Circuit Analysis Bw

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ME2143/ME2143E Sensors and Actuators

Review of Electrical Circuits Theory

Chew Chee Meng

Department of Mechanical EngineeringNational University of Singapore

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Outline

IntroductionBasic Electrical ElementsKirchhoff’s LawsMethod of SuperpositionEquivalent CircuitsPractical Considerations

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IntroductionAll mechatronic and measurement systems contain electrical circuits and components

Typical elements of electrical circuits

IntroductionBasic mechanical quantities

DisplacementVelocityForce

What about electrical domain?

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Introduction

Basic electrical quantitiesChargeCurrentVoltage

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Introduction

Charge Fundamental electric quantity Unit: coulombs (C)

Atomic structure of matter:Consists of a nucleus (neutrons and protons) surrounded by electrons

Elementary chargesA proton has a charge of 1.6 × 10-19 CAn electron has a charge of -1.6 × 10-19 C

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IntroductionElectrical current (charge in motion)

time rate of flow of electrical charge through a conductor or circuit elementunit: amperes, A (or C /s)

q(t) : quantity of charge flowing through a cross-section of the circuit element.

Current flow direction

Electrons

( )( ) dq ti tdt

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IntroductionDirect Current vs Alternating Current

direct current (dc): constant with time.alternating current, (ac): varies with time, reversing direction periodically (typically sinusoidal).

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IntroductionCurrent Measurements

How to measure current in a circuit?

Refer to: http://www.youtube.com/watch?v=y_o34SY77yo

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IntroductionVoltage (potential difference, electromotive force (emf))

difference in electrical potential between 2 pointsSI unit: volt, V (or J/C)

Let Va be the electrical potential at point A and Vb at point B, then the voltage across A and B, Vab (A wrt B) is

Vab=Va-VbAlso, Vba=-Vab

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IntroductionNotations

When vab is positive (negative), electric potential at a is higher(lower) than that at b

When v is positive(negative), electric potential at arrow end is higher (lower) than that at the non-arrow end

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IntroductionHow to measure voltage?

http://www.youtube.com/watch?v=t0Zzoz4nM0I&amp

4.889V

A

B

Vab

Voltmeter

-ve+ve

Vab

+

-

IntroductionDigital Multimeter (DMM) can be used to measure:

VoltageCurrentResistance

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IntroductionGround

Typical reference for electric potentialSymbol

“Voltage at point A, Va = 3.8V” means potential at point A is 3.8V with respect to ground potential

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IntroductionWhen current flows through an element and voltage appears across the element, energy is transferred.When positive charge or current entersthrough positive (negative) polarity into an element, energy is absorbed (supplied) by the element

Energy

supplied

by the

element

Energy

absorbed

by the

element+

+

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IntroductionPower and Energy

Power absorbed by an element:

Energy absorbed from time t1 to t2:

( ) ( ) ( )p t v t i t

2

1

( )t

t

w p t dt t1 < t2

*Remark: This formula is based on convention that current reference i enters the positive polarity of the voltage . Positive => energy is absorbed by the element.

*

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Basic Electrical Elements

An electrical circuit is an interconnection of electrical elements and energy sources.Energy sources

Voltage source (Vs), current source (Is)Ideal energy sources: Contain no internal

resistance, inductance, or capacitance.Three basic passive* electrical elements

Resistors (R), capacitors (C) , inductors (L)

*Passive elements: Require no additional power supply (compared with integrated circuits (ICs))

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Ideal voltage source (Vs)

Ideal independent voltage source

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Ideal voltage source (Vs)

Ideal dependent voltage source (rhomboidal shape symbol)

• Depends on a current or voltage that appears elsewhere in the circuit

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Ideal current source (Is)

Ideal independent current source

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Ideal current source (Is)

Ideal dependent current sourceDepends on a current or voltage that appears

elsewhere in the circuit

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ResistorA dissipative element: Converts electrical energy into heat

Symbol

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ResistorIdeal resistor

Voltage-current characteristics defined by Ohm’s law:

where R is a constant called resistance (SI Unit: Ohm, )

v Ri

v

i

R = v / i

Resistor

Method of reading resistor’s value of wire-lead resistors

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(%)10 toleranceabR c

10 k5%

ResistorVariable resistors

Potentiometer (pot)Trim pot

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Schematic symbols

Three terminals

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ResistorResistance related to physical parameters

ALR

is called the resistivitywhich is a property of the material

http://www.youtube.com/watch?v=wUgJgK2aTG0&feature=related

How to measure resistance?

E.g. Resistance of a homogeneous material of length L and with uniform cross-sectional area, A:

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CapacitorA passive element that stores energy in the form of an electric field

Dielectricmaterial

Conducting plates

Dielectric material (an insulator): increases capacitance

i ++++++++

--------

Current flow* results in opposite charge built up on the conducting plates

Symbol:

*Strictly, current does not flow through a capacitor

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CapacitorCapacitor and its fluid-flow analogy

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Capacitor

where q (unit: coulombs, C): amount of accumulated chargeappearing on each capacitor plate C (unit: farads, F (coulombs/volts)): capacitancev: voltage across the capacitori : current flowing into the positive polarity of the capacitor

Cvq dtdvCi

v(t)+ -

i(t)

C

or

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Capacitor

Voltage across a capacitor cannot change instantaneously (why?)

Can be used for timing purposes in electrical circuits (e.g. RC circuit)

Used in low-pass filter

With DC sources, capacitor behaves like an open circuit during steady state condition

Vin

Vin

Vc

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Capacitor

Capacitance: A property of dielectric materialplate geometry and separation

Typical values: 1pF (picofarads, 10-12) to 1000F (microfarads, 10-

6)

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Capacitor

Primary types of commercial capacitors:Electrolytic (polarized, have a positive and negative ends)TantalumCeramic diskMylar

Capacitance codes: Three-digit code, e.g. 102, implies 10x102

pF = 1 nFTwo-digit code, e.g. 22, implies 22 pF

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Inductor

A passive energy storage element that stores energy in the form of a magnetic field.

Symbol:

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Inductor

Inductor’s characteristics governed by Faraday’s lawof induction:

where λ = total magnetic flux (webers, Wb) through the coil windings due to the current

dtdtV

)(

Fig 2.10, p15 of Alciatore and Histand, 2003

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InductorFor ideal coil, ,

hence,

where L is the inductance (henry, H (=Wb/A)) of the coil.

or

=> Current cannot change instantaneously

dtdiLtv

0

0

1 tidttvL

tit

t

Li

Note: With DC sources, Inductor behaves like a short circuit during steady state condition

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Inductor

Source: http://www.wikipedia.org/

Fuel injectorElectric motors

Typical inductor values: 1H to 100mHPresent in motors, relays, power supplies, oscillators circuits, etc

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Branches, Nodes and LoopsBranch: Any portion of a circuit with two terminalsconnected to it

Node: Junction of two or more branches

Loop: Any closed connectionof branches

Node

Circuitelements

Loop 1 Loop 2

Loop 3

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Kirchhoff’s Circuit LawsAnalysis of circuits: calculate voltagesand currents anywhere in a circuit*Kirchhoff’s circuit laws: essential for analysis of circuits which involve various electrical elements ranging from basic elements to semiconductor components like transistors, op amps, etc

Kirchhoff’s current law (KCL)Kirchhoff’s voltage law (KVL)

*Named after Gustav Kirchhoff (1824-1887)

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Kirchhoff’s Current LawSum of currents flowing into a node is zero:

I1+I2-I3 = 0

N

iiI

10

Eg:

(I3 has negative signbecause it is flowing away from the node)

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Kirchhoff’s Current LawAlternatively, the sum of the currents entering a node equals to the sum of the currents leavingthe node

I1+I2 = I3

Eg:

(Sum of currents entering node)

(Sum of currents leaving node)

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Kirchhoff’s Voltage LawSum of voltages around a closed loop is zero:

N

iiV

10

Start from a node (e.g. A) and end at the same node

Either clockwise or anti-clockwise is fine

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Kirchhoff’s Voltage Law

Loop 1: -va + vb + vc = 0Loop 2: -vc – vd + ve = 0Loop 3: va – vb + vd – ve = 0

Eg:

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Analysis of circuitsProcedure:

• First, assign current variable to each branch and assume its flow direction.

• Then assign appropriate polarity to the voltage across each passive element (current entering into +ve polarity).

• Apply KVL for loops or apply KCL for nodes to generate sufficient equations together with constitutive equations of the elements (eg. Ohm’s law) to solve the unknown current and voltage variables

i1 i2 i3

-

vA

+

+ vB - - vD +

-

vE

+

Passive

element

+

vC

-

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Systematic Circuits Analysis Methods

For resistive circuits Node-voltage methodMesh-current method

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Series Resistance Circuit

Apply KVL to the closed loop starting from node A (clockwise):

-Vs+VR1+VR2 = 0Constitutive equations, Ohm’s law:

VR1 = IR1

VR2 = IR2

=> -Vs+IR1+IR2 = 0

Fig 2.13, p18, Alciatore and Histand, 2003

?

??

Circuit: R1 and R2 connected in series with a voltage source Vs

To find: I, VR1 and VR2 (need three equations to solve)

Hence, I = Vs/(R1+R2) , VR1= IR1 , VR2 = IR2

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Series Resistance CircuitSince Vs = I(R1+R2)=IReq where Req=R1+R2

Vs

I

Req

+

-

i.e. the two resistors can be replaced by a single resistor Reqof value R1+R2.


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Series Resistance Circuit

In general, N resistors connected in series is equivalent to a resistor with resistance:

N

iieq RR

1

where Ri is the resistance of ithresistor connected in series

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Series Resistance CircuitVoltage divider

1

11

1 2R s

RV IR VR R

2

22

1 2R s

RV IR VR R

iR RViand,

In general, voltage across the resistor Ri of N series connected resistors branch is given by:

sN

jj

iR V

R

RVi

1


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Series Resistance CircuitVoltage divider : Create different reference voltages by selecting appropriate resistors.Question:

Given a 12 V battery, is it appropriate to use the voltage divider to directly create a voltage source or supply of say, 5 V, for a device directly?

Vout

Vin=12VR1

R2

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Parallel Resistance Circuit

Constitutive equation, Ohm’s law:I1 = Vs/R1

I2 = Vs/R2

Applying KCL at node A:I - I1 - I2 = 0

=> 1 2

1 2

s sV VI I IR R


?

? ?

Circuit: R1 and R2 connected in parallel with a voltage source Vs

To find: I, I1 and I2 (need three equations to solve)

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eq

ss

ss

RV

RRV

RV

RVI

2121

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i.e. the two resistors can be replaced by a single resistor Req of value =

1 2

1 2

1 2

11 1

R RR R

R R

Vs

I

Req

+

-


Since where1 2

1 1 1

eqR R R

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In general, N resistors connected in parallel is equivalent to a resistor of resistance, Req , given by:

1

1 1N

ieq iR R

where Ri is the resistance of ith resistor

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Parallel Resistance CircuitCurrent divider

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1 1 2

SV RI IR R R

12

2 1 2

SV RI IR R R


1 2

1 2S eq

R RV IR IR R

That is, and1 2I R 2 1I R

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Series Capacitors/Inductors Circuit

By applying KVL, it can be shown that:

1

1 1N

ieq iC C

In general

1

N

eq ii

L L

In general

L1 L2

21 LLLeq

C1 C2

21

21

CCCCCeq

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Parallel Capacitors/Inductors Circuit

By applying KCL, it can be shown that:

1

N

eq ii

C C

In general

1

1 1N

ieq iL L

In general

C2

C1

21 CCCeq 21

21

LLLLLeq

L2

L1

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Principle of Superposition

Apply to linear circuits (for example, those which consist of multiple ideal sources and passive elements)For a linear system:

SystemInput, u1 Output, y1

Input, u2 Output, y2

Input

au1+ bu2

Output

ay1+by2

where a and b are some constantsSystem

System

Given:

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Principle of Superposition

If more than one independent voltage or current source is present in any given circuit, each branch voltage and current is the sum of the independent voltages or currentswhich would arise from each voltage or current source acting individually when all the other independent sources are zero*.

*To zero a source, current source replaced by open circuit and voltage source by short circuit.

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Example: Superposition

Ans: I=II1+II2+IV

=I1-I2

RI2 I1

V

I

RI2

II2

RV

IV

RI1

II1

To find I

(a)(b)

(c)

?

II1: Portion of Iarising from I1II2: Portion of Iarising from I2IV: Portion of Iarising from V

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Equivalent Circuits

Equivalent circuits

Portion of circuit to be replaced with an equivalent circuit

Equivalent circuit

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Equivalent Circuits

Equivalent circuit - one that has identical V-I relationship as viewed from a given pair of terminals

Equivalent circuit

V V

Portion of circuit to be replaced with an equivalent circuit

II

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Thévenin EquivalentThévenin’s theorem: Given a pair of terminals in a linear resistive network, the network may be replaced by an independent voltage source VOC in series with a resistance RTH.:

I

VVOC

RTHI

V

VOC - Thévenin voltage RTH - Thévenin resistance

Vin

R1

R2

Linear resistivenetwork

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Thévenin Equivalent (procedures)

Thévenin voltage - open circuit voltageacross the terminals.Thévenin resistance – equivalent resistance across the terminals when independent voltage sources are shorted and independent current sources are replaced by open circuit.(Applicable only if there is no dependent sources in the circuit)

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Example: Thévenin Equivalent

Vin

R1

R2

21

2

RRRVV inoc

Find VOC by voltage dividerformula,

A

B

Find the Thevenin equivalent circuit as seen from terminals A and B

Solution:

VOC

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R1

R21 2

1 21 2

||THR RR R R

R R

Find RTH across the terminals A & B after replacing the voltage source with a short circuit:

A

B

Solution (cont):

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Thévenin Equivalent:

VOCRTH

+Vin

R1

R2

A

B

A

B

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Norton EquivalentNorton equivalent: Linear resistive network can be replaced by an independent current source ISC and Thevenin resistance RTH in parallel with the source.

ISCRTH V

ILinear

resistive network

I

V

ISC - Norton current RTH - Thevenin resistance

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Norton Equivalent (procedures)ISC - current that would flow through the terminals if they were shorted together.To convert to Thevenin equivalent circuit, we can compute Thevenin voltage VOC as follows:

ISCRTH V

I

VOC

RTHI

V

Thevenin equivalentNorton equivalent

THSCOC RIV

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Example: Find Norton Equivalent circuit across A and B

First, find the short circuit current (ISC) across AB:

Vo Io

R1

R2

A

B

Applying KCL at node X:

1

0o Xo SC

V V I IR

Vo Io

R1

R2

A

BISC

oo

SC IRVI

1

X

(since VX = 0)

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Example - Norton Equivalent (cont.)Next, find the Thevenin resistance:

Replace voltage source with short circuit and current source with open circuit and inspect the equivalent resistance across the terminals. R1

R2

A

B

RTH= RAB= R1

Thus the Norton equivalent circuit would be:

R1

A

Bo

o IRV

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Practical ConsiderationsBreadboard

For prototyping circuits

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Points are internally connected as shown

Instruments for powering and making measurements in circuits

Practical Considerations

Impedance (AC concept of resistance) matchingMaximum power transmission

In order to transmit maximum power to a load from a source, the load’s impedance should match the source’s impedance (see textbook for proof).

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For example, when you select speakers, the audio amplifier output impedance should be considered for maximum power transmission to a load (speaker).

Practical Considerations

GroundingVery important to provide a common ground defining a common voltage reference among all instruments and voltage sources used in a circuit or system.

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Not to confuse the signal ground with the chassis ground. The chassis ground is internally connected to the ground wire on the power cord and may not be connected to the signal ground (COM).

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Review of Electrical Circuits Theory

IntroductionBasic Electrical ElementsKirchhoff’s LawsPrinciple of Superposition Equivalent CircuitsPractical Considerations

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Circuit Analysis Bw