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*THE NEGATIVE-FEEDBACK LOOP - U of S 323 -Feedback and stability Page 16 of 27 *** BANDWIDTH...*

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EE 323 -Feedback and stability Page 1 of 27

FEEDBACK AND STABILITY

THE NEGATIVE-FEEDBACK LOOP

xS = input signal xOUT = AxIN xF = feedback signal xIN = xS xF linear feedback: xF = xOUT =feedback factor ( constant)

xOUT = AxIN = A(xS xF) xOUT = A(xS xOUT)

xIN XOUT A A

xS xIN xOUT

xF

+

_ Feedback network

Output Signal source

Open loop Closed loop

EE 323 -Feedback and stability Page 2 of 27

xOUT depends upon itself a property intrinsic to the nature of a feedback path

xOUT(1+A) =AxS or A1A

xx

AS

OUTfb +

==

Afb = closed-loop gain (gain with feedback) A>>1

1AAAfb =

Closed-loop gain, Afb is independent of A in the limit A>>1, depends only on the feedback factor .

This feature is important. It allows Afb to be precisely set regardless of the exact value of A.

EE 323 -Feedback and stability Page 3 of 27

Feedback network is generally made from passive (and easy-to-control) circuit elements and factors that affect A (component variations, temperature, and circuit non-linearity) become much less important to the closed-loop circuit.

Worth the price of reducing gain from AOL to ACL.

Example: Non-inverting op-amp configuration

EE 323 -Feedback and stability Page 4 of 27

GENERAL REQUIRMENTS OF FEEDBACK CIRCUITS

Signals at summing node must be the same type (i.e., all voltages or currents).

The output, xOUT, needs not be of the same signal type as its input. The amplification factor, A, can have dimension units:

Av=Volt/Volt or Ai=Ampere/Ampere Ar=Volt/Ampere or Ag=Ampere/Volt.

Feedback function, , must have units reciprocal to those of A, (i.e., product A is dimensionless -- ensures xF is the same signal type as xS and xIN).

In general, feedback network is made from passive components only and never exceeds unity.

In the feedback loop, xF is subtracted from xS, making the feedback negative.

EE 323 -Feedback and stability Page 5 of 27

If xF is added to xS at the summation node, the feedback becomes positive (oscillator, active filters.)

Negative feedback benefits (desirable in amp. design): Reducing amplifier non-linearity, Improving input and output impedance, Extending amplifier bandwidth, Stabilizing gain, and reducing amplifier sensitivity to transistor

parameters. .

EE 323 -Feedback and stability Page 6 of 27

FOUR TYPES OF NEGATIVE FEEDBACK

Four basic amplifier types

EE 323 -Feedback and stability Page 7 of 27

a. A voltage amplifier with gain Av

b. A current amplifier with gain Ai

c. A transconductance amplifier or voltage-to-current converter. The amplification factor, Ag, or

gm = iOUT/vIN, (A/V or conductance).

d. A transresistance amplifier or current-to-voltage converter. The amplification factor, Ar or

rm, = vOUT/iIN (V/A or resistance).

EE 323 -Feedback and stability Page 8 of 27

The four types of negative feedback In Out Circuit zin zout Converts Ratio Symbol Type of Amplifier

V V VCVS 0 - vo/vi Av Voltage amplifier I V ICVS 0 0 i to v vo/ii rm Trans-resistance amplifier V I VCIS v to i io/vi gm Trans-conductance amplifier I I ICIS 0 - io/ii Ai Current amplifier

vi vo Avvi HIGH

LOW

VCVS

ii

vo riii LOW

LOW

ICVS

io

gmvi HIGH HIGH

VCIS

~ ~

io

Aivi LOW HIGH

ii

ICIS

vi

EE 323 -Feedback and stability Page 9 of 27

VOLTAGE-CONTROLLED VOLTAGE SOURCE (VCVS)

High input impedance Low output impedance Stiff voltage source

Feedback fraction:

Closed loop gain: Loop gain:

Error between ideal and exact values:

211

out1R

RRR

vv

+==

12

121

OL

OL

OL

OLCL R

R1R

RR1AA

A1AA +=+=

+=

vin vout + _

R1

+VCC

-VEE R2

A1%100Error%OL+

=

A1Gain OLCL +=

EE 323 -Feedback and stability Page 10 of 27

Impedances:

Output voltage: vin=Avout v Negative feedback: Stabilizes voltage gain, Increases input impedance,

Decreases output impedance, Reduces nonlinear distortion of the amplified signal.

a. Gain stability:

The gain is stabilized because depends only on the external resistances (i.e., can be precision resistors).

The gain stability depends on having a low percent error between the ideal and the exact closed-loop voltage gains.

The smaller the percent error, the better the stability. The worst-case error of closed-loop voltage gain occurs when the open-loop

voltage gain AOL is minimum.

A1%100errorMaximum%(min)OL+

=

A1RZR)A1(Z

OL

outoutinOLin +

=+=

EE 323 -Feedback and stability Page 11 of 27

b. Nonlinear distortion:

non-linear distortion will occur with large signals. input/output response becomes non-linear. Nonlinear also produces harmonics of the input signal.

Total harmonic distortion:

%100voltagelFundamenta

voltageharmonicTotalTHD =

EE 323 -Feedback and stability Page 12 of 27

Negative feedback reduces harmonic distortion (closed-loop harmonic distortion):

A1THDTHD

OL

OLCL +

=

Quantity 1+AOL has a curative effect. When it is large, it reduces the harmonic distortion to negligible levels, (ex.., high-fidelity sound in audio amplifier system).

Example 19-1, 19-2, 19-3, 19-4 (page 667)

EE 323 -Feedback and stability Page 13 of 27

CURRENT-CONTROLLED VOLTAGE SOURCE (ICVS)

Low input impedance, Low output impedance. Stiff voltage source from a current input. Trans-resistance (rm) (i.e., output voltage is proportional to the current by a

resistance).

R2 can be selected to have different conversion factors (trans-resistances).

Input and output impedances:

Example: inverting amplifier, 19-5, 19-6 (page 674)

2inOL

OL2inout RiA1

ARiv =+

=

OL

out)CL(out

OL

2)CL(in A1

RZA1

Rz+

=+

=

iin vout _

+

+VCC

-VEE

R2

EE 323 -Feedback and stability Page 14 of 27

VOLTAGE-CONTROLLED CURRENT SOURCE (VCIS)

Transconductance, gm, (i.e., 1/R) Both input and output impedances are high Stiff current source.

Input and output impedances:

Example 19-7 (page 677)

1minm

1

inout

OL

211

inout

R1gwherevg

Rvi

A)RR(R

vi

===

++=

1OL)CL(out

inOL)CL(in

R)A1(Z

R)A1(Z

+=

+=

vin iout + _

R1

+VCC

-VEE RL=R2

EE 323 -Feedback and stability Page 15 of 27

CURRENT-CONTROLLED CURRENT SOURCE (ICIS)

Low input impedance, High output impedance . Stiff current source. Current gain factor Ai.

Input and output impedances:

Example 19-8 (page 678)

iin

iout _ +

R1

+VCC

-VEE RL

R2

1RR

RAR)RR(AA

1

2

1OLL

21OLi ++

+=

21

1

OL

2)CL(in RR

RwhereA1RZ

+=

+= 1OL)CL(out R)A1(Z +=

EE 323 -Feedback and stability Page 16 of 27

BANDWIDTH

Negative feedback increases the bandwidth of an amplifier. Because of the roll-off in open-loop voltage gain means less voltage is fed

back, which produces more input voltage as a compensation. Closed-loop cutoff frequency is higher than the open-loop cutoff frequency.

The closed-loop cutoff frequency:

Gain Bandwidth Product

Gain bandwidth product is constant

unity)CL(2CL f