ENGR 202 Electrical Fundamentals II

First, where have we been and where are we going!

ENGR 201 ---DC Circuits and Analysis Techniques ENGR 202 ---Apply AC Circuit Analysis to solve

engineering problems

ENGR 203 Develop mathematical tools (namely the Lapace and Fourier Transform) to simplify more complex circuits and analyses

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Quick Review of ENGR 201

st1 Ohms Law IRV = (constant (dc), V & I upper case for dc, Ave, rms)

iRv = (time varying (ac), v & i lower case) nd2Kirchoffs Laws (What circuit analysis is based on) 1. Kirchoffs Current Law (KCL) = outin '' sisi , cba iii += or 0entering' = si , 0= cba iii or 0leaving' = si , 0=++ cba iii (We use this for Node Voltage)

biai

ci

2

Quick Review of ENGR 201

2. Kirchoffs Voltage Law (KVL) The sum of the voltages ( sv ' ) around any closed loop equals zero. (Defn of Loop Closed path - Begin at a given node and trace a path through the circuit back to the original node without passing through any intermediate node more than once) (resistors) (sources) e.g., = risesdrops VV

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Quick Review of ENGR 201The systematic Approaches to Circuit Analysis

(valid for dc and ac analysis which we simplify with phasors)

1. Circuit Reduction

combining elements (R1 + R2) in series (share the same current) / parallel (share the same voltage) [(R3*R4)/(R3 + R4)], Y transformations (

3=

ZZ y ), Source transformations (above), Thev. and Norton Equivalent

(R above is replaced by Req), for when only interested in terminal behavior, e.g., with a power-supply, we know there is a variety of circuitry in there, but we mainly want to know what voltage, current i.e., power it can supply

a

bRVV

R

R

b

a

1V 2V

sV sI

1R 2R

3R 4R

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Quick Review of ENGR 2012. Circuit Recognition (of common configurations)

voltage / current dividers, bridges

11

1 IRV

= , parallel comb.

[(R1*R2)/(R1 + R2)]

Vi

R1

Rout+Vo

-

I

R2R1I1

II2

I V1

V1/R1 V2/R2 )(,

11 out

i

out

outio RR

VIRR

RVV+

=+

= 21

12

21

2111

21

211

)(,)(,RR

RIIRR

RIIRIRR

RIRV+

=+

==+

=

3. Node Voltage Method (need 1en node voltage (KCL) equations) en essential node where 3 or more circuit elements join

We write KCL at the essential nodes ( en ) minus the reference node. (typically end up with fewer equations than for Mesh Current Method) (end up with 2 equations in the example above) -If circuit not reduced

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Quick Review of ENGR 2011V 2V

sV sI

4. Mesh Current Method ( need )1( ee nb mesh current (KVL) equations)

eb essential branch that connects essential nodes without passing through an essential node.

We write KVL for each mesh current loop (need to solve 3 equations)

5. Superposition ---kill( short voltage source, open current source) all independent sources but one, and sum the response for each.

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3

4 5

5 2

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Ch. 9: Sinusoids and Phasors This chapter will cover alternating current (AC). A discussion of complex numbers is included

prior to introducing phasors. Applications of phasors and frequency domain

analysis for circuits including resistors, capacitors, and inductors will be covered.

The concept of impedance and admittance is also introduced.

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Alternating Current (AC)AC circuits are used for generation, transmission, distribution and utilization of electric energy, also the dominant form of signals in communications. So time varying, we deal mostly with sinusoidal excitation (input) and response (output). Favored by Edison favored by Westinghouse and Tesla After a bitter battle in electrical power generation and transmission, ac won over because: (in 1890 ) i) It is easier to generate; ii) It is easier to change the voltage (through a transformer) for efficient transmission of electric power. I.e., higher voltage, lower RI 2 losses, more suitable for higher power motors.

"dc"

IV or

t

iv or+

""Sinusoidal

ac

t

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Alternating Current (AC)Disadvantage: ac cannot be stored as easily as dc (no ac equivalent of batteries or capacitors). Why do we use sinusoidal time variation? - Sin and cos are simple periodic, continuous functions, easy to deal with

(derivative and integral of a sinusoid is also a sinusoid, see below) - Cosine is sine advanced by 90 tt

dtd cos)(sin = ; tdtt

cos1sin = ;

- Any periodic waveform (no matter how complicated) can be represented

as a sum of sinusoids of different frequencies (Fourier Series in 203) (power supply example)

- From the superposition principle: we find the circuit response for each individual frequency and sum the response to get the full behavior

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Steady State Sinusoidal Analysis f=1/T Hz, =2f (rad/sec) (angular freq.) Most often we are interested in the steady-state response of a circuit, which exists for t , rather than the transient, which dies off as t . The time depends on the time constant RL= (smaller dies away quicker). What we find is that the sinusoidal excitation (input/source) i.e.,

)cos( += tVv m produces a steady-state (s.s.) sinusoidal response, i.e., )cos( += tIi m of the same frequency ( f 2= ). Theta () is the phase shift

due to capacitors or inductors, etc. Our ac circuit problems are a matter of determining:

1. magnitudes (i.e., mI )

2. phase angles (i.e., )(tan 1RL = for RL circuit)

v iR

L

mV

)(ti

T

)(tv

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Characteristics of a Sinusoidal Function

)cos( += tVv m ; )cos( += tIi m ; specified by 3 parameters: Vm = amplitude = angular freq. (rad/sec) , = phase angle (or troughs) - Period (T ) ssec = the time between successive peaks of the same sign - (The function repeats itself every T seconds, Period is inversely related to Frequency 1f

T= , cycles per second ( HzHertz, )

ex: U.S. electric utility frequency fHz =60

msT 67.16601period ===

The angular equivalent of the period is 360 or 2 radians the angular frequency is =

(sec))(2

Trad , (

Tf 1= ),

which we refer to as omega: sec/22 radfT

==

v

tmV

mV

mV

mV

TPeriod

2T

=

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More generally, we need to account for relative timing of one wave versus another.

This can be done by including a phase shift, (degrees or radians)

Consider the two sinusoids below

( ) ( ) ( )1 2sin and sinm mv t V t v t V t = = +

Characteristics of a Sinusoidal Function

If is positive, sinusoid shifts to left If is negative, sinusoid shifts to right.

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Sinusoids If two sinusoids are in phase, then this

means that the reach their maximum and minimum at the same time.

Sinusoids may be expressed as sine or cosine.

The conversion between them is:( )( )( )( )

sin 180 sin

cos 180 cos

sin 90 cos

cos 90 sin

t t

t t

t t

t t

=

=

=

=

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Complex Numbers A powerful method

for representing sinusoids is the Phasor: an effective way of dealing with ac circuit problems, and much simpler than dealing with sinusoidal quantities.

Phasor: a complex number that carries the magnitude and phase angle information of a sinusoidal function.

V=+= mm VtVv )cos( in phasor form (angular or polar) (bold in text) Vector in the Complex Plane We can put this in rectangular form as follows:

cosmVa = , sinmVb = , (from Eulers identity) where ba j+=V in rectangular form j is complex 90 operator that comes into play because of the phase angles introduced by sinusoidal sources, inductors and capacitors, etc. ( 12 =j ) then 22 baVm += , a

b1tan=

Re

Im

a

b

1

mV

cos sinje j =

-The length of the vector is the amplitude of the sinusoid.-The vector,V, in polar form, is at an angle with respect to the positive real axis.

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Complex Numbers A powerful method for representing sinusoids is the

Phasor: an effective way of dealing with ac circuit problems, and much simpler than dealing with sinusoidal quantities.

Phasor: a complex number that carries the magnitude and phase angle information of a sinusoidal function.

A complex number z can be represented in rectangular form as:

It can also be written in polar or exponential form as:

z x jy= +

jz r re = =

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Complex Numbers

Again, the different forms can be interconverted.

Starting with rectangular form, one can go to polar:

Likewise, from polar to rectangular form goes as follows:

2 2 1tan yr x yx

= + =

cos sinx r y r = =

In general, a complex number z can be represented in rectangular form as:

And it can be written in polar or exponential form as:

z x jy= +

jz r re = =

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Complex Numbers The following mathematical operations are

important

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 11 2

2 2

*

1 1 / 2

j

z z x x j y y z z x x j y y z z r r

z r z rz r z r

z x jy r re

+ = + + + = + = +

= = =

= = =

Addition Subtraction Multiplication

Division Reciprocal Square Root

Complex Conjugate

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Sinusoid-Phasor Transformation

Here is a handy table for transforming various time domain sinusoids into phasor domain:

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Phasor Relationships for Resistors

Each circuit element has a relationship between its current and voltage.

These can be mapped into phasor relationships very simply for resistors capacitors and inductors.

For the resistor, the voltage and current are related via Ohms law.

As such, the voltage and current are in phase with each other - see the Complex Plane/Phasor Diagram.

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Phasor Relationships for Inductors

Inductors on the other hand have a phase shift between the voltage and current.

In this case, the voltage leads the current by 90. (ELI the ICE man)

Or one says the current lags the voltage, which is the standard convention.

This is represented on the phasor diagram by a positive phase angle between the voltage and current.

is the angular frequency 20

Phasor Relationships for Capacitors

Capacitors have the opposite phase relationship as compared to inductors.

In their case, the current leads the voltage.

In a phasor diagram, this corresponds to a negative phase angle between the voltage and current.

21

Overview of Representing Complex Numbers in the Complex Plane

Re

Im

x

y

VV

= VV with angular frequency back in sinusoidal form

)cos( += tVv

)cos( += tVv Polar: = VV Rectangular: YX j+=V ; sin,cos VYVX == so related back to polar: 22 YXV +=

)(1tanXY

=

22

Voltage current relationships

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(Complex) Impedance

We now use complex arithmetic for all of the Systematic Approaches developed for dc analysis. For example, we introduce the concept of impedance Z , and Ohms law becomes:

ZIV = , where )(j)( XRZ += R =real (resistive) component X =imaginary (reactive) component

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Impedance In frequency domain, the values

obtained for impedance are only valid at that frequency.

Changing to a new frequency will require recalculating the values.

As a complex quantity, the impedance may be expressed in rectangular form (real part is the resistance R, imaginary component is the reactance, X).

The impedance of capacitors: inductors:

reactance )(= LX L )(1 =

CX C

( ) j( )L cZ R X X = + +25

Impedance and Admittance

When the overall reactance X is positive, we say the impedance is inductive, and capacitive when it is negative.

The impedance of a circuit element is the ratio of the phasor voltage to the phasor current.

Admittance is simply the inverse of impedance.

orVZ V ZII

= =

( ) j( )L cZ R X X = + +

reactance )(= LX L )(1 =

CX C

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Impedance and Admittance Admittance, being the reciprocal of the impedance,

is also a complex number. It is measured in units of Siemens The real part of the admittance is called the

conductance, G The imaginary part is called the susceptance, B These are all expressed in Siemens or (mhos) The impedance and admittance components can be

related to each other:

2 2 2 2R XG B

R X R X= =

+ +

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Impedance and Admittance

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Kirchoffs Laws in Frequency Domain

A powerful aspect of phasors is that Kirchoffs laws apply to them as well.

This means that a circuit transformed to frequency domain can be evaluated by the same methodology developed for KVL and KCL.

One consequence is that there will likely be complex values.

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Impedance Combinations Once in frequency domain, the impedance

elements are generalized. Combinations will follow the rules for

resistors:

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Impedance Combinations Series combinations will result in a sum of

the impedance elements:

Here then two elements in series can act like a voltage divider

1 21 2

1 2 1 2

Z ZV V V VZ Z Z Z

= =+ +

1 2 3eq NZ Z Z Z Z= + + + +

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Parallel Combination Likewise, elements combined in parallel will

combine in the same fashion as resistors in parallel:

1 2 3

1 1 1 1 1eq NZ Z Z Z Z= + + + +

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Admittance Expressed as admittance, though, they are

again a sum:

Once again, these elements can act as a current divider:

1 2 3eq NY Y Y Y Y= + + + +

2 11 2

1 2 1 2

Z ZI I I IZ Z Z Z

= =+ +

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ENGR 202 Electrical Fundamentals IIQuick Review of ENGR 201Quick Review of ENGR 201Quick Review of ENGR 201Quick Review of ENGR 201Quick Review of ENGR 201Ch. 9: Sinusoids and PhasorsAlternating Current (AC)Alternating Current (AC)Steady State Sinusoidal AnalysisCharacteristics of a Sinusoidal Function Characteristics of a Sinusoidal Function SinusoidsComplex NumbersComplex NumbersComplex NumbersComplex NumbersSinusoid-Phasor TransformationPhasor Relationships for ResistorsPhasor Relationships for InductorsPhasor Relationships for CapacitorsOverview of Representing Complex Numbers in the Complex PlaneVoltage current relationships(Complex) ImpedanceImpedanceImpedance and AdmittanceImpedance and AdmittanceImpedance and AdmittanceKirchoffs Laws in Frequency DomainImpedance CombinationsImpedance CombinationsParallel CombinationAdmittance