Excerpt of Sedra/Smith Chapter 1: Signals and Ampliﬁer …...Excerpt of Sedra/Smith Chapter 1: Signals and Ampliﬁer) Concept 30 Transfer Function The transfer function T(s) of

INF3410 — Fall 2016

Excerpt of Sedra/Smith Chapter 1: Signals andAmplifier Concept

Content

Signals and Spectra

Amplifiers

Excerpt of Sedra/Smith Chapter 1: Signals and AmplifierConcept 2

Content

Signals and Spectra

Amplifiers


Book Sections Covered

1.1-1.3


Signal Sources

Two equivalent models of signal sources (e.g. a sensoror other transducer): a) is called the The Thévenin formand b) the Norton form. Note: Outputresistance RS isthe same in both models while vS(t) = iS(t)RS. A moregeneral model would consider an outputimpedancerather than just a resistance ...Excerpt of Sedra/Smith Chapter 1: Signals and Amplifier

Concept 5

Arbitrary SignalsAny arbitrary signal can beexpressed as an (infinite)sum or (infinite) integral ofsine wave signals ofdifferent frequencies andphases by means of theinverse Fourier transform:

vS(t) =1

2π

∫ ∞

−∞v̂s(ω)e

iωtdω

Where the Fourrier transform v̂s is a complex number.How does the above equation represent a ’sum of sinewaves’?Excerpt of Sedra/Smith Chapter 1: Signals and Amplifier

Concept 6

Cyclic Signals

Cyclic signals can beexpressed as discreet sumsof sine wave signals, moreprecisely with harmonics ofthe cycle frequency.


Example: Approximating Square Wave

Here is how a square wavecan be approximated withthe sum of 3 sine waves.

v(t) =4V

π

�

sinω0t +1

3sin3ω0t +

1

5sin5ω0t + ...

�

(1.2)


Spectrum of Square Wave

Thus the spectrum of a square wave (or any cyclicsignal) is non-zero at the harmonics (multiples of thefundamental frequency) only. It can thus be expressedas a sum.Excerpt of Sedra/Smith Chapter 1: Signals and Amplifier

Concept 9

Spectrum of Arbitrary Function

whereas the spectrum of a non-cyclic function can benon-zero for any frequency, e.g. the function in figure1.3.


Discrete Time/Sampled


Discrete Value/Digital

(for example binary)


ADC


Content

Signals and Spectra

AmplifiersFrequency Response


Book Sections Covered

1.4-1.6


Amplifier Basic Concept


Example: Voltage Amplifier with LoadResistance


Gain in Decibels

Av =vO

vI

Ap =pO

pL=

vOiO

vIiI= AvAi

Ap,dB = 10 log10 Ap [dB]

Av,dB = 20 log10 Av [dB]


Increasing Signal Power

(and thus in need of a power supply)

η =PL

Pdc× 100 (1.10)


Non-Ideal Behaviour (1/many)

Linear range:

L−

Av≤ vI ≤

L+

Av

(near Fig. 1.14)


Symbol Convention

iC(t) = IC + ic(t) (1.11)Excerpt of Sedra/Smith Chapter 1: Signals and AmplifierConcept 21

A First Voltage Amplifier Model


Voltage Gain Dependence

Av ≡vo

vi= Avo

RL

RL +Ro(1.13)

vo

vs= Avo

Ri

Ri +RS

RL

RL +Ro(below 1.13)


Example: Cascaded Amplifiers

vo

vs=

Ri1

Ri1 +RSAvo1

Ri2

Ri2 +Ro1Avo2

Ri3

Ri3 +Ro2...AvoN

RL

RL +RoN


4 Equivalent Models


Determining Ri and Ro


Frequency Response

(This behaviour is not explained by the simple model!)


Linear Amplifier

Linear here means that there is no distortion of a fixedfrequency sinusoid. An amplifier composed of but linearelements will behave like that, including somewhatmore complicated models than our first purely resistivemodel ...


Single Time Constant Networks

When the input voltage source provides a signal theSTC network is a filter with a specific transfer function,i.e. a frequency dependent complex number.Excerpt of Sedra/Smith Chapter 1: Signals and Amplifier

Concept 29

Transfer Function

Transfer functions T(s) for linear electronic circuits canbe written as dividing two polynomials of s (for us s issimply short for jω).

T(s) =a0 + a1s+ ...+ amsm

1+ b1s+ ...+ bnsn

T(s) is often written as products of first order terms inboth nominator and denominator in the following rootform, which is conveniently showing some properties ofthe Bode-plots. More of that later.

T(s) = a0(1+ sz1 )(1+

sz2)...(1+ szm )

(1+ sω1 )(1+sω2

)...(1+ sωn )Excerpt of Sedra/Smith Chapter 1: Signals and AmplifierConcept 30

Transfer Function

The transfer function T(s) of a linear filter isÉ the Laplace transform of its impulse reponse h(t).É the Laplace transform of the differential equation

describing the I/O realtionship that is then solvedfor Vout(s)Vin(s)

É (easiest!!!) the circuit diagram solved quitenormally for Vout(s)Vin(s) by putting in impedances Z(s) forall linear elements according to some simple rules(next page).


Impedances of Linear Circuit Elements

resistor: Rcapacitor: 1sCinductor: sL

Ideal linearly dependent sources (e.g. the id = gmvgssources in small signal models of FETs) are left as theyare.


Single Time Constant Transfer Functions


Bode Plot

1st Order Low-Pass Filter


Bode Plot

1st Order High-Pass Filter


Example


Characterizing Amplifiers by TransferCharacteristics


Capacitively Coupled Two Stage Amplifiers


Signals and SpectraAmplifiersFrequency Response

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Excerpt of Sedra/Smith Chapter 1: Signals and Ampliﬁer …...Excerpt of Sedra/Smith Chapter 1: Signals and Ampliﬁer) Concept 30 Transfer Function The transfer function T(s) of