ETEN3002 Electronics 1 Background(1)

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Electronics Design (ETEN3002) Dr John Siliquini

1: Background Fundamentals

Notes:

- These notes are provided to assist your education. At a minimum you shouldaugment these notes, as appropriate.

- Your education is your responsibility. Your future will be affected by the extent thatyou develop independent learning skills, independent problem solving skills, theability to critically assess material and the skill of paying appropriate attention todetail. The development of such skills is facilitated by individual study, reflection andsignificant effort.

- The expected standard: You are expected to understand the presented theory in itsown right. Attempting to solve relevant problems will give you feedback on yourunderstanding of the theory.

- With respect to problem solving the expected standard is: First, you know why youranswer/solution is correct. Second, your answer is in the best form to facilitateunderstanding/clarity etc.

- Clarity follows from rigour.

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1.0 Background

References for Circuit Theory

Alexander, C. K. & Sadiku, M. N. O., Fundamentals of Electric Circuits, McGrawHill, 2000, 2007, 2009 ch 15.

Chua, L. O., Desoer, C. A. & Kuh, E. S. Linear and Nonlinear Circuits, McGrawHill, 1987, ch. 10.

Thomas, R. E. & Rosa, A. J., The Analysis and Design of Linear Circuits, Wiley,2004, ch 9, 10, 12.

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2.0 Background - Mathematics

Laplace Transform

Definition: Laplace Transform

For a real signal, defined on the interval , theLaplace transform of , denoted , is, by definition

is defined for values of its argument where the integral isfinite.

x:R R 0 ,( )x X

X s( ) x t( )e st td0

= s C

X s

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.

Table 1: Laplace Transforms

x t( ) X s( )u t( ) 1 se

ptu t( ) 1 p

1 s p+------------------

wct[ ]u t( )sin wcs2

wc2

+-----------------

wct[ ]cos u t( ) ss2

wc2

+-----------------

tdd

x t( ) sX s( ) x 0( )

x ( ) d0

t

X s( )

s-----------

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Taylor Series

Reference: Grossman, S. I., Multivariable Calculus.

Definition: Taylor Series: One Dimensional Case

If is continuous, and its first derivatives are contin-uous, then the order Taylor series of , based on a point , is

A first order Taylor series, i.e.

is a linear approximation to the function based on the point

and is such that the slope of the linear approximating functionequals the slope of at the point .

f:R R n 1+nth f xo

fT n x,( ) f xo( ) x xo( ) xdd f x( )

x xo=

x xo( )n

n!----------------------

xn

n

dd f x( )

x xo=

+ + +=

fT 1 x,( ) f xo( ) x xo( ) xdd f x( )

x xo=

+=

f xo

f xo

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Definition: First Order Taylor Series for Two Dimensional Case

If is continuous, and its first order partial derivativesare continuous, then a first order Taylor series of , based on apoint , is

A first order Taylor series of a two dimensional function is a planeapproximation to the function at the point . The approxi-

f x( )

x

f xo( )

xo

fT 1 x,( )

f:R2 Rf

xo yo,( )

fT x y,( ) f xo yo,( ) x xo( ) x f x yo,( )

x xo=

y yo( ) y f xo y,( )

y yo=+ +=

xo yo,( )

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mation is such that the slope of the function at the point ,

and in the direction, matches the slope of the plane in the direction. The slope of the function at the point , and in the

direction, matches the slope of the plane in the direction.

xo yo,( )x x

xo yo,( )y y

y

f xo y,( )

xo

f x y,( )

x

f x yo,( )

yo

lines defining fT x y,( )

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3.0 Background - Circuit Theory

Basic Circuit Theory - Relationships for Basic Components

The following relationships apply for resistors, capacitors andinductors assuming causal signals (signals commencing at ). t 0=

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+

v t( )-

R

i t( )v t( ) Ri t( )= V s( ) RI s( )=

+

v t( )

-

Ci t( ) v t( ) v 0( ) 1C---- i ( ) d

0

t

+= V s( ) v 0( )s----------I s( )sC---------+=

i t( ) Ctd

dv t( )=

+

v t( )

-

Li t( ) i t( ) i 0( ) 1L--- v ( ) d

0

t

+= I s( ) i 0( )s---------V s( )sL-----------+=

v t( ) Ltd

d i t( )=

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Notation:

Usually Laplace transformed variables are used and the argument is omitted:

s

+

V

-

LI+

V

-

CI+

V

-

R

I


Definition: Impedance

The impedance of a component is the ratio

assuming zero initial conditions.

The impedance of a resistor, capacitor and inductor, respectively,are:

Z s( ) V s( )I s( )-----------=

Z s( ) R= resistorZ s( ) 1

sC------= capacitor

Z s( ) sL= inductor


Rationale for Impedance Definition

Consider the relationships

For the case where it follows, for the resistor andinductor cases, that

For the capacitor case where it follows that

The sinusoidal steady state is consistent with .

v t( ) Ri t( )= i t( ) Ctd

dv t( )= v t( ) L

tdd i t( )=

i t( ) Aest=

v t( ) ARest= v t( )i t( )--------- R=

v t( ) AsLest= v t( )i t( )--------- sL=

v t( ) Aest=

i t( ) AsCest= v t( )i t( )---------1

sC------=

s jw=


4.0 Kirchhoffs Laws (KCL & KVL)

The two laws which allow circuits to be analysed so that the volt-ages and currents at all points in a circuit can be determined areKirchhoffs current law (KCL) and Kirchhoffs voltage law(KVL).

Kirchhoffs Current Law

Basis: A node, by definition, cannot store charge. Conservation ofcharge then implies that the charge entering a node must equal thecharge leaving a node.

Implication: Consider the charge flowing into a node during a time from to :t t t t+


At the node charge conservation implies

Thus

i1 t( )i3 t( )i2 t( )

q1q2

q3component 1

component 2

component 3

q1 q2 q3+ + 0=

q1t-----

q2t-----

q3t-----+ + 0= i1 t( ) i2 t( ) i3 t( )+ + 0=


Theorem: Kirchhoffs Current Law

The algebraic sum of currents entering a node, or leaving a node, iszero, i.e. with currents entering, or leaving, a node it

is the case that

Convention: Analysis is simplified if the sum of the currents leav-ing a node is used.

Example: Find the current :

N i1 t( ) iN, ,

ii t( )i 1=

N

0=

i4 t( )

i1 t( ) 0.1A=

i2 t( ) 0.2A=i3 t( ) 0.4A=

i4 t( )

KCL i1 t( ) i2 t( ) i3 t( ) i4 t( )+ + + 0= i4 t( ) i1 t( ) i2 t( ) i3 t( ) 0.1( ) 0.2 0.4 0.5A= = =


4.1 Kirchhoffs Voltage Law (KVL)

Basis: Conservation of energy of a charge as it moves around aclosed loop.

The energy required to move a positive charge through a poten-tial of volts is Joules:

Consider a closed loop of an electrical circuit at a time :

qv21 qv21

E

+

+- v21

q

t

v1 t( )+

-

v4 t( )+

-v2 t( ) +- v3 t( ) +-


For a positive charge moving around this loop, at time , conser-vation of energy implies:

Theorem: Kirchhoffs Voltage Law

The algebraic sum of voltages around a closed loop is zero, i.e. with entities in a closed loop it is the case that

Example

Find in the following circuit:

KVL implies:

q t

qv1 t( ) qv2 t( ) qv3 t( ) qv4 t( )+ + + 0= v1 t( ) v2 t( ) v3 t( ) v4 t( )+ + + 0=

N

vi t( )i 1=

N

0=

v2 t( )


15V+

-

10V+

-

v2 t( ) +- 2V +-v4 t( )

+

-

v3 t( ) +-

v1 t( )+

-

v1 t( ) v2 t( ) v3 t( ) v4 t( )+ + + 0 = 15 v2 t( ) 2 10( )+ + + 0= v2 t( ) 10 15 2 7V= =


Note: To apply KVL it is easier to use the form:

This form arises by traversing the loop in a set direction and byassociating the sign (+ or -) first encountered with each voltage.Consider the following example:

v 1 t( ) v2 t( ) v3 t( ) v4 t( ) 0=

v1 t( )+

-

v4 t( )+

-v2 t( ) +- v3 t( ) +-

v1 t( )+

-

v4 t( )+

-v2 t( )+ - v3 t( )+ -

KVL v1 t( ) v2 t( ) v3 t( ) v4 t( )+ + 0=


4.2 KCL & KVL: Laplace Transformed Case

Theorem: Kirchhoffs Current Law

The algebraic sum of the Laplace transform of the currents enter-ing a node, or leaving a node, is zero, i.e. with currents

entering, or leaving, a node it is the case that

Theorem: Kirchhoffs Voltage Law

The algebraic sum of the Laplace transform of the voltages arounda closed loop is zero, i.e. with entities in a closed loop it is thecase that

Ni1 t( ) iN, ,

Ii s( )i 1=

N

0= ii t( ) Ii s( )

N

Vi s( )i 1=

N

0= vi t( ) Vi s( )


4.3 Further Examples

As a review of Kirchhoffs current law (KCL) and Kirchhoffsvoltage law (KVL) consider the following circuits and subsequentanalysis:

KCL implies

Thus

L

v1 t( )

CR

i1 t( )

i3 t( )i2 t( ) i4 t( )

v4 t( )v3 t( )v2 t( )

i1 t( ) i2 t( ) i3 t( ) i4 t( )+ + + 0= I1 s( ) I2 s( ) I3 s( ) I4 s( )+ + + 0=


Laplace transformation yields:

Notation: omitting the argument yields:

i1 t( )v1 t( ) v2 t( )

R------------------------------ C

tdd

v1 t( ) v3 t( )[ ]+ + +

i4 0( )1L--- v1 ( ) v4 ( )[ ] d0

t

+ 0=

I1 s( )V1 s( ) V2 s( )

R---------------------------------- sC V1 s( ) V3 s( )[ ]

i4 0( )s

------------V1 s( ) V4 s( )

sL----------------------------------+ + + + 0=

s

I1V1 V2

R------------------- sC V1 V3[ ]

i4 0( )s

------------V1 V4

sL-------------------+ + + + 0=


KVL implies

Thus

Laplace transformation yields:

L

CR iR t( )

iL t( )

iC t( )vR t( )

+- vS t( )

+ - vC t( )+ -

vL t( )+

-

vS t( ) vR t( ) vC t( ) vL t( )+ + + 0= VS s( ) VR s( ) VC s( ) VL s( )+ + + 0=

vS t( ) iR t( )R vC 0( )1C---- iC ( ) d0

t

L tdd iL t( )+ + + + 0=


Notation: omitting the argument yields:

VS s( ) IR s( )RvC 0( )

s--------------

IC s( )sC

------------- sLIL s( )+ + + + 0=

s

VS IRRvC 0( )

s--------------

ICsC------ sLIL+ + + + 0=


4.4 Notation

Transistor circuits are powered by DC voltages. The followingnotation is used to represent a DC voltage source:

This notation is used for convenience and means the following:

Similar notation is also used for node voltages.

VCC

VCC+

-VCC


5.0 Solving Circuits

Consider a linear time-invariant circuit and assume zero initialconditions. Systematic application of Kirchhoffs current law andKirchhoffs voltage law underpins, respectively, nodal analysis andmesh analysis of circuits. Consider a general circuit with nodes:

Applying Kirchhoffs current law at each of the nodes yields thefollowing matrix of equations

N

VS

V1 VN 1 VN

+-

V2

N


where is the negative of the admittance between the and

node for , is the admittance of the elements connected to

the node and is the admittance between the source and the

node.

Matrix inversion yields

or

Y11 Y12 Y1NY21 Y22 Y2N

YN1 YN2 YNN

V1V2

VN

Y1VSY2VS

YNVS

=

Yij ith jthi j Yii

ith Yiith

V1V2

VN

Y11 Y12 Y1NY21 Y22 Y2N

YN1 YN2 YNN

1Y1VSY2VS

YNVS

=


for appropriately defined .

Hence, any of the desired node voltages can be determined. If therelationship between the node voltage and is required then

it follows that

Hence, for any given input signal , whose Laplace transform

is , it follows that the Laplace transform of the output signal

is

V1V2

VN

Z11 Z12 Z1NZ21 Z22 Z2N

ZN1 ZN2 ZNN

Y1VSY2VS

YNVS

=

Zij

Nth VS

VNVS------- ZN1Y1 ZN2Y2 ZNNYN+ + +=

vS t( )VS s( )

VN s( ) ZN1Y1 ZN2Y2 ZNNYN+ + +[ ]VS s( )=


Taking the inverse Laplace transform yields .

Alternatively, mesh analysis could be used to generate a matrix ofequations for the loop currents.

vN t( )


6.0 Small Signal Characteristics of Amplifiers

In general, the following small signal characteristics of an ampli-fier are of interest:

a) Forward Transfer Function:

At low frequencies the transfer function yields the low frequencygain:

RS

vS+ +

vo

-

RL

+

vi

-

ii io

H s( ) Vo s( )VS s( )--------------=


b) Input Impedance:

At low frequencies the input impedance is usually resistive consist-ent with

c) Output Impedance:

AVvovS-----=

Zin s( )Vi s( )Ii s( )-------------=

Rinviii----=

Zo s( )Vo s( )Io s( )--------------

VS s( ) 0==


At low frequencies the output impedance is usually resistive con-sistent with

Note: The definition for the output impedance is consistent with anideal voltage source with a voltage being placed at the output.

Consequently the resistance does not affect the output imped-

ance.

Rovoio-----

vS 0=

=

vo

RL


7.0 Review of System Theory

Consider a causal linear time invariant circuit which assumed tobe stable such that its impulse response is integrable, i.e.

Definition: System Transfer Function

The transfer function is defined by the one-sided Laplace trans-form according to

In terms of notation it is common to write

h t( )

h t( ) td0

HM f( ) H s( ) s j2pif= A1 j2pif p1+-----------------------------

A

14pi

2f2p12

--------------+

---------------------------= = =

A

1f2f12----+

------------------ f1p12pi------= =

H f( ) arg H s( ) s j2pif=[ ] argA

1 j2pif p1+-----------------------------= =

arg A[ ] arg 1 j2pif p1+[ ] arg A[ ] 2pif p1[ ]atan= =arg A[ ] f f1[ ]atan=


HM f( )

fH f( )

f

log-log graph

log-linear graph

asymptotic approxA

f1f1

pi 4pi 2 assumption: arg A[ ] 0=


9.0 Linear Systems: Basic Definitions

Consider a linear system with a transfer function defined accord-ing to

Definition: Poles and Zeros

The roots of are the called the zeros of the transfer function.The roots of are called the poles of the transfer function.

H s( ) N s( )D s( )-----------=

N s( )D s( )


9.1 Low Pass Systems

Many circuits in electronics are low pass systems.

Definition: Low Pass System

A system with a transfer function which is of the form

where , and for , is a system with a

low pass transfer function, i.e. a low pass system.

The zeros of this transfer function are . The poles of

this transfer function are .

Definition: Low Frequency Gain

For a low pass system, with transfer function defined above,the low frequency gain, or simply the gain, is:

H s( )

H s( ) k 1 s z1+( ) 1 s zM+( )1 s p1+( ) 1 s pN+( )

--------------------------------------------------------------=

M N< Re pi[ ] 0> i 1 N, ,{ }

z1 zM, ,

p1 pN, ,

H s( )G H 0( ) k= =


Definition: Bandwidth

The bandwidth of low pass system with a transfer function

and for the case where (assuming

and ), is the frequency

where the magnitude of the transfer function has dropped to of the low frequency gain value, i.e.

H s( ) k 1 s z1+( ) 1 s zM+( )1 s p1+( ) 1 s pN+( )

--------------------------------------------------------------= M N< Re pi[ ] 0>,

Re p1[ ] Re z1[ ]>z1 z2 zM p1 p2 pM f3dB

1 2

f3dB f: HM f( )HM 0( )----------------- 1

2-------= =

f

HM f( )HM 0( )

HM 0( )2

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

f3dB


Interpretation of Bandwidth: The bandwidth of a system is ameasure of the information processing capability of the system.Needless to say, in the information age, the bandwidth of a systemis very important.

Notation: the notation is also used for bandwidth.

Bandwidth for a Single Pole Transfer Function

For a system with a single pole transfer function

the bandwidth, , is

This result is simple to prove - a simple application of the defini-tion of bandwidth - see Exercise 5.

BW

H s( ) k1 s p1+---------------------= p1 0>

BW

BWp12pi------=


9.2 Rise Time and Fall Time

The rise and fall time of a system are defined for a step input into asystem.

Definition: Rise Time

The rise time is the time taken for a signal to go from 10% aboveits lower stable level to 90% of its upper stable level.

Definition: Fall Time

The fall time is the time taken for a signal to go from 90% of itsupper stable level to 10% above its lower stable level.

The definitions of rise time and fall time of a signal are illustratedbelow:


rise time

fall time

upper steady state level

lower steady state level

100%

90%

10%

0%


10.0 Analysis vs Simulation

Simulation of circuits is facilitated by software packages such asPSPICE.

Analysis of circuits is facilitates by software packages such asMaple/Mathematica.

Note: in general, simulation of a circuit will give specific, not gen-eral, information about a circuit. The information given dependson the parameter values chosen. When the parameter valueschange the simulation needs to be redone and new informationgenerated.

In contrast, analysis leads to general results. Such results facilitateoptimization of performance etc.


11.0 Review of Basic Electromagnetics

The basis of electrostatics is the existence of forces betweencharges:

According to Coulombs formula the magnitude of the force thatcharge exerts on charge equals the magnitude of the force

that exerts on with the magnitude of the force being given

by:

Here is the distance between the two charges and is the per-

mittivity of free space .

rq1 q2

q2 q1q1 q2

F 12

F 21

14pio------------

q1q2r2

-----------= = N

r o

8.85x1012 C2N 1 m 2( )


11.1 Electric Field

Consider the case where there exists a fixed charge, or fixedcharges, at various point is space. Consider the case of a testcharge of Coulomb being placed at an arbitrary point.

This test charge will experience a force consistent with Coulombslaw. The magnitude of this force will depend on the magnitude ofthe test charge. A measure of this force is defined for the case of atest charge of 1 Coulomb and is called the Electric Field:

Definition: Electric Field

The electric field, denoted , at a point in space is the force perunit charge experienced at that point. That is

q

r

q1

q2

qF

r

( )

E


where is the test charge at the point being considered and isthe force that the test charge experiences.

E

Fq----= NC 1 or Vm 1

q F


11.2 Implications of Electrostatic Force

1. The first consequence of electrostatic force is the movement offree charge. The movement of free charge is consistent with cur-rent flow:

2. The second consequence of electrostatic force is that a charge ata given point has potential energy due to the effect of the forcesof other charges.

Definition: Potential

The potential at a set point is the potential energy that a chargeof one Coulomb would have at that point.

Thus for a charge of Coulomb at a point , which has a potentialenergy of Joules, the potential of the charge is

J

E

=

normalized forcecurrent density

conductivity(measure of resistance to force)

r

q rEP


Definition: Potential Difference

The potential difference, measured in volts, between two points isthe difference in potential at the two points. For the illustrationbelow

Interpretation: The potential difference is the gain in energy, perCoulomb of charge, as a charge moves from the initial point to thefinal point.

r

( ) EP r( )q--------------= JC1

or V Volt( )

V21 r2( ) r1( )=

+

- r1

( )

r2

( )

V21

Potential at point 2 with respect to point 1

r2

r1


11.3 Relationships Between Charge, Electric Field and Potential

a) The source of the electrostatic electric field is charge. ConsiderGausss law in differential form

where is the electric flux density is the charge density and

For a linear homogenous medium with a permittivity of it fol-lows that . For this case, and in one dimension, the relation-ship implies

D

=

D

i x

j y

k z

+ +=

D

E

= D

=

xdd EX x( )

x( )

-----------= EX x( ) ( )

----------- d

x

=


b) The relationships between electrostatic potential and the elec-trostatic electric field ,

for the one dimensional case, are:

where is the component of the field in the direction.

c) The potential difference between two points and (potential

at point with respect to point ) is, for the 1D case:

E

xEX xo( ) i xo( )

E

xo

EX x( ) xdd x( )= x( ) EX ( ) d

x

=

EX x( ) E x

x2 x1

x2 x1

V21 21 x2( ) x1( ) EX ( ) dx1

x2

E( ) i

( ) d

x1

x2

= = = =


12.0 Definitions for Resistance, Capacitance and Inductance

Definition: Resistance

The resistance between two equipotential surfaces is

Definition: Capacitance

The capacitance of an entity consisting of two separated conduc-tive plates

is defined as the charge required on the plates to establish a poten-tial difference between the plates of one volt, i.e.

R potential difference between surfacescurrent flow between surfaces

----------------------------------------------------------------------------------------------=

+++++++++++++++++----------------------------

QQ


Definition: Inductance

The self inductance of a loop carrying a current

is

C Q----=

I

I

B

B

L magnetic flux through loop generated by II---------------------------------------------------------------------------------------------------------------=


Capacitance and Inductance of 2 Wires

Using the above general definitions it can be shown that the capac-itance and inductance of two wires in free space

is

d wire radius ro=

Cpio

1dro----+ln

-------------------------= Fm 1

Lopi------ 1

dro----+ln= Hm 1


13.0 Device Modelling: Large and Small Signal Operation

It is the case that many electronic devices are operated in a mannersuch that two distinct current (or voltage) components can bedefined:

a) The first component is a DC component and this componentdetermines the region of operation of the device. This component isusually the dominant component.

b) The second component, usually a fraction of the level of the firstcomponent, is usually a time varying signal that contains informa-tion to be modified (usually amplified) by the device. This compo-nent results in small variations in the properties of the device; theproperties essentially being determined by the first DC compo-nent.

To illustrate the two components consider a diode which is drivenby a DC voltage source and a AC signal source as illustratedbelow:


First, consider the case where and . The diode is

operating on the vs curve as illustrated below. The diode

voltage equals the bias voltage and the resulting current flow,

, is determined by the diode characteristic curve.

+

-

VD

ID

VB

v t( )

VB for Bias Voltage

VB 0> v t( ) 0=ID VD

VBIB


Definition: Operating Point, Bias Point

The operating point, or bias point, of a device is the current-volt-age pair - for the diode circuit illustrated above - that is

determined by the DC conditions applied to the device.

For most electronic devices correct operation is dependent on acorrect, or appropriate, bias point being established. The majorexception are devices that are operated digitally (these devices canbe considered to be operating at either of one of two possible oper-ating points).

ID

VDVB

IBOperating Point

IB VB,( )


Second, consider the case of and . For this case

the voltage results in small changes, as illustrated below,around the operating point. In this diagram .

VB 0> v t( ) VBv t( )

i t( ) ID t( ) IB=

ID

VDVB

IB t

t

v t( )

i t( )

linear approx.


Note: for the case where the maximum magnitude of is smallrelative to the bias voltage the diode characteristic curve, as

given by the relationship, is close to being affine (linear)

around the bias point defined by .

Definition: Small Signal Operation (Linear Operation)

A device with a set bias point is said to be operating in a small sig-nal manner, i.e. small signal operation, if the signal variationaround the operating point is small enough such that linear opera-tion is valid.

Small signal operation is consistent with linear operation and asfar as the small input signal is concerned the device can bereplaced by an equivalent small signal model.

v t( )VB

ID VDVB IB,( )


14.0 Modelling of Power Supply

The following is a model for an ideal power supply:

Consistent with this model the output of the power supply is a setvoltage, independent of its load. Hence, there is no variation - nosmall signal variation - at the output of the power supply. As far assmall signals in a circuit are concerned the power supply terminalsact as a point where there is no variation, i.e. a ground point.

Thus: All power supply nodes are treated as ground points forsmall signal analysis of a circuit.

VCC+

-VCC


15.0 Modelling of Diode

15.1 Large Signal Model

The following is a large signal model for a diode that is valid at lowfrequencies. More complicated models that incorporate non-linearcapacitances are required to predict high frequency performance.

cathode

anode+

-

VD

ID

circuit symbol

+

-

VD

ID

model

ID IS eVD VT 1( )=


First order low frequency models for restricted regions of opera-tion are:

The following definition is used:

Definition: Thermal Voltage

The thermal voltage is defined as where is Boltz-

manns constant, is the absolute temperature, and is the elec-tronic charge.

At . It is usual to use a value of

(a value of is also commonly used).

+

-

VD

ID

forward bias

ID ISeVD VT=

+

-

VD

ID

reverse bias

ID 0=

VTkTq-------= k

T q

300K VT 0.025875= VT 0.0259=VT 0.026=


15.2 Small Signal Model for Diode

Assuming the diode is biased with an operating point defined by the following small signal models for a diode are valid:

In these models

VB IB,( )

rD CD CDif+

forward bias

CD

reverse bias

rDVTIB-------=

CD VD( )CD 0( )

1VDj-------

---------------------=VD j 2 > >

IB4IB3IB2IB1

0.3

Saturation Region

Forward Active Mode

Cutoff Mode


2. Cutoff occurs when the B-C junction is reverse biased and

is such that the collector current flow is negligible

3. In the forward active mode the relationship between the collec-tor current and the base current, as indicated on the above dia-grams is

4. The base current, and hence the collector current, is dependenton the B-E voltage via the relationship:

VBE

IC IB=

IBIS-----e

VBE VT= IC ISeVBE VT=

IB IC,

VBE0.7V


PNP Transistor

The characteristics of a pnp transistor are analogous to those ofthe npn transistor provided the E-C voltage is considered and thecurrent directions are taken as those defined for a pnp transistor:

The idealized characteristics are shown below:

IB

IC

PNP

IE

VEC

ICIB4

IB1

IB2

IB3

IB4 IB3 IB2 IB1> > >

IB4IB3IB2IB1

0.3


16.5 Small Signal Model for a BJT

The major interest is in the small signal model for a BJT when it isoperating in the forward active region. The npn case is considered.The small signal model for a pnp transistor is identical to the npnsmall signal model.

The small signal model utilizes the following relationships:

Fundamental Definitions

Definition: Transconductance

The transconductance, denote , of a BJT, when operating in the

forward active mode with a collector current , is defined

according to

gmIC VBE( )

gmIC VBE( )

VT---------------------= IC VBE( ) ISe

VBE VT=


Definition:

The resistance is defined as the ratio of the small signal change in thebase emitter voltage to the small signal current change when the base cur-rent is . Hence,

Fundamental Relationship

From the definitions for and , and the definition , itfollows that

rpi

rpi

IB VBE( )

rpi

VTIB VBE( )---------------------= IB VBE( )

IS-----e

VBE VT=

gm rpi IC IB=

gmrpi =


Small Signal Equivalent Model

A detailed analysis can be used to show the following:

1) A small change in to , results in a corresponding

linear change in the collector current, , according to

2) A small change in the B-E voltage from to ,

results in a corresponding linear change in the base current, ,

according to

Noting that the base-collector voltage has no influence on theserelationships the following model can be proposed consistent withthe changes in the base voltage, base current and collector current:

VBE VBE v+ic

icvVT-------IC VBE( ) gmv= =

VBE VBE v+ib

ibvVT-------IB VBE( )

vrpi------= =


Notes:

1. This model implies a base current of .

2. This model implies a collector current of .

3. In this model the terminal currents and voltages are independ-ent of the B-C voltage

4. The small signal parameters are dependent on the bias condi-tions. Specifically, depends on the bias voltage and

depends on .

+

gmvv

-

rpi

B C

E

ib ic

ibvrpi------=

ic gmv=

rpi IB VBE( ) gmIC VBE( )


5. The model (i.e. linear operation) is valid for the case where thechange in B-E voltage around the bias point of is much less

than the thermal voltage (25.9 mV).

6. The model is called the (low frequency) Hybrid-Pi model for aBJT.

VBEVT


Complete Small Signal Model - The Hybrid-Pi Model

The complete small signal model for a BJT (the model is suitablefor both the npn and pnp transistor), named the Hybrid-Pi model,is shown below:

In this model accounts for the diffusion and depletion capaci-

tance of the Base-Emitter junction and accounts for the deple-

tion capacitance of the Base-Collector junction.

+

gmvv

-rpi

B C

E

ib ic

Cpi

C

ro

rx

gmIC VBE( )

VT---------------------=

gmrpi =

CpiC

C VBC( )C 0( )

1VBC

j----------

m

-----------------------------= m 0.7 j 0.7,


It is common to define, for a BJT, the unity gain bandwidth

according to

Values of are usually given on a data sheet for a BJT. Once

and have been determined can be ascertained, e.g.

fT

fTgm

2pi Cpi C+( )-------------------------------=

fT Cgm Cpi

Cpigm2pifT------------ C=


The resistance accounts for the resistance in the narrow base

region:

Typical values for are in the range of 20-100 Ohms.

The resistance accounts for the non-zero slope of the vs

curves:

where is the early voltage. A typical value for is 100 .

Typical Parameter Values: See Exercise 11.

rx

np

n

E B C

rx is the resistance of this path

rx

ro IC VCE

ro

VAIC VBE( )---------------------=

VA ro k


16.6 Three Amplifier Structures

The three basic single stage transistor amplifiers are the commonemitter amplifier, the common collector amplifier and the commonbase amplifier:

VCC

Vo

VCCRC

REvS

+

CE

decoupling capacitor

common emitter amplifier


VCC

Vo

VCC

REvS

+

VCC

Vo

VCC

CC

vS+ IE

RC

common collector

common base


17.0 Physical Constants

Table 2: Fundamental Constants

Parameter Value

proton mass kg

- electron mass kg

- effective electron mass for Si

- effective hole mass for Si

- electronic charge

- Boltzmanns constant

- Plancks constant

- Permittivity of free space

1 Electron Volt

1.67x1027

m 9.11x1031

me* 1.18m

mh* 0.81m

q 1.6x10 19 Ck 1.38x10 23 JK 1

h 6.62x10 34 Jso 8.85x10

12 C2N 1 m 2

1.6x1019 J


- velocity of light in free space

Light wavelength - visible

Table 3: Constants for Silicon

Parameter Value at 300K

- Energy gap

- Permittivity

- Nominal Intrinsic Carrier Concentration

- Electron mobility (low doping levels)

- Hole mobility (low doping levels)

and - Diffusion Length

Table 2: Fundamental Constants

Parameter Value

c 3.0x108

ms1

400 to 700 nm

EG 1.12 eV

11.8o

ni 1010

cm3

n 1360 cm2V 1 s 1

h 460 cm2V 1 s 1

ln lp 2 m


Recombination time typically

Typical doping Density in CMOS

Table 3: Constants for Silicon

Parameter Value at 300K

1 sec

NAND

3x1015

cm3

1x1015

cm3


Appendix 1: Proof of Theorem: Interpretation of TF

The Laplace transform of is

As it follows that

With the assumptions it follows that can be written in the fol-lowing partial fraction form:

x t( ) A wct[ ]u t( )sin=

X s( ) Awcs2

wc2

+-----------------=

Y s( ) H s( )X s( )=

Y s( ) Awcs2

wc2

+-----------------H s( )=

H s( )

H s( ) h1s p1+--------------

hNs pN+---------------+=

100 R. M. Howard 2013

where is the degree of the denominator polynomial,

are the roots of this polynomial, and are the coefficients

that can be determined by the partial fraction expansion.

As the partial fraction form for

for appropriately defined constants and , it follows that a full

partial fraction expansion for is

It then follows that

N p1 pNh1 hN, ,

AwCs2

wc2

+-----------------

x1s jwc+-----------------

x1s jwc----------------+=

x1 x2

Y s( )

Y s( ) k1s jwc+-----------------

k1s jwc----------------

g1s p1+--------------

gNs pN+---------------+ + +=

101 R. M. Howard 2013

Hence

Using the inverse Laplace transform result

k1 s jwc+( )Y s( )s jwclim

Awcs jwc----------------H s( )

s jwclim= =

Awc2jwc

--------------H jwc( )A2j-------H jwc( )= =

k2 s jwc( )Y s( )s jwclim

Awcs jwc+-----------------H s( )

s jwclim= =

Awc2jwc-----------H jwc( )

A2j-----H jwc( )= =

Y s( ) A2j-----

H jwc( )s jwc+

------------------------H jwc( )s jwc------------------+

g1s p1+--------------

gNs pN+---------------+ +=

1s p+----------- e

pt

102 R. M. Howard 2013

it follows that

For the case, as assumed, of all roots of the denominator polyno-mial having negative real parts, consistent with

, it follows that

To simplify this expression can be written in polar form

according to

y t( ) A2j----- H jwc( )e

jwct H jwc( )ejwct+[ ]=

g1ep1t

gNepNt+ + + +

Re p1[ ] 0> Re pN[ ] 0>, ,

ySS t( ) y t( )t lim A

H jwc( )ejwct H jwc( )e

jwct2j-----------------------------------------------------------------------= =

H jwc( )

H jwc( ) H jwc( ) ej wc( )=

103 R. M. Howard 2013

To find the relationship between and consider the

definition of :

It then follows that

and that

or (this property is called Hermitian symme-

try).

H jwc( ) H jwc( )H s( )

H s( ) h t( )e st td0

=

H jwc( ) h t( )ejwct td

0

= H j wc( ) h t( )ejwct td

0

=

H* j wc( ) h t( )ej wct td

0

H jwc( )= =

H j wc( ) H* jwc( )=

104 R. M. Howard 2013

Hence

Thus

as required.

H j wc( ) H jwc( ) ej wc( )=

ySS t( ) AH jwc( )e

jwct H jwc( )ejwct

2j-----------------------------------------------------------------------=

A H jwc( ) ej wc( )

ejwct

ej wc( )

ejwct

2j------------------------------------------------------------------=

A H jwc( )e

j wct wc( )+[ ]e

j wct wc( )+[ ]2j-----------------------------------------------------------------------=

A H jwc( ) wct wc( )+[ ]sin=

105 R. M. Howard 2013

18.0 Exercises

The following exercises are provided to assist your education. It isexpected that you are proactive with respect to your education andare progressing towards the standard where you learn independ-ently, attempt problems prior to a tutorial, and know why youranswer to a set problem is correct.

Exercise 1

If

then evaluate an expression for and . Assume

. Sketch and for the case of ,

and .

H s( ) k 1 s z1+( )1 s p1+( )

----------------------------=

HM f( ) H f( )k z1 p1 0>, , HM f( ) H f( ) k 10=z1 2pi10= p1 2pi=

106 R. M. Howard 2013

Exercise 2

If


. Sketch and for the case of ,

, and .

H s( ) k 1 s z1+( )1 s p1+( ) 1 s p2+( )----------------------------------------------------=

HM f( ) H f( )k z1 p1 0>, , HM f( ) H f( ) k 10=z1 2pi10= p1 2pi= p2 2pi100=

107 R. M. Howard 2013

Exercise 3

If


and Sketch and

for the general case consistent with .

Specify the maximum gain.

H s( ) k 1 s z1+( ) 1 s z2+( )1 s p1+( ) 1 s p2+( ) 1 s p3+( )-------------------------------------------------------------------------------=

HM f( ) H f( )k z1 z2 p1 p2 p3, , 0>, , , z1 p1 z2 p2 p3 HM f( )H f( ) z1 p1 z2 p2 p3

108 R. M. Howard 2013

Exercise 4

Establish the transfer function between the input and output of thefollowing op. amp. circuit.

Use the following model for the op. amp.

vo t( )vS t( ) +-

Rf

R1

Vo

+

-

++

-

A s( )ViVi

+

-

A s( ) Ao1 s so+--------------------=

109 R. M. Howard 2013

Exercise 5

Determine bandwidth expressions for the following transfer func-tions:

a)

b)

H s( ) k1 s p+------------------= p 0>

H s( ) k1 s p+( )2--------------------------= p 0>

110 R. M. Howard 2013

Exercise 6

Consider the following circuit:

a) Determine an expression for the impedance, , of the circuit:

b) Write this impedance in the form

Specify and in terms of and .

c) Given an impedance , determine the values of

and consistent with the above circuit.

R C

Z

Z k1 s p+------------------=

k p R C

Z k1 s p+------------------= R

C

111 R. M. Howard 2013

Exercise 7

Determine an expression for . as defined in the following circuit. VY

L

CR

i2 t( )

VB

+-

i1 t( )

VY

112 R. M. Howard 2013

Exercise 8

Consider the common base amplifier:

Specify how the operating point , as well as the output

voltage , can be determined.

VCC

Vo

VCC

CC

vS+ IEE

RC

IB IC IE, ,Vo

113 R. M. Howard 2013

Exercise 9

Using the approximation of establish, for the follow-

ing circuit, for the BJT, and , and the voltages

and .

Assume: , , , ,

and .

VEB 0.7V=IB IC IE, , I1 I2 VB

Vo

RE

VCC

RC

Vo

VEB +-

RB2

RB1

VB

I1

I2

50= RB1 5k= RB2 10k= RE 500=RC 1k= VCC 12V=

114 R. M. Howard 2013

Exercise 10

a) Complete the design, i.e. specify appropriate resistance values,for the following circuits when the requirements are:

Assume , , forward active mode of operation

where when .

b) Comment on the temperature stability of each of these circuits.

IC 1mA=

Vo 8V=

VCC 12V= 100=VBE 0.7V= IC 1mA=

115 R. M. Howard 2013

VCC

RCVo

RB2

RB1

circuit 1

VCC

RCVo

RB1

circuit 2

RE

VCC

RCVo

RB2

RB1

RE

VCC

RCVoRB1

VCCcircuit 3circuit 4

116 R. M. Howard 2013

Exercise 11

Consider the following BJT amplifier:

Typical parameter values are: , ,

, , , , and

.

Determine , , and . Assume ,

and .

RE

VCC

RCVoRS

VCC

vS+

IC 1mA= VCC 12V=RS 50= RC 5k= RE 11.3k= VBE 0.7= 100=fT 100MHz=

gm rpi C Cpi C 0( ) 3.3pF=j 0.7V= m 0.5=

117 R. M. Howard 2013

19.0 Solution to Exercises

ETEN3002 Electronics 1 Background(1)

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