Microelectronic Circuits II Ch 9 : Feedback

Microelectronic Circuits II

Ch 9 : Feedback

9.1 General Feedback structure

9.2 Some properties of Negative feedback 9.3 The Four basic Feedback topologies

9.4 The Feedback Voltage Amplifier (Series-Shunt)

CNU EE 9.1-2

Feedback

Theory of negative feedback

- Electronics engineer, Harold Black invented the feedback amplifier in 1928

- Search for a methods for the design of amplifiers with stable gain for use in telephone repeaters - Feedback : negative (degenerative) or positive (regenerative)

- Almost all op-amp circuits employ negative feedback Properties of Negative feedback

- Desensitize the gain

the value of gain is less sensitive to variations in the value of circuit components (Temperature change) - Reduce nonlinear distortion

the output is proportional to the input & gain is constant, independent of signal levels - Reduce the effect of noise

minimize the contribution to the output of unwanted electric signals generated - Control the input and output impedance

raise or lower the input and output impedance - Extend the bandwidth of the amplifier

Expense of the desirable properties of negative feedback - Reduction in gain

- gain-reduction factor = amount of feedback

ß circuit is desensitized, input impedance of a voltage amplifier is increased, bandwidth is extended - The basic idea of negative feedback is to trade off gain for other desirable properties

The General Feedback Structure

loop gain

b

A A x

A x

s o

f

º = +

1

if A

b

b

= 1 Af

- Gain of the feedback amplifier is determined by the feedback network

- Feedback network = passive components à accurate, predictable & stable gain

s

f x

A x A

b b

= +

1 & if Ab >>1, x_f » x_s The gain of feedback amp.

Signal-flow diagram of the feedback amplifier - open-loop gain A, feedback factor b

- negative feedback reduces the signal that appears at the input of the basic amplifier

i

o Ax

x =

o

f x

x = b

f s

i x x

x = -

) (

)

( _{s f s o}

i

o Ax A x x A x x

x = = - = - b

Subtraction makes the feedback negative

Gain-with-feedback A_f is smaller than the open-loop gain A by the amount of feedback, 1+Ab

A_f : closed-loop gain

s

i x

x A

b

= + 1

1

CNU EE 9.1-4 - the feedback amplifier has a midband gain of A_M/(1+ A_M b) and upper 3-dB frequency w_Hf :

Some properties of Negative feedback

§Gain Desensitivity : sensitivity reduction property

The percentage change in A_f (due to variations in some circuit parameters) is smaller than the percentage change in A by the amount of feedback (b is constant)

§Bandwidth Extension

- Amplifier whose high-frequency response is characterized by a single pole

( ) ^>

>

1 b 1 A b

dA dA A

A

A

= +

= +

( ) ^A

dA A

A dA

f f

b

= + 1

1

( )

( b )

w

b b

w

M M

f f

H M

A s

A s A

s A A

s s A

s A s A

A + +

= +

= +

= +

1 /

1

1 ) /

) ( ( 1

) ) (

(

&

) 1

( >

) 1

( b

w

w

=

+ A

Desensitivity factor

w_Hf : Upper 3dB freq.

A_M: midband gain, w_H : upper 3-dB freq.

b: freq.-independent factor

Midband gain

- If the open-loop gain has a dominant low-frequency pole giving rise to a lower 3-dB frequency w_L, then the feedback amplifier have a lower 3-dB frequency w_Lf :

A_f(s) : closed-loop gain

b w w

M L

Lf = + A

1

à w_Hf is increased by the amount of feedback

à w_Lf is decreased by the amount of feedback

Some properties of Negative feedback

§Bandwidth Extension

Amplifier bandwidth is increased by the amount of feedback by which its midband gain is decreased, maintaining the gain-bandwidth product at a constant value

Negative feedback works to minimize the change in gain magnitude, including its change with frequency

CNU EE 9.1-6

Some properties of Negative feedback

§Noise reduction

- reduces the noise or interferences in an amplifier = increase the ratio of signal to noise

n s

V I V S / =

b

b

1 2

1 2 1

1

1 A A

V A A

A A V A

V_{o s n}

+ +

= +

A

V V I

S

n

=

\

- Improvement in S/I ratio by the application of feedback is possible only if one can precede the interference-prone stage by a (relatively) interference-free stage

- Application : reduction of the power supply hum in the output power-amplifier stage of an audio amplifier à hum at the output is reduced by the amount of the voltage gain of this added preamplifier

- Amplifier w/ gain A₁, input signal V_s, noise or interference V_n

- Signal-to-interference Ratio (=S/N ratio)

- Amplifier w/ gain A₂, that does not suffer from the noise problem à clean amplifier - apply negative feedback b to keep the

overall gain constant

A₂ times higher than in the original case

Some properties of Negative feedback

§Reduction in nonlinear distortion

- Amplifier transfer characteristic is considerably linearized (i.e., made less nonlinear) because

negative-feedback reduces the dependence of the overall closed-loop amplifier gain on the open-loop gain of the basic amplifier

- Change in open-loop gain : 1000 to 100 - b = 0.01

- Resulting transfer characteristic of the closed-loop amplifier

9 . 01 90

. 0 1000 1

1000

1 =

´

= + Af

01 50 . 0 100 1

100

2 =

´

= + Af

b

A A_f A

= + 1

- Order-of-magnitude change in slope is considerably reduced - Reduction in voltage gain à preamplifier should be added

- Negative feedback can do nothing at all about amplifier saturation, since in saturation the gain is small (almost zero) and hence the amount of feedback is also very small (almost zero)

CNU EE 9.1-8

Four basic Feedback topologies

Voltage-mixing voltage-sampling (series-shunt)

Current-mixing current-sampling (shunt-series)

Voltage-mixing current-sampling (series-series)

Current-mixing voltage sampling (shunt-shunt)

Based on the quantity to be amplified (voltage or current) and on the desired form of output (voltage or current)

Voltage Amplifiers

- Amplify an input voltage signal and provide an output voltage signal

- Voltage controlled voltage source ( high input impedance & low output impedance are required)

- Since output is voltage in the voltage amplifier, the feedback network should sample the output voltage - Because of the Thevenin representation of the source, feedback signal x_f should be a voltage

à voltage x_f is mixed with the source voltage in series

- Suitable feedback topology for the voltage amplifier = voltage-mixing, voltage-sampling

- Series connection at the input and parallel or shunt connection at the output à Series-shunt feedback - Stabilize the voltage gain

- High input resistance caused by the series connection at the input - Low output resistance caused by the parallel connection at the output

Sample the output voltage Mixed with the source

voltage in series

§ Voltage Amplifiers

CNU EE 9.1-10

Voltage Amplifiers

- Increased input resistance : V_f subtracts from V_s à smaller V_i at the input of basic amplifier à smaller input current à larger resistance seen by V_s

- decreased output resistance : feedback works to keep V_o as constant as possible

à current change DI_o lowers the change DV_o in V_o à lower output resistance DV_o/DI_o

§Example of series – shunt feedback amplifiers - Noninverting op-amp configuration

- Feedback network = voltage divider (R₁, R₂) develops V_f à negative input terminal of op-amp

- Negative feedback : V_f must be of the same polarity as V_s à a smaller signal at the input of the basic amplifier

- As V_s , V_o & voltage divider à V_f : change in V_f is of the same polarity as the change in V_s à negative feedback

- Two MOSFET amplifier stages in cascsade

- Feedback network = voltage divider (R₁, R₂) develops V_f à source terminal of Q₁

- Subtraction by applying V_s to the gate of Q₁& V_f to its source à amplifier input signal V_i = V_gs = V_s – V_f

- As V_s , drain voltage of Q₁ = gate of Q₂ à drain voltage V_o à feedback network voltage divider à V_f : change in V_f is of the same polarity as the change in V_s à negative feedback

Current Amplifiers

Sample the output current Mixed with

the source in shunt

- Input : Norton equivalent current source

- Feedback network samples the output current

- Feedback signal = current à mixed in shunt with the current source - Current-mixing current-sampling

- parallel (or shunt) connection at the input and series connection at the output à Shunt-series feedback - Stabilize the current gain

- Lower input resistance : I_s – I_f à lower input voltage across the current source I_s

- Higher output resistance : negative feedback keeps I_o as constant & if the voltage across R_L is changed, the resulting change in I will be lower than it would have been without the feedback

CNU EE 9.1-12

Current Amplifiers

§A CG stage Q₁ followed by a CS stage Q₂ - Load current I_o is fed to a load resistance R_L - A small resistance R_M in series with R_L

à a sample of I_o

- The voltage developed across R_M is fed via a large resistance R_Fto the source node of Q₁ - The feedback current I_f that flows through R_F is

subtracted from I_s at the source node à input current I_i = I_s – I_f

- For negative feedback, I_f must have the same polarity as I_s

Current sampling Reference

direction of I_f: I_fsubtracts from I_s

§ Qualitative check for feedback polarity I_s à I_i à drain voltage of Q₁ = gate of the p-channel device Q₂ à drain current of Q₂, I_o à voltage across R_M à I_f

:: the same polarity assumed for the initial change in I_s à negative feedback

Transconductance Amplifiers

Sample the output current Mixed with

the source in series

- Input signal : voltage - Output signal : current

- Voltage-mixing current-sampling

- Series-series feedback (series connection at both the input and the output)

- Series connection at input à increased R_in - Output current sampling à increased R_o

§Differential amplifier A₁ followed by a CS stage Q₂

- Output current I_o is fed to R_L & series resistance R_F à R_F develops V_f à V_f is applied to the positive input terminal of A₁ - Subtraction of V_s – V_f by differential action

§Check that V_f & V_s have the same polarity - Positive change in V_s à negative change at

the gate of Q₁ à I_o à Positive change in V_f :: same polarity assumed for the change in V_s à negative feedback

CNU EE 9.1-14

Transresistance Amplifiers

Sample the output voltage Mixed with the

source in shunt

- Input signal : current - Output signal : voltage

- Current-mixing voltage-sampling - Shunt-shunts feedback (parallel (or

shunt) connection at both the input and the output)

- Shunt connection at input à reduces R_in - Shunt connection at output à reduced R_o

§Op-amp w/ a feedback resistance R_F

- R_F senses V_o & provides a feedback current I_f - I_f is subtracted from I_s at the input node

§Qualitative check for negative feedback - I_s à input current I_i à voltage of the

negative input terminal à output voltage à I_f :: I_f & I_s have the same polarity à negative feedback

Feedback Voltage Amplifier (Series-Shunt )

§Ideal case

i o

V A º V

b A A V

A V

s o

f º = +

1

( )

( )

i s if

i s i

i i

s i

R I A

R V

R A V R

I V A

V V

b

b b

+

= º

= + + =

=

1

1 1

- Unilateral open-loop amplifier (A circuit) à input resistance R_i, voltage gain A,

output resistance R_o

à source & load resistances are included inside the A circuit

- Ideal voltage-mixing voltage-sampling feedback network (b circuit)

à does not load the A circuit à does not change the value of A

Ideal structure

§Input resistance with feedback (ideal case)

Series-mixing feedback increases the input resistance by a factor equal to the amount of feedback, (1+Ab)

Increased R_if is independent of the type of A_f : open-circuit voltage gain of the feedback amplifier

CNU EE 9.1-16

§Ideal situation : Output Resistance with feedback

x x

of I

R = V

Apply V_x between the output terminals

o i x

x

R

AV I V -

=

Set V_s = 0

x o

f x

f

i V V from V V V

V = - = -b = b = b

( )

o x

x R

A

I V +

b

=

\ 1

b

A R_of R^o

= + 1

- Shunt sampling (or voltage sampling) at the output decreases the amplifier output resistance by a factor equal to the amount of feedback, (1+Ab)

- The reduction of R_of does not depend on the method of mixing

From the input loop

Feedback Voltage Amplifier (Series-Shunt )

§Practical situation

- Feedback network is not an ideal voltage controlled voltage source

à resistive and hence, load the basic amplifier

à affect A, R_i & R_o

- Source resistance R_s & load resistance R_L affect A, R_i & R_o

- Derivation of A circuit & b circuit from a given series-shunt feedback amplifier (Practical series-shunt feedback amplifier) Ideal structure

•Practical series-shunt feedback amplifier

- Source and load resistances should be lumped with the basic amplifier - Two-port feedback network is

represented in terms of h parameters

Feedback Voltage Amplifier (Series-Shunt )

R & R vs. R & R vs. R & R

CNU EE 9.1-18

§Derivation of A circuit & β circuit

use of h parameters (appendix C)

ú û ù ê ë é ú û ù ê ë

= é ú û ù ê ë é

2 1 22

21

12 11

2 1

V I h

h

h h

I V

Feedback Circuit Input Impedance (w/ Output short)

Feedback Circuit Output Admittance (w/ input open)

negligible

β

Practical series-shunt feedback amplifier w/ the feedback network represented by its h parameter Feedback network is

represented by a series network at port 1 and a parallel network at port 2

Feedback Voltage Amplifier (Series-Shunt )

R_i & R_o vs. R_in & R_outvs. R_if & R_of

§Derivation of A circuit and β circuit

º 1

= V

b

amplifier basic network

feedback

h

h

<<

network feedback amplifier

basic

h

h

<<

- Current source h₂₁I₁ : forward transmission of the feedback network

- Passive feedback network à forward transmission is neglected in comparison to the much larger forward transmission of the basic amplifier

à Omit the controlled source h₂₁I₁

- Includesh₁₁and h₂₂with the basic amplifier - If the basic amplifier is unilateral,

- Loading effect of feedback network on basic amplifier is represented by h₁₁& h₂₂

- h₁₁ : impedance looking into port 1 of the feedback network with port 2 short-circuited - h₂₂ : conductance looking into port 2 of the feedback network with port 1 open-circuited - loading effectof the feedback to basic Amp. : If connection is shunt, short-circuit the port;

if connection is series, open-circuit it

-bshould be found w/ port 1 open-circuited

Feedback Voltage Amplifier (Series-Shunt )

CNU EE 9.1-20

§Summary : the Rules for Finding A circuit and β circuit

- R_i & R_o : input & output resistance of the A circuit - R_if & R_of : input & output resistance of the feedback amplifier, including R_s & R_L - Actual input & output resistance of the feedback amplifier usually exclude R_s

& R_Là R_in & R_out

in if s

R = R - R

1 1

out 1

of L

R R R

æ ö

= çç - ÷÷

è ø

Feedback Voltage Amplifier (Series-Shunt )

1 0 1 11

2=

º I V

R V

2 0 2 22

1=

º V I

G I

Example 9.3

Op-amp connected in a noninverting configuration (open-loop gain m=10⁴, differential input resistance R_id=100kW, output resistance r_o=1kW). Use the feedback method to analyze the circuit taking both R_id and r_o into account. Find A, b, closed-loop gain V_o/V_s, input resistance R_in & output resistance R_out.

Feedback Voltage Amplifier (Series-Shunt )

1 0 1 11

2=

º I V

R V

2 0 2 22

1=

º V I

G I

A circuit

b circuit

2 0 1

1=

º V I

b V

Feedback network samples the output voltage V_o and provides a voltage signal (across R₁) that is mixed in series with the input signal V_s

Loading effect of the feedback network at input/ output side : R & G R_L=2kW, R₁=1kW,

R₂=1MW, R_s=10kW

CNU EE 9.1-22

( )

[ ]

( )

[ ]

(

)

R r

R R

R

R R

R V

A V

s id

id o

L L i

o 6000 /

//

//

//

2 1

2 1

2

1 »

+

× + + +

= +

º

m

V A V

A V

A V

s o

f 857 /

7 6000

1 = =

= +

º

b

V R V

R R V

V

o

f 10 ³ /

2 1

1 -

+ »

=

b

( )

W

=

´

=

+

=

k A

R R_{if i}

777 7

111

1

b

( )

+ +

= R R R R k

R_{i s id 1} // ₂ 111

W

= -

= R R k

R

739

W + =

= 95.3

1 A

b

(

)

= r //R // R₁ R₂ 667 R_{o o L}

L out

of

R R

R = //

Feedback Voltage Amplifier (Series-Shunt )

Input resistance of A circuit

Input resistance of feedback amplifier

Output resistance of A circuit

Output resistance of feedback amplifier