2. electric circuits

8/2/2019 2. Electric Circuits

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2. Electric Circuits

Circuit elementsR,L and C.

Applications as sensors.

Equivalent circuits of some devices. Kirchhoffs Laws

Writing circuit equations for resistive circuits.

Circuit simplifications (Series, parallel, Star-deltatransformations, Superposition, Thevinin)

Writing state space equations for RLC circuits


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Mathematical Modelling of

Electrical Devices Field models

Requires geometry of device, material properties,charge distribution etc.

Difficult and time consuming.

Not covered in this course.

Circuit models

Black box model.

Relationship between voltage and current.

Easier.

Does not work at high frequency.

Approach used in this course.


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4/36

High Frequency Effects

Electromagnetic wave radiation

Radio waves

Microwaves

Light

X-Ray


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ResistanceOhms Law

R

VRIVIp

IRV

22

LawsOhm'

driftingelectron

vibrating

nucleus

Electrons collide with nuclei.

Vibration sensed as heat

I I

metal conductor

V+ -

A

l

i

V

Symbol

R

V

I

R

R = Resistance in ohms W


6/36

Resistors

Metal film or carbon resistor for electronic circuits

l

metal conductor

A

myResistivit W

A

l

A

lR

Strain gauges l andA vary with mechanical

stress. Used for measuring mechanical

force.

Wound large metal resistors rheostats, heaters.

Variable resistors vary length

used in electronic circuits and asposition sensors.


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Capacitance

Energy (voltage) is required

to drive charges against repulsion

force from charges on plates

source

-

+++

+

--

-

-

V

i

d

metal plate

areaA

+q

-q+ -

idtC

Vdt

dVCi

dt

dqiCVq

1or

,

q

V

C

C = capacitance in Farads (F)

V

i

symbol


8/36

Capacitors Component capacitors

Ceramic, electrolytic, tantalum Used in electronic circuits. Filter circuits.

ypermitivit

(Farads)F

d

A

C

dArea A

Energy storage capacitors Known as super capacitors Used in Electric vehicles

2

2

1

CVVdtCdtdt

dVVCEnergy

dt

dVCi

VidtpdtEnergy

Variable capacitors Position, speed, acceleration or

force sensors. Radio and TV Tuning circuits.


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InductanceFaradays Law

Energy (voltage) is required togenerate the magnetic field

Nf

i

L

LiN f

dt

diL

dt

dNV

f

LawsFaraday'

2

2

1

Energy

LiidiLdtdt

di

L

Vidtpdt

L

i

V

symbol

f

iV

Nturns

Edt

dNE

f

Note


10/36

Inductors

Component inductor

Radio frequency circuits(radio, TV, mobile)

materialcoreoftypermeabili

coilsolenoidlongveryaFor

2

lANL

Area A

l

Variable inductors

Tuning circuits.

Position sensors.


11/36

Circuit Elements

Resistance Sources

Inductance Zero resistance

Wires

Capacitance Switch

iRV

dt

diLV

dt

dVCi


12/36

Electric Circuits

These represent a number of devices

connected together.

Equivalent circuit

Battery

Actual devices

Objective: find current given battery voltage

and values of resistors


13/36

Equivalent circuits of devices

E

R

Battery/PV/

thermocouple

E

L R

AC Synchronous generator

Piezoelectric device

I R C

Short cable

L R

Medium length cable

C

L R

C

Lamp

R


14/36

Kirchhoffs Laws

Kirchhoffs Current Law (KCL)

Conservation of charge

Kirchhoffs Voltage Law

Conservation of energy


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Kirchhoffs Current Law (KCL)

Conservation of Chargecurrent going into a node

= current leaving a node

54321 iiiii

i1

i2

i3

i4

i5

node

054321 iiiii

0node

i

The algebraic sum of current into a node is zero.


16/36

Kirchhoffs Voltage Law (KVL)

Conservation of EnergyEnergy available from the supply

is consumed by the load.

Remember voltage is

energy per unit charge.

Vs VL

Vs = VLor

Vs+V

L=

0

0loop

V

loop

The algebraic Sum of voltages around a loop is zero

source load


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KVL application to a single loop

circuit

V

V1

V2loop

0loop

V

021 VVV

R1

R2

i

2211 andBut iRViRV

21 VVV

21 iRiRV

21 RR

Vi

V=10 V,R1=R2=5W,

i 1 A


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i1 i3

i2

R1

R2

R3 i4

R4V

V1

V3

V2 V4

Two Loop Circuit

loop Bloop A

loop C

node

x

node

y

Apply KVL to loop A

021 VVV

222

111

Substitute

RiV

RiV

(1)02211 RiRiV

Apply KVL to loop B

(2)0443322

RiRiRi

Apply KVL to loop C

(3)0443311 RiRiRiV

Apply KCL to node x

(4)0321 iii

Apply KCL to node y

(5)043 iiWe have 4 unknowns i

1, i

2, i

3and i

4We need 4 independent equation (1), (2), (4), (5)


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Branch and Loop Currents

i1 i3

i2

R1

R2

R3 i4

R4V

V1

V3

V2 V4IA IB

i1, i2, i3, i4 are branch currents

Define Loop Currents as

IA,IB such that

BA

B

A

IIi

iii

iii

Iii

Ii

2

312

321

43

1

This has the advantage of reducing the number of unknowns!


20/36

Analysis using Loop Currents

i1 i3

i2

R1

R2

R3 i4

R4V

V1

V3

V2 V4IA IB

Apply KVL to loop A

0)( 21 RIIRIV BAA

Rearrange

(1))( 221 VIRIRR BA

The above equation has the format

(sum of loop resistances) x loop current(mutual

resistance) x neighbouring loop current = source voltageApply to loop B

(2)0)( 2432 AB IRIRRR

Satisfy yourself that the above equation is correct!

We now have only 2 equations instead of 4.


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ExampleWriting Circuit

Equations

10V

2W

4W

2W

3W

4W

i1 i2

i3

i4

I1 I2

I3

(1)1024)42(

1looptoKVLApply

321 III

(2)02)432(4

2Loop

321 III

(3)0)422(22

3Loop

321 III

Solve (1), (2) and (3) to find loop currents

34

213

322

311

currentsBranch

Ii

IIi

IIi

IIi


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Circuit Simplifications

Writing circuit equations is easy, but the

resulting equations are hard to solve by

hand. A computer is often needed. Often interested in calculating just one or

two currents.

Circuit can be simplified to make thecalculations easier.


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Resistors Connected in Series

=RR1 R2 R3

i i i i

V V

i

321 iRiRiRiRV

321

RRRR

n

k

kRR

n

1

seriesinresistorsforgeneralIn


24/36

Resistors in Parallel

=R1

V

i

V

i

i1

i2

i3

R2

R3

R

321 iiii

321 R

V

R

V

R

V

R

Vi

321

1111

RRRR

n

k kRR

n

1

11

resistorsFor

21

21

resistorsFor two

RR

RRR


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Voltage Divider

V

V1

R2

R1

i

(1)11 iRV

(2))( 21 RRiV

VRR

RV

21

11

ruledividervoltageobtain the

we(2)and(1)From


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Current Divider

i

ii1 i2

R1R2

V(1)

KCL

21 iii

(2),

LawsOhm'

2

2

1

1R

Vi

R

Vi

(3)

(1)into(2)sSubstitute

21

21

21

RR

RRV

R

V

R

Vi

i

RR

Ri

21

21

ruledividercurrenttheobtainwe(3)and(2)From


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Example

10V

5W

10W

5W

5W

i

Find the currents in the following circuit:

series

parallel

serie

10V

5W

5W

i

A155

10

LawsOhm'

i

i1

i2

10W

5W10W10V

i

i1

i2

A5.011010

10

DividerCurrent

21

ii


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Star-Delta Transformation

star delta

A A

BB

C C

Ra

Rc Rb

RA

RCRB

CBA

BAc

CBA

ACb

CBA

CBa

RRR

RRR

RRR

RRR

RRRRRR

StartoDelta

c

accbbaC

b

accbbaB

a

accbbaA

R

RRRRRRR

R

RRRRRRR

RRRRRRRR

DeltaStar to


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Superposition

Superposition means

that we can consider the

effect of each supply onits own.

We can apply

superposition because

electric circuits arelinear systems.

V1

V2

iiV1

Calculate iV1

V1

V2

iV2

Calculate iV2

V1

V2

21 VV iii

i

Calculate i


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Superposition Example

Find i in the following circuit:

V1=10VV2=10V

5W

5W5W

i


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Thevinins Theorem

In a complex circuit

we wish to calculate

the current i in theresistorR.

R

i

Replace the complex

circuit with a

simpler Thevinins

equivalent circuit.

RRthVth

i

RR

Vi

th

th


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Simple Thevinins Example

Find the current i in the following circuit

10V

5W

5W 10W

i


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RCCircuit

V

R

C VC

i

(1)0

KVL

CViRV

(2)dtdVCi C

(3)

rearrangeand(1)into(2)Substitute

VVdtdVRC CC

Equation (3) is a first order ordinary differential equation, which

can be solved by separation of variables and integration (try that

at home).


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RLCCircuit

R L

CV

i

VC


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Circuit State-Space Equations

C

L1 L2

R2

R1

V VC

i1 i2


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Summary

Kirchhoffs current law is the law of conservationof charge: sum of currents into a node is zero.

Kirchhoffs voltage law is the law of conservationof energy: sum of voltages around a loop is zero.

For resistive circuits use loop currents to minimizethe number ofalgebraic equations.

Circuit simplifications: parallel, series, currentdivider, voltage divider, star-delta transform,

superposition, Thevinin.

Case study: strain gauges and bridge circuits.

RLCcircuits are described by differentialequation. The number of unknowns equal the

b f l i h i i

2. electric circuits

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