EIE 211 Electronic Devices and Circuit Design II

04/19/2023 1 King Mongkut’s University of Technology Thonburi

EIE 211 Electronic Devices and Circuit Design II

EIE 211 : Electronic Devices and Circuit Design IILecture 9: Two-port Networks & Feedback



Example: Design a 2nd order high pass active filter based on the inductor replacement



Second Order Active Filters based on the Two-Integrator-Loop Topology

To derive the two-integrator loop biquadratic circuit, or biquad, consider the high-pass transfer function

We observe that the signal (ωo/s)Vhp can be obtained by passing Vhp through an integrator with a time constant equal to 1/ωo. Furthermore, passing the resulting signal through another identical integrator results in the signal (ωo

2/s2)Vhp. The block diagram on the next page shows a two-integrator arrangement.



From

It suggests that Vhp can be obtained by using the weighted summer in Fig b. Now we combine blocks a) and b) together to obtain:



If we try to look at the Fig c. more carefully, we’ll find that

22

2

)/( ooi

hphp Qss

Ks

V

VT

And the signal at the output of the first integrator is –(ωo/s)Vhp, which is a band-pass function, with the center-frequency gain of –KQ,

Therefore, the signal at the output of the first integrator is labeled Vbp. In, a similar way, the signal at the output of the second integrator is (ωo

2/s2)Vhp, which is a low-pass function,

Thus, the output of the second integrator is labeled Vlp. Note that the dc gain of the low-pass filter is equal to K. Hence, the 2-integrator-loop biquad realizes 3 basic 2nd order filtering functions simultaneously, that’s why it’s called a universal active filter.

04/19/2023 6


Circuit ImplementationWe replace each integrator with a Miller integrator circuit having CR = 1/ωo and we replace the summer block with an op amp summing circuit that is capable of assigning both positive and negative weights to its inputs. The resulting ckt, known as the Kerwin-Huelsman-Newcomb or KHN biquad.



We can express the output of the summer Vhp in terms of its inputs, Vbp = –(ωo/s)Vhp and Vlp = (ωo

2/s2)Vhp, as

To determine all the parameters, we need to compare it to the original eq:

We can match them up, term by term, and will get:



The KHN biquad can be used to realize notch and all-pass functions by summing weighted versions of the three outputs, LP, BP, and HP as shown.

Substitute Thp, Tbp and Tlp that we found previously, we’ll get the overall transfer function

from which we can see that different transmission zeros can be obtained by the appropriate selection of the values of the summing resistors. For instance, a notch is obtained by selection RB = ∞ and



Two-Port Network Parameters



Characterization of linear, two-port networksBefore we begin a discussion on the topic of oscillators, we need to study feedback. However, in order to understand how the feedback works, we also need to first learn the two-port network parameters.

A two-port network has four port variables: V1, I1, V2 and I2. If the two-port network is linear, we can use two of the variables as excitation variables and the other two as response variables. For example, the network can be excited by a voltage V1 at port 1 and a voltage V2 at port 2, and the two current I1 and I2 can be measured to represent the network response.

There are four parameter sets commonly used in electronics. They are the admittance (y), the impedance (z), the hybrid (h) and the inverse-hybrid (g) parameters, respectively.



Two-Port Network (z-parameters)(Open-Circuit Impedance)

2121111 IzIzV

2221212 IzIzV

021

111

II

Vz

012

112

II

Vz

Open-circuit input impedance

At port 1

Open-circuit reverse transimpedance 012

222

II

Vz

021

221

II

Vz

At port 2

Open-circuit forwardtransimpedance

Open-circuit output impedance

V1

+

I1

V2

+

I2

z11z22

z12I2 z21I1+ +

2

1

2221

1211

2

1

I

I

zz

zz

V

V





Two-Port Network (y-parameters)(Short-Circuit Admittance)

2121111 VyVyI

2221212 VyVyI

021

111

VV

Iy

012

112

VV

Iy

Short-circuit input admittance

At port 1

Short-circuit reverse transadmittance 012

222

VV

Iy

021

221

VV

Iy

At port 2

Short-circuit forwardtransadmittance

Short-circuit output admittance

V1

+

V2

+

I1 I2

1/y11 1/y22

y12V2 y21V1

2

1

2221

1211

2

1

V

V

yy

yy

I

I





Two-Port Network (h-parameters)(hybrid)

2121111 VhIhV

2221212 VhIhI

021

111

VI

Vh

012

112

IV

Vh

Short-circuit input impedance

At port 1

Open-circuit reverse voltage gain 012

222

IV

Ih

021

221

VI

Ih

At port 2

Short-circuit forwardcurrent gain

Open-circuit output admittance

V2

+

I2

1/h22

h21I1

V1

+

I1

h11

h12V2 +

2

1

2221

1211

2

1

V

I

hh

hh

I

V





Two-Port Network (g-parameters)(inverse-hybrid)

2121111 IgVgI

2221212 IgVgV

021

111

IV

Ig

012

112

VI

Ig

Open-circuit input admittance

At port 1

Short-circuit reverse current gain 012

222

VI

Vg

021

221

IV

Vg

At port 2

Open-circuit forwardcurrent gain

Short-circuit output impedance

V2

+

I2

g22

g21V1

V1

+

I1

1/g11

g12I2 +

2

1

2221

1211

2

1

I

V

gg

gg

V

I



19


z-parameter examplesI1 I2

V1

+ V2

+6

I1 I2

V2

+V1

+

12

3V2

+V1

+

I1 I2312

6

30

012

00

00

312

21

221

12

112

2211

Z

II

VZ

II

VZ

ZZ

66

66

60

60

66

21

221

12

112

2211

Z

II

VZ

II

VZ

ZZ

96

618

66

0

66

0

918

1

1

21

221

2

2

12

112

2211

Z

I

I

II

VZ

I

I

II

VZ

ZZ

Note: (1) z-matrix in the last circuit = sum of two former z-matrices

(2) z-parameters is normally used in analysis of series-series circuits

(3) Z12 = Z21 (reciprocal circuit)

(4) Z12 = Z21 and Z11 = Z22 (symmetrical and reciprocal circuit)



y-parameter examples

I1 I2

V1

+ V2

+

0.05S

V2

+V1

+

I1 I20.2S0.1S

0.025S

05.005.0

05.005.0

05.005.0

0

05.005.0

0

05.005.0

1

1

21

221

2

2

12

112

2211

y

V

V

VV

Iy

V

V

VV

Iy

ySy

S

S

S

0769.00615.0

0615.00692.0

S0615.0 ,reciprocalBy

S0615.0

0615.08.0025.01.0

0769.0But

0

S0769.0025.01.0

1

2.0

1

S0692.0025.02.0

1

1.0

1

1221

12

221

121

22222

12

112

1

22

1

11

y

yy

y

VII

III

VVyI

VV

Iy

y

y



Example: figure below shows the small-signal equivalent-ckt model of a transistor. Calculate the values of the h parameters.





Summary: Equivalent-Circuit Representation



Feedback


Feedback

What is feedback? Taking a portion of the signal arriving at the load and feeding it back to the input.

What is negative feedback? Adding the feedback signal to the input so as to partially cancel the input signal to the amplifier.

Doesn’t this reduce the gain? Yes, this is the price we pay for using feedback. Why use feedback? Provides a series of benefits, such as improved

bandwidth, that outweigh the costs in lost gain and increased complexity in amplifier design.

XoXi

Xf

Xs +-

βf



Feedback Amplifier Analysis

AA

A

X

XA

X

XA

XX

AX

X

XA

bygivenisfeedbackwithgainsamplifierThe

sourcethefromsignaltheX

amplifierbasicthetosignalinputnettheisXwhereXXX

gainvoltagegegainsamplifiertheisAwhereAXX

factorfeedbackthecallediswhereXX

f

i

of

i

ffi

i

s

of

s

ifsi

io

foff

111

'

,

..,'

XoXi

Xf

Xs +-

βf



Summary: General Feedback Structure

A

Source Load+

-

Vs

Vf

V V A : Open Loop Gain A = Vo / V : feedback factor = Vf / Vo

VAV

VVV

VV

VVV

o

oS

of

fs

1 :Note

1 :feedback ofAmount

:Gain Loop

)1

(1

1 :gain loop Close

Af

s

of

A

A

AT

T

T

A

A

V

VA

The product Aβ must be positive for the feedback network to be the negative feedback network.



* Gain desensitivity - less variation in amplifier gain with changes in β (current gain) of transistors due to dc bias, temperature, fabrication process variations, etc.

* Bandwidth extension - extends dominant high and low frequency poles to higher and lower frequencies, respectively.

* Noise reduction - improves signal-to-noise ratio* Improves amplifier linearity - reduces distortion in signal due to gain

variations due to transistors* Impedance Control - control input and output impedances by applying

appropriate feedback topologies

* Cost of these advantages:æ Loss of gain, may require an added gain stage to compensate.æ Added complexity in design

Advantages of Negative Feedback

AAf

LLfHfHf

1

1



Gain Desensitivity

Feedback can be used to desensitize the closed-loop gain to variations in the basic amplifier. Let’s see how.

Assume β is constant. Taking differentials of the closed-loop gain equation gives…

Divide by Af

This result shows the effects of variations in A on Af is mitigated by the feedback amount. 1+Aβ is also called the desensitivity amount

We will see through examples that feedback also affects the input and resistance of the amplifier (increases Ri and decreases Ro by 1+Aβ factor)

21 AdA

dAf

A

dA

AA

A

A

dA

A

dA

f

f

1

11

1 2

AA

Af

1



Bandwidth Extension

We’ve mentioned several times in the past that we can trade gain for bandwidth. Finally, we see how to do so with feedback… Consider an amplifier with a high-frequency response characterized by a single pole and the expression: Apply negative feedback β and the resulting closed-loop gain is:

•Notice that the midband gain reduces by (1+AMβ) while the 3-dB roll-off frequency increases by (1+AMβ)

H

M

s

AsA

1

MH

MMf As

AA

sA

sAsA

11

1

1



Finding Loop Gain

Generally, we can find the loop gain with the following steps:

– Break the feedback loop anywhere (at the output in the ex. below) – Zero out the input signal xs

– Apply a test signal to the input of the feedback circuit – Solve for the resulting signal xo at the output

If xo is a voltage signal, xtst is a voltage and measure the open-circuit voltage

If xo is a current signal, xtst is a current and measure the short-circuit current

– The negative sign comes from the fact that we are apply negative feedback

A

xs=0

xf

xi

xoxtst

Ax

x

AxAxAxx

xx

xx

tst

o

tstfio

fi

tstf

gain loop

0



* There are four types of feedback amplifiers. Why?æ Output sampled can be a current or a voltageæ Quantity fed back to input can be a current or a voltageæ Four possible combinations of the type of output sampling and input

feedback* One particular type of amplifier, e.g. voltage amplifier, current amplifier,

etc. is used for each one of the four types of feedback amplifiers. * Feedback factor βf is a different type of quantity, e.g. voltage ratio,

resistance, current ratio or conductance, for each feedback configuration.* Before analyzing the feedback amplifier’s performance, need to start by

recognizing the type or configuration.* Terminology used to name types of feedback amplifier, e.g. Series-shunt

æ First term refers to nature of feedback connection at the input.æ Second term refers to nature of sampling connection at the output.

Basic Types of Feedback Amplifiers



Basic Feedback Topologies

Depending on the input signal (voltage or current) to be amplified and form of the output (voltage or current), amplifiers can be classified into four categories. Depending on the amplifier category, one of four types of feedback structures should be used.

(Type of Feedback) (Type of Sensing)

(1) Series (Voltage) Shunt (Voltage)

(2) Series (Voltage) Series (Current)

(3) Shunt (Current) Shunt (Voltage)

(4) Shunt (Current) Series (Current)


Figure 8.4 The four basic feedback topologies: (a) voltage-mixing voltage-sampling (series–shunt) topology; (b) current-mixing current-sampling (shunt–series) topology; (c) voltage-mixing current-sampling (series–series) topology; (d) current-mixing voltage-sampling (shunt–shunt) topology.


Basic Feedback Topologies

Depending on the input signal (voltage or current) to be amplified and form of the output (voltage or current), amplifiers can be classified into four categories. Depending on the amplifier category, one of four types of feedback structures should be used (series-shunt, series-series, shunt-shunt, or shunt-series)

Voltage amplifier – voltage-controlled voltage source Requires high input impedance, low output impedance Use series-shunt feedback (voltage-voltage feedback)

Current amplifier – current-controlled current source Use shunt-series feedback (current-current feedback)

Transconductance amplifier – voltage-controlled current source Use series-series feedback (current-voltage feedback)

Transimpedance amplifier – current-controlled voltage source Use shunt-shunt feedback (voltage-current feedback)

series-shunt

shunt-series

series-series

shunt-shunt

04/19/2023 36


Series-Shunt Feedback Amplifier - Ideal Case* Assumes feedback circuit does not load down the basic

amplifier A, i.e. doesn’t change its characteristics® Doesn’t change gain A® Doesn’t change pole frequencies of basic

amplifier A

® Doesn’t change Ri and Ro

* For the feedback amplifier as a whole, feedback does change the midband voltage gain from A to Af

* Does change input resistance from Ri to Rif

* Does change output resistance from Ro to Rof

* Does change low and high frequency 3dB frequencies

A

AA

ff

1

ARR fiif 1

A

RR

f

oof

1

AAf

LLfHfHf

1

1

Basic Amplifier

Feedback Circuit

Equivalent Circuit for Feedback Amplifier



Series-Shunt Feedback Amplifier - Ideal CaseMidband Gain

Vf

V

i

of

V

i

f

V

fi

iV

s

oVf A

A

V

VA

V

VA

VV

VA

V

VA

111

Input Resistance

Vfi

ii

ofi

i

fi

i

sif AR

RV

VV

I

VV

I

VR

1

Output Resistance

Vt

It

fV

o

t

tof

o

fVt

o

tfVt

o

fVtt

tfoff

fis

o

iVtt

A

R

I

VRso

R

AV

R

VAV

R

VAVI

soVVVand

VVsoVBut

R

VAVI

1

1

0



Series-Shunt Feedback Amplifier - Ideal Case

of

LLf

of

ofo

Lf

fo

of

L

of

o

ofL

o

L

of

L

o

ff

L

o

AA

AAwhere

s

A

sA

A

A

As

A

s

A

s

A

A

AAthen

s

AAFor

11

11

11

1

1

11

1

11

Low Frequency Pole

High Frequency Pole

ofHHfof

ofo

Hf

fo

ofH

of

o

ofH

o

H

of

H

o

ff

H

o

AA

AAwhere

s

A

A

s

A

A

As

A

sA

sA

A

AAthen

sA

AFor

11

11

1

1

1

11

1

11

Low 3dB frequency lowered by feedback.

Upper 3dB frequency raised by feedback.



39

* Feedback networks consist of a set of resistors æ Simplest case (only case considered here)æ In general, can include C’s and L’s (not

considered here)æ Transistors sometimes used (gives variable

amount of feedback) (not considered here)* Feedback network needed to create Vf feedback

signal at input (desirable)

* Feedback network has parasitic (loading) effects including:

* Feedback network loads down amplifier inputæ Adds a finite series resistanceæ Part of input signal Vs lost across this series

resistance (undesirable), so Vi reduced

* Feedback network loads down amplifier outputæ Adds a finite shunt resistanceæ Part of output current lost through this shunt

resistance so not all output current delivered to load RL (undesirable)

Practical Feedback Networks

Vi

Vf

Vo

* How do we take these loading effects into account?



* Need to find an equivalent network for the feedback network including feedback effect and loading effects.

* Feedback network is a two port network (input and output ports)

* Can represent with h-parameter network (This is the best for this particular feedback amplifier configuration)

* h-parameter equivalent network has FOUR parameters

* h-parameters relate input and output currents and voltages

* Two parameters chosen as independent variables. For h-parameter network, these are input current I1 and output voltage V2

* Two equations relate other two quantities (output current I2 and input voltage V1) to these independent variables

* Knowing I1 and V2, can calculate I2 and V1 if you know the h-parameter values

* h-parameters can have units of ohms, 1/ohms or no units (depends on which parameter)

Equivalent Network for Feedback Network



* Feedback network consists of a set of resistors* These resistors have loading effects on the basic

amplifier, i.e they change its characteristics, such as the gain

* Can use h-parameter equivalent circuit for feedback network æ Feedback factor βf given by h12 since

æ Feedforward factor given by h21 (neglected)

æ h22 gives feedback network loading on output

æ h11 gives feedback network loading on input

* Can incorporate loading effects in a modified basic amplifier. Basic gain of amplifier AV becomes a new, modified gain AV’ (incorporates loading effects).

* Can then use feedback analysis from the ideal case.

Series-Shunt Feedback Amplifier - Practical Case

fo

f

IV

V

V

Vh

02

112

1

'1

'1

'1'1

'1

'

Vf

LLfHVfHf

fV

oofVfiif

Vf

VVf

AA

A

RRARR

A

AA

sifin RRR )11

/(1Lof

out RRR

04/19/2023 42

EIE 211 Electronic Devices and Circuit Design IISeries-Shunt Feedback Amplifier - Practical Case

* How do we determine the h-parameters for the feedback network?

* For the input loading term h11 æ Turn off the feedback signal by

setting Vo = 0.æ Then evaluate the resistance seen

looking into port 1 of the feedback network (also called R11 here).

* For the output loading term h22

æ Open circuit the connection to the input so I1 = 0.

æ Find the resistance seen looking into port 2 of the feedback network (also called R22 here).

* To obtain the feedback factor βf (also called h12 )æ Apply a test signal Vo’ to port 2 of the

feedback network and evaluate the feedback voltage Vf (also called V1 here) for I1 = 0.

æ Find βf from βf = Vf/Vo’

Summary of Feedback Network Analysis



* Evaluate modified basic amplifier (including loading effects of feedback network)æ Including h11 at inputæ Including h22 at outputæ Including loading effects of source resistanceæ Including load effects of load resistance

* Analyze effects of idealized feedback network using feedback amplifier equations derived

* Note æ Av’ is the modified voltage gain including the

effects of h11 , h22 , RS and RL. æ Ri’, Ro’ are the modified input and output

resistances including the effects of h11 , h22 , RS and RL.

Summary of Approach to Analysis

'1

'1

'1

''1'

'1

'

Vf

LLfHVfHf

fV

oofVfiif

Vf

VVf

AA

A

RRARR

A

AA

Modified Basic Amplifier

Idealized Feedback Network

Practical Feedback Network

Basic Amplifier



Example: Find expression for A, β, the closed-loop gain Vo/Vs, the input resistance Rin, and the output resistance Rout. Given μ = 104, Rid =100 kΩ, Ro = 1 kΩ, RL = 2 kΩ, R1 = 1 kΩ, R2 = 1MΩ and Rs = 10 kΩ.





Documents

EIE 211 Electronic Devices and Circuit Design II