EE303_Topic1_Rahal

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Microelectronic Circuits - Fifth Edition Sedra/Smith 1Copyright 2004 by Oxford University Press, Inc. 1

Single-Time Constant Circuits,

s-Domain Analysis: Poles, Zeros,

and Bode Plots

EE300 Electronics II

First Week:

Lectures 1-2

Dr Mohamad Rahal


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Microelectronic Circuits - Fifth Edition Sedra/Smith 2Copyright 2004 by Oxford University Press, Inc.

Outline of Lecture

• Evaluating the time constant

• Classification of STC circuits

• Frequency response of STC circuits• Step response of STC circuits

• Pulse response of STC circuits

• S-domain, Poles, Zeros, Bode plots and open-circuit time constant technique


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Figure D.1 The reduction of the circuit in (a) to the STC circuit in (c) through the repeated application of Thévenin’s theorem.

STC Example 1


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Figure D.2 Circuit for Example D.2.

STC Example 2


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Figure D.3 The response of the circuit in (a) can be found by superposition, that is, by summing the responses of the circuits in (d) and (e).

STC Example 3


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Figure D.3 (Continued)


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Classification of STC Circuits


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Figure D.4 STC circuits of the low-pass type.

Classification Examples of STC

Circuits (low-pass)


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Figure D.5 STC circuits of the high-pass type.

Classification Examples of STC

Circuits (high-pass)


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Figure ED.1

Time Constants?

R L L /)//( 21 )///()//( 2121 R R L L


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Figure D.6 (a) Magnitude and (b) phase response of STC circuits of the low-pass type.

Low-pass Type

)/(1)( 0 s

K

sT

)/(1)(

0

j

K jT

/10

2

0)/(1)(

K jT

)/(tan)( 01


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Low-pass Type: Bode Plots

2

0)/(1)(

K jT

For > 1

At )( 0 dB K jT

32

1

log20

)(

log20

At the phase is equal to -45 and at >>1 the

phase is equal to -90 degrees with an a maximum error of 5.7

degrees

)( 0 )/( 0


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High-pass Type

0

)(

s

Ks sT

)/(1)(

0

j

K jT

/10

2

0 )/(1)(

K jT

)/(tan)( 01

Figure D.8 (a) Magnitude and (b) phase response of STC circuits of the high-pass type.


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Frequency Response Comparison


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Figure ED.3

Exercise: Try @ Home

Find the dc transmission and the corner frequency for the circuit

Show above?


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Figure D.9 A step-function signal of height S .

Step Response of STC Circuits

)1()( t

eS t y

t

Set y

)(


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Pulse Response of STC (Low-pass)

Figure D.12 A pulse signal with height P and width T .

0

35.0

2.2 f t t f r


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Figure D.14 Pulse responses of three STC high-pass circuits.

Pulse Response of STC (High-pass)

T P

P

100

T Percentage Sag


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S-Domain Analysis

)()()(

sV sV sT

i

o

where both N(s) and D(s) are polynomials with real coefficients

and an order of m and n respectively

The order of the network is equal to n

For real systems, the degree of N(s) (or m) is always less

than or equal to that of D(s)(or n).


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Poles and Zeros

where am is a multiplicative constant; Z1, Z2, …, Zm are the roots of the

numerator polynomial (N(s)); P1, P2, …, Pn are the roots of the denominator

polynomial (D(s)).

Poles — roots of D(s)=0 {P 1 ,P 2 ,…, P n } are the points on the s- plane where |T| goes to ∞.

Zeros —

roots of N(s)=0 {Z 1 ,Z 2 ,…, Z m } are the points on the s-pl ane where |T| goes to 0.

• The poles and zeros can be either real or complex. However, since the polynomial

Coefficients are real numbers, the complex poles (or zeros) must occur in conjugate pairs.

• A zero that is pure imaginary (± jωz) cause the transfer function T(jω) to be exactly zero

(or have transmission null) at ω=ωz.

• Real zeros will not result in transmission nulls.

• For stable systems all the poles should have negative real parts.

• For s much greater than all the zeros and poles, the transfer function may be approximated

T(s)≅ am/sn-m . Thus the transfer function have (n-m) zeros at s=∞.


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Figure E.1 Bode plot for the typical magnitude term. The curve shown applies for the case of a zero. For a pole, the high-frequency asymptote should

be drawn with a – 6-dB/octave slope.

Bode Plot Magnitude


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Figure E.2 Bode plots for Example E.1.

Bode Plots: Example 1


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Figure E.3 Bode plot of the typical phase term tan – 1 ( /a) when a is negative.

Bode Plot Phase


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Figure E.4 Phase plots for Example E.2.

Overall Phase of Example 1


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High Frequency Gain

Figure 6.12 Frequency response of a direct-coupled (dc) amplifier. Observe that the gain does not fall off at low

frequencies, and the midband gain A M extends down to zero frequency.

Dc coupled


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Dominant Pole

As a rule of thumb, a dominant pole exists if the lowest-

frequency pole is at least two octaves (a factor of 4) away from the nearest pole or zero.


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Determining the Dominant Pole

2 pole-2 zeros example

NeglectingHigh-order

terms


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Open-Circuit (OC) Time Constants 1


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Using Open-Circuit Time Constants

CS Example

Figure 6.14 Circuits for Example 6.6: (a) high-frequency equivalent circuit of a MOSFET amplifier; (b) the equivalent circuit at midband

frequencies; (c) circuit for determining the resistance seen by C gs; and (d) circuit for determining the resistance seen byC gd .


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CS Example OC Time Constant

At G

At D


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CS Example


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End of First Week

Lectures

Home work: D.1, D.2, D.4, D.6, D.9,

DD.16, E.5, E.7, E.10. Deadline 13/03/2010

Materials can be found on the Y:\EE303-001

Please check this folder on a daily/weekly basis

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