2ports, feedback and filters - circuits and...

Imperial College London – EEE 1R-V0.0 Autumn 2006 Electronics Revision - CP

Revision of intermediate electronics

2ports, feedback and filters


Generalised Thevenin + Norton Theorems:2-port parameters

• Amplifiers, filters etc have input and output• Input can be voltage or current• Output can be voltage or current• By convention current positive into positive terminal• Negative terminals usually considered connected together• General form of amplifier or filter:

Thevenin or Norton

Thevenin or Norton

Amplifier

+ +- -

V1 V2

I1 I2


The voltage amplifier – G parametersAlso called the reverse hybrid parameters

1

2

1 1

2 2

11 11 11 12

22 22 21 22

11 12

21 22

1 1 2

2 1 2

1 1

2 2

v v ii v i

v vi

g i g gv g g g

i

i

g g

v

i iv g g v

= + = + ⎫⇒⎬= + = + ⎭

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= ⇒ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦G

Formal description:g11: Input admittanceg12: Reverse current gaing21: Voltage gaing22: Output impedance

• A voltage amplifier exhibits non-zero reverse current gain!• The port-reversed amplifier is a current “amplifier”• Reverse path has less than unity power gain.• If the reverse current gain is zero, the amplifier is called Unilateral

i11 =g12i2

i1

v1 g11 v2

i2

v22=g21v1

g22TheveninNorton


The current amplifier – H parametersAlso called the hybrid parameters

11 11 11 11 1 2

2 1 2

1

1

2

1

2

2

22

1

2

22 21 22

11 1

21 2

2

22

i i vv

h v h hi i vh h h

h hh

vi

vv vh

ii

i

= + = + ⎫⇒⎬= + = + ⎭

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦H=

Formal descriptionh11: Input impedanceh12: Reverse voltage gainh21: Current gainh22: Output admittance

• A current amplifier exhibits a reverse voltage gain!• The port-reversed amplifier is a voltage “amplifier”• Reverse path has less than unity power gain.• If the reverse voltage gain is zero, the amplifier is called Unilateral

i22=h21i1

i2

v2h22v1

i1

v11=h12v2

h11NortonThevenin


The transconductance amplifier – Y parametersAlso called the short circuit parameters

11 11 11 11 1 2

2 1 2

1

1

2

1

2

2

22

1

2

22 21 22

11 1

21 2

2

22

v v vv

y i y yi v vy y y

y yy

ii

iv vy

vi

v

= + = + ⎫⇒⎬= + = + ⎭

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦Y=

Formal descriptiony11: Input admittancey12: Reverse admittance gainy21: trans-admittance (gain)y22: Output admittance

• A transconductance amplifier exhibits a reverse transconductance gain!• The port-reversed amplifier is also a transconductance “amplifier”• Reverse path has less than unity power gain.• If the reverse gain is zero, the amplifier is called Unilateral

i2

v2y22

i22=y21v1i11 =y12v2

i1

v1 y11Norton Norton


The transresistance amplifier – Z parametersAlso called the open circuit parameters

11 11 11 11 1 2

2 1 2

1

1

2

1

2

2

22

1

2

22 21 22

11 1

21 2

2

22

i i ii

z v z zv i iz z z

z zz

vv

vi iz

iv

i

= + = + ⎫⇒⎬= + = + ⎭

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦Z=

Formal descriptionz11: Input impedancez12: Reverse impedance gainz21: transimpedance gainz22: Output impedance

• A transresistance (also called a transimpedance) amplifier exhibits a reverse transimpedance gain!• The port-reversed amplifier is also a transresistance “amplifier”• Reverse path has less than unity power gain.• If the reverse gain is zero, the amplifier is called Unilateral

v2

i2

v22=z21i1

z22v1

i1

v11=z12i2

z11 TheveninThevenin


Gain of a fully loaded voltage amplifier

( )( )

1 11 1 12 2

1 11 1 12 22 21 1 22 2

1 1 2 21 1 22 2

2 2

s s L

s s s s L

L

i g v g ii g v i Z g v Yv g v g i

v v i Z v g v i Z g v Yi v Y

= + ⎫⎪ = − −⎧= + ⎪ ⎪⇒⎬ ⎨= − = − −⎪⎪ ⎩⎪= − ⎭

We start with the amplifier definition,plus the source-load boundary conditions:

After a lot of algebra we conclude that:

G12i2

i1

v1 G11 v2

i2

G21V1

G22

ZS

YLVS

2 2111 22 21 12

11 22

, 1 g

s S L g L s

v g g g g gv g Z g Y Y Z

= ∆ = −+ + + ∆


Cascade connection: Transmission Parameters

X1 X2 X3=X1X2

In a cascade connection,• V1 of network X2= V2 of network X1• I1 of network X2 = -I2 of network X1

We can define a new set of parameters so that we have a simple way to calculate the response of cascades of amplifiers.A suitable definition is:

1 2

1 2

v vA Bi iC D⎡ ⎤ ⎡ ⎤⎡ ⎤

=⎢ ⎥ ⎢ ⎥⎢ ⎥ −⎣ ⎦⎣ ⎦ ⎣ ⎦With this definition, the ABCD parameters of a cascade of two networks are found from the matrix product of the individual ABCD matrices ports labelled for clarity): 1 2 3 1 3

1 1 1 2

1 1 1 2 3 3 3 31 1 1 2 2 1

3 3 3 31 1 1 2 2 132 2 2

32 2 2

v A B vi C D i v A B vv A B A B v

i C D ii C D C D ivv A Bii C D

⎫⎡ ⎤ ⎡ ⎤ ⎡ ⎤= ⎪⎢ ⎥ ⎢ ⎥ ⎢ ⎥− ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎪⇒ = ⇒ =⎬ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎪= ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎪−−⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎭


Transmission (or ABCD) parameters (2)

• Note the sign of i2 and also the reverse sense of signal flow. The sign is chosen so the ABCD matrix of a cascade of two networks is just the matrix product of the individual ABCD matrices (compare this to the messy loading calculation before!)

• The reverse sense of signal flow is to keep the matrix finite if an amplifier is unilateral.

• The conversion from, say, Y to ABCD follows the same logic as the Y(H) calculation:

1 2 1 1

1 2 11 12 2 21 22 2

2211 22 21 12

1121

1 0 0 1

11 , yY

v v v vA B A Bi i Y Y v Y Y vC D C D

YA BY Y Y Y

YC D Y

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= ⇒ = ⇒⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − −⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤ −= ∆ = −⎢ ⎥⎢ ⎥ ∆⎣ ⎦ ⎣ ⎦

1 2

1 2

v vA Bi iC D⎡ ⎤ ⎡ ⎤⎡ ⎤

=⎢ ⎥ ⎢ ⎥⎢ ⎥ −⎣ ⎦⎣ ⎦ ⎣ ⎦

Note that all ABCD parameters are inversely proportional to the gain. This is the reason for formally choosing port 2 as the input port. The intuitive choice of input at port 1 would make all parameters inversely proportional to the reverse gain, which is small, and often not very accurately determined.


2-port Connection rules summary

Y1

Y2

Y1+Y2

Z1

Z2

Z1+Z2

G1+G2

G1

G2

H1

H2

H1+H2

X1 X2 X1X2

For the exact calculation of circuit interconnections we can use 2-port matrix algebra:

shunt-shunt: add Y matrices shunt-series: add G matrices

series-series: add Z matrices series-shunt: add H matrices

cascade connection: multiply ABCD matrices


more on 2-port parameters

1 11 1 12 2

2 21 1 22 2

i y v y vi y v v v= += +

We can determine 2-port parameters from the definitions.For example, the Y parameter description states that:

These relations imply that the y parameters are partial derivatives:

2 1

2 1

1 111 12

1 20 0

2 221 22

1 20 0

v v

v v

i iy yv v

i iy yv v

= =

= =

∂ ∂= =∂ ∂

∂ ∂= =∂ ∂

Note that these are small signal parameters, so, e.g. v1=0 means that v1 is kept constant.

The y21 parameter of a transistor is the familiar transconductance.


• We often use “Impedance” or “Admittance” to imply which derivative we have in mind. The input impedance is the z11 or g11 while the input admittance the y11 or h11 parameter.

• For example, if the output is open circuited:

• On the other hand, if the output is shorted, then:

• From our discussion about parameter conversions we know that:

• Note that only if the amplifier is unilateral (y12=0) we have z11=1/y11• Similarly the output impedance of an amplifier depends on whether the amplifier is driven by a

voltage or a current source, and indeed, on the value of the source impedance in the general case.• The argument can be reversed: If the input or output impedance of an amplifier does not

depend on the load or the source impedance respectively, then the amplifier is necessarily unilateral

2

111 110

1 0

1/L

input in Yi

vZ Z z gi=

=

∂= = = =

∂

Input and output impedance of an amplifier

2

111 110

1 0

1/L

input in Zv

iY Y y hv=

=

∂= = = =

∂

( ) 2211 11

11 22 21 12 11

1yZy y y y y

−= = ≠−

1Y


Amplifiers: modelling summary

ZYHG

Parameters

TheveninNorton

TheveninNorton

Input

ImpedanceImpedanceTheveninTransimpedanceAdmittanceAdmittanceNortonTransconductanceVoltageCurrentNortonCurrentCurrentVoltageTheveninVoltage

Reverse gain

Forward gainOutputName /

Representation

VIIV

Output

CCVSVCCSCCCSVCVS

Idealform

LowLow00ITransimpedanceHighHigh∞∞VTransconductanceHighLow∞0ICurrentLowHigh0∞VVoltage OutputInputOutputInput

RealIdealTerminal impedance

InputName / Representation

Notes: 1. Choice of representation is arbitrary2. Representation emphasises the intended function3. Can convert one representation into any other by Thevenin Norton transforms


Conversion between amplifier representations•Some are obvious matrix inversions

from the matrix equations:

Recall that the inverse of a 2x2 matrix A is:

•Other conversions, e.g. to express y in terms of h, start with the definitions and express the variables of the y description in

terms of h parameters. The resulting equation is an identity valid for all values of the h description independent variables.

All this can also be done by re-arranging the terms in the equations. This may be easier at the beginning.

111 12 22 12

21 22 21 11

1

a

a a a aa a a a

−

−

−

=

−⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥−∆⎣ ⎦ ⎣ ⎦

-1

1

1

Z = YG H

A

1 11 12 1 1 111 12

2 21 22 2 21 22 2 2

11211 12

11 22 12 212121 22 11

1 00 1

11 0 1 , 0 1 h

h

i y y v i ih hi y y v h h v v

hh hh h h h

hh h h

−

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤= ⇒ = ⇒⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

−⎡ ⎤⎡ ⎤ ⎡ ⎤= = = ∆ = −⎢ ⎥⎢ ⎥ ⎢ ⎥ ∆⎣ ⎦⎣ ⎦ ⎣ ⎦

Y

Y


The non-inverting amplifier

1 2

1 2 11

1 2

1lim 11

1

in outout in out out in G

in

Gv vR RGv G v v v vR R GH v H RRG

R R

→∞

⎛ ⎞= − ⇒ = = ⇒ = = +⎜ ⎟+ +⎛ ⎞⎝ ⎠ + ⎜ ⎟+⎝ ⎠

Assume finite op-amp gain G. Treat the network connecting the output and the input as an ideal “amplifier” with gain H=R1/(R1+R2) from output to input

R2R1H=R1/(R1+R2)

Gvin

Feedbacknetwork


Negative feedback: A “control systems”perspective

• The signals on the network must be self-consistent

( ) 11 0 1o o

v Gv G v v H vGH

= − ⇒ =+

If GH is large, Taylor expansion gives:( )2

1 1 11o

i

vv H GH GH

⎛ ⎞= − + −⎜ ⎟

⎜ ⎟⎝ ⎠

GH is called the LOOP GAIN GL

Forward Gain=G

+-

Feedback Gain=H

v1 vo

1GGH+

V1 Vo


The inverting amplifier

Note: vout and vin are applied by superposition on the circuit. This way we obtainthe 2 voltage dividers K,HThere are two negative signs on the summing junction, since both forward and feedback signals are applied on the inverting input

( )1 0 1

1 21

1

11lim lim 1 ...

o o

oG G

KGv G Kv v H v vGH

v K Rv vH GH R→∞ →∞

= − − ⇒ = −+

− ⎛ ⎞= − + = −⎜ ⎟⎝ ⎠

R2

R1

G

( ) ( )2 1 2 1 1 2/ , /K R R R H R R R= + = +


The inverting amplifier:A simpler way to calculate

limG→∞

Op-amp Gain=G--

Feedback v-dividerGain=H=R1/(R1+R2)

v1

vo

Input v-dividerGain=K=R2/(R1+R2)

-K G/(1+GH)

-KG/(1+GH)

-KG/(1+GH) = -K/H


Feedback in electronics

• There is both a voltage and a current at every terminal• Precise definitions of measurements:

– Voltage is measured with voltmeters. Voltmeters are connected in parallel to the circuit, and have infinite internal resistance (VM draw no current)..

– Current is measured with ammeters. Ammeters are connected in series to the circuit and have zero internal resistance (AM develop no voltage).

• There are 4 ways to implement electronic feedback:– We may “sample” (measure) the output:

• Voltage, by connecting the input (port 2!) of the FB network in shunt (parallel)• Current, by connecting the input (port 2!) of the FB network in series

– We can then “mix” (feed back) the signal to the input as:• Voltage, by connecting the output (port 1!) of the FB network in series• Current by connecting the output (port 1!) of the FB network in shunt (parallel)

• Exact description of electronic feedback involves 2-port matrix addition.• This is very tedious, we usually use approximations.


The non-inverting amplifier: series – shunt feedback

The op-amp acts like a voltage amplifierThe feedback network samples the output voltage, voltage divides it and feeds back a voltage into the input, so that vin is the sum of input and fed back v.

The feedback network shares input current and output voltage with the op-amp

R2R1

H=R1/(R1+R2)

Gvin

FeedbackNetwork

Amplifier


Feedback on voltage amplifiersSeries-Shunt connection: Add H parameters

+

-V1

I1

V2+

-

I2+

-+

-

+

-+

-

VoltageAmp A

FeedbackNet B

Port 1 Port 2

Port 1 Port 2Port 2 Port 1

Electrical portsFunctional ports

The feedback network is functionally a voltage amplifier from Port2 Port1Electrically both networks must be treated as current amplifiers P1 P2to account for the shared (input) electrical variables.In the calculation we consider the ELECTRICAL description:

Input: I1 , V2

Output: V1 , I2

Shared electrical variables:

( )1 1 1 1 1 1 1

2 2 2 2 2 2 2

A B A BB

A B A B

V V V I I I II I I V V V V⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= + = + = + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

A A B A+BH H H H H

• Add H parameter representations of amplifier and feedback network• Convert back to G parameter representation for composite V-amp.

Function of feedback net:Measure output VCorrect (mix) input Vi.e. it improves a G-amp


The inverting amplifier: Shunt – Shunt feedback

Amplifier and feedback network have identical input and output voltages

The feedback network samples the output voltage and contributes a current to correct the input. The amplifier G functions as a CCVS (butthis should not confuse us, the representation is arbitrary!)

Since the amplifier and the feedback network share voltages they must be treated as transconductors!

R2

R1

G

FeedbackNetwork

Amplifier


Feedback on transimpedance amplifiersThe Shunt-Shunt connection: Add Y parameters

V2

I2

V1+

-

I1+

-+

-

+

-+

-

TransimpAmp A

FeedbackNet B

Port 1 Port 2



The feedback network is functionally a transconductance amplifier from Port2 Port1Electrically both networks must be treated as transconductance amplifiers P1 P2to account for the shared (input) electrical variables.In the calculation we consider the ELECTRICAL description:

Output: I1 , I2

Input: V1 , V2


( )1 1 1 1 1 1 1

2 2 2 2 2 2 2

A B A B

A B A B

I I I V V V VI I I V V V V⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= + = + = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

A B A B A+BY Y Y + Y Y

+

-

• Add Y parameter representations of amplifier and feedback network• Convert to Z parameter representation for composite Z-amp

Function of feedback net:Measure output VCorrect (mix) input Ii.e. it improves a Z-amp


• Consider a shunt admittance connected between the input and output of an inverting voltage amplifier of gain G.

• Looking from the input, the current going through the feedback element is:

• Likewise, looking from the output, the amplifier has a gain=-1/G, so the extra current going into the feedback element is:

• These considerations lead to the equivalence of the two diagrams above in terms of their input and output admittance.

• NOTE: Only if the amplifier is ideal its gain will not change!!!

The Miller theorem: shunt-shunt feedback

( ) ( ) ( ), ,1 1in F in out in M ini v v Y G Yv Y G Y= − = + ⇒ = +

, ,1 11 1out F out M outi Yv Y YG G

⎛ ⎞ ⎛ ⎞= + ⇒ = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

In Out

-G

YIn Out

-G(1+G)Y (1+1/G)Y


Negative Impedance converter

1

22

1

11 1

inRRR RZ

G RRR

−= = =

− ⎛ ⎞− +⎜ ⎟⎝ ⎠

R1 R2

R

Vout

Vin

This method is used to •synthesise negative resistances, C’s, L’s•Invert a given impedance (think of a capacitor in the position of R1)•Multiply or divide impedance magnitudes (note the ratio R2 /R1)

From the Miller Theorem,

Since the op-amp with R1 and R2 forma voltage amplifier of gain

2

1

1 RGR

= +


The emitter degenerated CE amplifier: series-series feedback

We can now apply the feedback equations: (the limit is for large transconductance)

VCC

ZL

ZE

ZS

VS

V=V(ZE)

I=IE

IE

( )( )

// //1 1 / 1

m L CE CEout m L L

T m E m E E

g Z R CV g Z ZV g R g R Rβ β

− −= → −

+ + +

The closed loop amplifier behaves an amplifier with a reduced transconductance

( )( ) ( )0 1 1 / 1out out m E CE m EZ Z g R R g Rβ β= + + +

( )1

1 1 / 1m m

mm E m E E

g ggg R g R Rβ β

′ = →+ + +

The input impedance can easily be calculated (note we have included the shuntMiller effect) :

The output impedance is

( )( ) ( )( )0 01 1 / with // 1in in m E in BE V BCm

Z Z g R Z C A Cgββ β= + + = + +

The frequency response is again calculated from the input and output voltage dividers.


Feedback on transconductance amplifiersSeries - Series connection: Add Z parameters

The feedback network is functionally a transimpedance amplifier from Port2 Port1Electrically both networks must be treated as transimpedance amplifiers P1 P2to account for the shared (input) electrical variables.In the calculation we consider the ELECTRICAL description:

Input: I1 , I2Output: V1 , V2


( )1 1 1 1 1 1 1

2 2 2 2 2 2 2

A B A BB

A B A B

V V V I I I IV V V I I I I⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= + = + = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

A A B A+BZ Z Z + Z Z

+

-V1

I1+

-+

-

+

-+

-

TranscondAmp A

FeedbackNet B

Port 1 Port 2



+

-V2

I2

We will study this type of connection when we study transistor amplifiers

Function of feedback net:Measure output ICorrect (mix) input Vi.e. it improves a Y-amp


The 2nd form of the Miller theorem – series feedback• consider an impedance connected in series with the common terminal of a current

amplifier of gain H.

• Looking from the input, the voltage developed on the feedback element is:

• Likewise, looking from the output, the amplifier has a gain=1/H, so the voltage developed on the feedback element is:

• These considerations lead to the equivalence of the two diagrams above in terms of their input and output impedance.

• NOTE: Only if the amplifier is ideal its current gain will not change as a result of the series-series feedback.

( ) ( ) ( ),1 1Z in out in M inV i i Z H Zv Y H Z= + = + ⇒ = +

,1 11 1Z out M outV Zi Z ZH H

⎛ ⎞ ⎛ ⎞= + ⇒ = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


Feedback on current amplifiersThe Shunt - Series connection: Add G parameters

+

-V2

I2

V1+

-

I1+

-+

-

+

-+

-

CurrentAmp A

FeedbackNet B

Port 1 Port 2



The feedback network is functionally a current amplifier from Port2 Port1Electrically both networks must be treated as voltage amplifiers P1 P2to account for the shared (input) electrical variables.In the calculation we consider the ELECTRICAL description:

Output: I1 , V2

Input: V1 , I2Shared electrical variables:

( )1 1 1 1 1 1 1

2 2 2 2 2 2 2

A B A B

A B A B

I I I V V V VV V V I I I I⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= + = + = + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

A B A B A+BG G G G G

We will study this type of connection when we study transistor amplifiers

Function of feedback net:Measure output ICorrect (mix) input Ii.e. it improves an H-amp


2-port network feedback connection rules

Y1

Y2

Y1+Y2

Z1

Z2

Z1+Z2

Shunt – Shunt: add Y

Series – Series: add Z

G1+G2

G1

G2

Shunt – Series: add G

H1

H2

H1+H2

Series – Shunt: add H


Series-shunt feedback: Effect on input – output impedance

A real op-amp has:• Finite input impedance• Finite output impedance• Finite GainTreat the feedback network as if it draws no current. This is equivalent to:The input impedance is derived from:

The output impedance is:

[ ]1 2,o iZ R R Z

( )( )

( )0

1

1out

in i out o in in i

in i in out o

inin i

in i

i Z G i Z H v i Z

i Z GH v i Z H

vZ Z GHi

=

+ = − ⇒

+ = − ⇒

∂= = +∂

( ) ( )01

in

out oin out out out o out

out v

v Zv v H G v i Z Zi GH

=

∂− = − ⇒ = =

∂ +

+

-

V0

Vi

R2R1

Zi G+-

Zo

Iin

Iout

Ideal amplifier


Positive feedbackSame analysis as negative feedback, apart for H -H

Positive feedback an be used to do things negative feedback cannot do:• Introduce hysteresis (e.g. Schmitt Trigger)• Generate negative impedances• Invert an impedance• Note that under positive feedback we can have F=1-GH=0• If F=0 we (in theory) can turn an amplifier into an ideal version by a suitable

feedback connection and GH=1. • GH=1, when it occurs at a finite (i.e. non zero) frequency, is the

Barkhausen condition for oscillation

The op-amp is called “operational” precisely because it can be used to perform mathematical operations – on signals (addition, subtraction, integration, differentiation,

multiplication by a scalar,…)– on operators (inversion)– on impedances (negation, inversion, multiplication, division,…)


The dominant pole approximation

•an op-amp has never an infinite gain at DC• the op-amp gain as a function of frequency is adequately described by:

Where ADC is the DC gain of the amplifier, typically 104 – 106

fp is the dominant pole frequency, typically 10 Hz.The product ADCfp is called the gain-bandwidth product (GBW).The GBW is a characteristic constant of the op-amp, typically 106-108

When we do AC analysis we must consider the finite complex gain ofThe amplifier, especially when we try to get high gain at high frequencies.

( )1 1 /

DC pDC DCv

p p

A fA AA fs jf f f jfτ

= = =+ + +


Invariance of the gain – bandwidth product

Consider a non-inverting amplifier, and a dominant pole op-ampApplying the feedback theory we get the closed loop gain:

1

DC p

p DC pV

DC p p DC p p

p

A ff jf A f GBWA A f f jf A f H f jf GBW HHf jf

+= = =

+ + + + ⋅++

The DC gain is: ( )0Vp

GBWA ff GBW H

= =+ ⋅

The pole of this amplifier is at: 0 pf f GBW H= + ⋅

It follows that the product of the DC gain of a non-inverting amp and its pole position equals the gain bandwidth product of the op-amp!This is only true for a dominant pole op-amp! (i.e. most voltage mode amplifiers)


Conclusions

• Feedback reduces gain• Feedback reduces component and environmental sensitivity• Feedback increases linearity• There are 4 ways to apply electronic feedback• Feedback can be used to modify input and output impedances:

– A series negative feedback increases impedance– A series positive feedback decreases impedance– A shunt negative feedback increases admittance– A shunt positive feedback decreases or zeroes admittance

• Positive feedback can lead to dynamic instability• Op-amps are modelled as dominant pole systems

– Op-amps have finite DC gain– Op-amps have a low frequency pole.– Amplifiers built with op-amp are subject to a constant GBW


2nd order filter transfer functions: Review

Second order filter transfer functions are all of the following form:

H0 is the overall amplitude, ω0 the break (or peak) frequency, and ζ the damping factor

ζ isrelated to the quality factor Q by: Q=1/2ζ

The 3dB bandwidth of an underdamped 2nd order filter is approx 1/Q times the peak frequency.

The coefficients A, B, C determine the function of the filter:

( ) ( )( )

20 0

0 20 0

/ 2 / 1 , 2/ 2 / 1

C s B s AH s H Q

s sω ζ ω

ζω ζ ω

+ += =

+ +

1-11All Pass101Band Stop010Band Pass100High Pass001Low PassCBAFunction


Tee – Pi transformations

1 3 3 1 3 3 1

3 2 3 3 2 3

32 11 2 3

1 2 1 3 2 3 1 2 1 3 2 3 1 2 1 3 2 3

32 11 2 3

1 2 1 3 2 3 1 2 1 3 2 3 1 2 1 3 2 3

, ,

, ,

, ,

Tee Pi Pi Tee

Z Z Z Y Y YZ Z Z Y Y Y

YY YZ Z ZYY YY Y Y YY YY Y Y YY YY Y Y

ZZ ZY Y YZ Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z

−+ + −⎡ ⎤ ⎡ ⎤= = = ⇒⎢ ⎥ ⎢ ⎥+ − +⎣ ⎦ ⎣ ⎦

⎧ = = =⎪ + + + + + +⎪⇒ ⎨⎪ = = =

+ + + + + +⎩

Z Y Y Z

⎪

When analysing active band pass or band stop active filters we often encounter the “twin-tee” passive notch filter topologyThis requires quite a bit of algebra to compute, so we prove a, useful for simplifying networks, lemma:

Z1 Z2

Z3Y1 Y2

Y3

Proof: write the z matrix of the Tee< and the y matrix of the Pi and require that the two circuits are representations of same network:

is equivalent to


Higher order filter synthesis using 2nd order sections• A general filter transfer function is of the form:

• P(s) and Q(s) have real coefficients. To make a higher order filter:– factor P(s) and Q(s) into quadratic and linear factors– Implement factors as biquads– Cascade biquad sections to obtain the original transfer function– Note that the roots of P, Q are real or come in conjugate pairs.

• The centre frequencies and damping factors of the sections required to implement standard forms (Butterworth, Chebyshev, Elliptic etc) are tabulated. Tables are included in CAD software for automatedsynthesis

( ) ( )( )

( )( ) ( )( )( ) ( )

0 10

0 1

0

nk

kn ni

mkm n

ki

a xP s s z s z s zH s

Q s s p s p s pb x

=

=

− − −= = =

− − −

∑

∑


A useful network transformation: Impedance inversion and the gyrator

A gyrator can perform• impedance inversion (L C)• Impedance scaling• series – parallel connection conversion!

“Proper” symbol of gyrator

Alternate symbolSimple active implementation (very popular by analogue CMOS designers.Each gm is made of a MOSFET or two!)


*Passive Gyrators• ¼ wavelength transmission line

• Pi and Tee networks with negative elements

negative values of components will be added to preceding and subsequent stage impedances resulting in overall positive impedances! Note that for narrowband signals, eg, –L is a capacitor!

Ladder LC filters can be synthesised only with capacitors and gyrators

Z, -Z is completely arbitrary, can be a filter transfer function and more…


•Two identical gyrators in series are the identity operator

•Two different gyrators in series perform direction sensitive impedance multiplication by a constant:

Gyrator function - basics

• A series (floating) component between two gyrators appears inverted and grounded

• A grounded component between two gyrators appears inverted and in series


Some more gyrator identities

or, how to make e.g. a series resonance circuit when you only have parallel resonators in your component box… and vice versa

2ports, feedback and filters - circuits and...

Documents