resistors and resistive circuits

EE101 CIRCUITS 1

1T SY 2014-2015

MCBLOYOLA

Resistors and Resistive Circuits

Week 2

MCBLOYOLA

Learning Outcomes

Solve for the total resistance, current through

an element, and voltage drop across an

element in a series, parallel, and series-parallel

connected dc network using voltage divider,

current divider, Ohms Law, and Kirchhoffs

Law.

Solve the total resistance, current through an

element, and voltage drop across an element

in a wye- or delta-connected dc network.

MCBLOYOLA

NODES, BRANCHES, AND LOOPS

A branch represents a single element such as a

voltage source or a resistor.

A node is the point of connection between two or

more branches.

A loop is any closed path in a circuit.

A network with b branches, n nodes, and l

independent loops will satisfy the fundamental

theorem of network topology:

MCBLOYOLA


Original circuit

Equivalent circuit

How many branches,

nodes and loops are

there?

MCBLOYOLA


How many branches, nodes and loops are there?

Should we consider it as one

branch or two branches?

MCBLOYOLA


Two or more elements are in series if they exclusively

share a single node and consequently carry the same

current.

Two or more elements are in parallel if they are

connected to the same two nodes and consequently have

the same voltage across them.

Is 5- resistor in series or in parallel with 2- resistor?

MCBLOYOLA

RESISTANCE

Materials in general have a characteristic behavior of

resisting the flow of electric charge. This physical property,

or ability to resist current, is known as resistance and is

represented by the symbol R.

The circuit element used to model the current-resisting

behavior of a material is the resistor.

MCBLOYOLA

OHMS LAW

Georg Simon Ohm (1787-1854), a German physicist, is

credited with finding the relationship between current

and voltage for a resistor

MCBLOYOLA

OHMS LAW

It states that the voltage across a resistor is directly

proportional to the current flowing through the resistor.

That is,

Ohm defined the constant of proportionality for a

resistor to be the resistance, R.

MCBLOYOLA

KIRCHHOFFS LAWS

Kirchhoffs laws were first introduced in 1847 by the

German physicist Gustav Robert Kirchhoff (1824-1887).

These laws are formally known as Kirchhoffs current law

(KCL) and Kirchhoffs voltage law (KVL).

MCBLOYOLA

KIRCHHOFFS CURRENT LAW (KCL)

Kirchhoffs current law (KCL) states that the

algebraic sum of currents entering a node (or a

closed boundary) is zero.

It is based on the law of conservation of charge, which requires

that the algebraic sum of charges within a system cannot

change.

The sum of the currents entering a node is equal to the sum of

the currents leaving the node.

MCBLOYOLA


Mathematically,

Where

N = number of branches

connected to the node

= nth current entering (or leaving) the node

= 0

=1

Current entering a node is (+) and

current leaving a node is (-) or vice-

versa

MCBLOYOLA


Current sources in parallel:

(a) Original circuit

(b) Equivalent circuit

MCBLOYOLA

KIRCHHOFFS VOLTAGE LAW (KVL)

Kirchhoffs voltage law (KVL) states that the algebraic

sum of all voltages around a closed path (or loop) is

zero. It is based on the principle of conservation of energy.

The sum of voltage drops is equal to the sum of voltage rises.

MCBLOYOLA


Mathematically,

Where

M = number of voltages in

the loop (or the number of

branches in the loop)

= nth voltage

= 0

=1

Sign on each voltage is the polarity

of the terminal encountered first as

we travel around the loop.

MCBLOYOLA


Voltage sources in series:

(a) Original circuit

(b) Equivalent circuit

MCBLOYOLA

ILLUSTRATIVE PROBLEM 1

Find the currents and voltages in the circuit shown.

= , = , = , = . , = , = .

MCBLOYOLA


Calculate in the circuit below.

MCBLOYOLA


Find and in the circuit below.

= =

MCBLOYOLA


Given the circuit shown, determine the following:

a)

b) c) 30

MCBLOYOLA

= = =

SERIES RESISTORS AND VOLTAGE

DIVISION

Recall: Two or more elements are in

series if they exclusively share a

single node and consequently carry

the same current.

Resistors in series behave as a

single resistor whose resistance is

equal to the sum of the

resistances of the individual

resistors.

For two resistors in series:

= 1 + 2 ++

=

=1

MCBLOYOLA


DIVISION

The equivalent conductance of resistors connected in

series is:

Voltage Division

=

1 + 2 ++

1

= 1

1+ 1

2+ +

1

MCBLOYOLA


DIVISION

1 = 1

1 + 2 + 3 1 =

11 + 2

Voltage Divider Circuit

2 = 2

1 + 2 + 3

3 = 3

1 + 2 + 3

2 = 21 + 2

MCBLOYOLA

PARALLEL RESISTORS AND CURRENT DIVISION

Recall: Two or more elements are in

parallel if they are connected to the

same two nodes and consequently

have the same voltage across them.

The equivalent resistance of a

circuit with N resistors in parallel

is:

For 2 resistors in parallel:

= 121 + 2

1

= 1

1+ 1

2+ +

1

MCBLOYOLA

PARALLEL RESISTORS AND CURRENT

DIVISION

The equivalent conductance of resistors connected in

parallel is:

Current Division

=

=

= 1 + 2 ++

MCBLOYOLA

PARALLEL RESISTORS AND CURRENT DIVISION

Current Divider Circuit

1 = 21 + 2

2 = 11 + 2

MCBLOYOLA

EQUIVALENT RESISTANCE

In general, it is often convenient and possible to combine

resistors in series and parallel and reduce a resistive

network to a single equivalent resistance.

Such an equivalent resistance is the resistance between

the designated terminals of the network and must exhibit

the same i-v characteristics as the original network at the

terminals.

MCBLOYOLA

POWER IN RESISTIVE NETWORKS

Total power is the sum of the power dissipated by each

resistor in the circuit:

= 1 + 2+ . . . +

= = 2

= 2

MCBLOYOLA

Consider the figure below. What are the corresponding

currents? What about equivalent resistances?

? ?

? ?

= 0

= 1

MCBLOYOLA

MORE ILLUSTRATIVE PROBLEMS

1. Two 115-V incandescent lamps A and B are connected

in series across 230-V source. If lamp A is rated 75

watts and lamp B is rated 50 watts, determine the

current drawn by the series connection.

2. R resistor is connected in series with two resistances

1 = 2 and 2 = 4 . The series combination is connected across a 36-V source. A voltmeter, placed

across 2, reads 12 V. Find the value of the resistor R.

MCBLOYOLA


3. Two resistances of 10 and 5 ohms are connected in

parallel and the combination is connected in series with

a 10-ohm resistance. If these are connected across a 48-

V battery, determine the current through the 5-ohm

resistance.

4. The equivalent resistance of three resistors A, B and C

connected in parallel is 1.714 ohms. If A is twice of B

and C is half as much as B, find the equivalent resistance

when the three of them are connected in series.

MCBLOYOLA


5. Three resistors R1, R2, and R3 are connected in series-

parallel with R1 in series with the parallel combination

of R2 and R3. The whole combination is connected

across a 120-V DC source. R1, R2, and R3 take 750 W,

250 W, and 200 W, respectively. Calculate the resistance

R2.

6. Resistances X, Y, and Z are connected in series. The

voltage across X and Y is 21 V, across Y and Z is 24 V,

and across X and Z is 27 V. Find the value of resistance

Y when the current is 1 A.

MCBLOYOLA


7. A 16- resistor is connected in series with a parallel combination of two resistors, one of which has an

ohmic value of 48 and the other is unknown. What is the resistance of R if the power taken by the 16- resistor is equal to the power taken by the parallel-

connected pair of resistors?

MCBLOYOLA


8. Calculate .

= 11.2

MCBLOYOLA

MCBLOYOLA


9. Find the equivalent resistance at terminals a-b.

Rab = R

MCBLOYOLA


10. Find the equivalent resistance at terminals a-b.

Rab = 54

MCBLOYOLA


11. Calculate Vo and Io in the circuit below.

MCBLOYOLA

VO = 8 V IO = 0.2 A


12. Determine V in the circuit below.

Confused if resistors are series- or parallel-connected?

They are neither in series nor in parallel. So, how do we combine them?

They can be simplified by using three-terminal equivalent network!

MCBLOYOLA

WYE (TEE)-CONNECTED

NETWORK

Two forms of the same network: (a) Y, (b) T

MCBLOYOLA

DELTA (PI)-CONNECTED NETWORK

Two forms of the same network: (a) , (b)

MCBLOYOLA

DELTA-WYE CONVERSION

Each resistor in the Y network is

the product of the resistors in the

two adjacent branches, divided by the sum of the three resistors.

Mathematical expression:

1

2

3

b c

a b c

a c

a b c

a b

a b c

R RR

R R R

R RR

R R R

R RR

R R R

MCBLOYOLA

WYE-DELTA CONVERSION

Mathematical expression: Each resistor in the network is the sum of all possible products of

Y resistors taken two at a time,

divided by the opposite Y resistor.

1 2 2 3 3 1

1

1 2 2 3 3 1

2

1 2 2 3 3 1

3

a

b

c

R R R R R RR

R

R R R R R RR

R

R R R R R RR

R

MCBLOYOLA


Determine V in the circuit below.

42.18 V

Can we now answer this?

MCBLOYOLA


Determine the current I as indicated in the circuit

shown below.

I = 0.375 A

MCBLOYOLA


Find in the four-way power divider circuit in the figure below. Assume each element is 1 .

2

References

Please refer to course syllabus.

MCBLOYOLA

resistors and resistive circuits

Documents