division is a circuit analysis tool that is used to find the voltage drop across a single resistance from a collection of series-connected resistances when the voltage drop across the collection is known:

svRR

Rv21

11

siRRRi

21

21

svRR

Rv21

22

siRRRi

21

12

vRR

veq

jj

Summary• Current division is a circuit analysis tool that is used to find the

current through a single resistance from a collection of parallel-connected resistances when the current into the collection is known:

• A voltmeter measures voltage and must be placed in parallel with the voltage being measured.

• An ammeter measures current and must be placed in series with the current being measured.

• A circuit with three resistors connected in a ∆ configuration can be transformed into an equivalent circuit in which the three resistors are Y connected.

iRjR

i eqj