FEEDBACK AND STABILITY

FEEDBCK ND STBILITY THE NEGTIVE-FEEDBCK LOOP x IN X OUT x S + x IN x OUT Σ Signal source _ β Open loop Closed loop x F Feedback network Output x S input signal x OUT x IN x F feedback signal x IN x S x F linear feedback: x F βx OUT βfeedback factor ( constant) x OUT x IN (x S x F ) x OUT (x S βx OUT ) EE 323 -Feedback and stability Page of 27

x OUT depends upon itself a property intrinsic to the nature of a feedback path x OUT (+β) x S or fb closed-loop gain (gain with feedback) β>> fb Closed-loop gain, fb is independent of in the limit β>>, depends only on the feedback factor β. fb x OUT x S + This feature is important. It allows fb to be precisely set regardless of the exact value of. EE 323 -Feedback and stability Page 2 of 27

Feedback network is generally made from passive (and easy-tocontrol) circuit elements and factors that affect (component variations, temperature, and circuit non-linearity) become much less important to the closed-loop circuit. Worth the price of reducing gain from to CL. Example: Non-inverting op-amp configuration EE 323 -Feedback and stability Page 3 of 27

GENEL EQUIMENTS OF FEEDBCK CICUITS Signals at summing node must be the same type (i.e., all voltages or currents). The output, x OUT, needs not be of the same signal type as its input. The amplification factor,, can have dimension units: v Volt/Volt or i mpere/mpere r Volt/mpere or g mpere/volt. Feedback function, β, must have units reciprocal to those of, (i.e., product β is dimensionless -- ensures x F is the same signal type as x S and x IN ). In general, feedback network is made from passive components only and β never exceeds unity. In the feedback loop, x F is subtracted from x S, making the feedback negative. EE 323 -Feedback and stability Page 4 of 27

If x F is added to x S at the summation node, the feedback becomes positive (oscillator, active filters.) Negative feedback benefits (desirable in amp. design): educing 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 5 of 27

FOU TYPES OF NEGTIVE FEEDBCK Four basic amplifier types EE 323 -Feedback and stability Page 6 of 27

a. voltage amplifier with gain v b. current amplifier with gain i c. transconductance amplifier or voltage-to-current converter. The amplification factor, g, or g m i OUT /v IN, (/V or conductance). d. transresistance amplifier or current-to-voltage converter. The amplification factor, r or r m, v OUT /i IN (V/ or resistance). EE 323 -Feedback and stability Page 7 of 27

The four types of negative feedback In Out Circuit z in z out Converts atio Symbol Type of mplifier V V VCVS 0 - v o /v i v Voltage amplifier I V ICVS 0 0 i to v v o /i i r m Trans-resistance amplifier V I VCIS v to i i o /v i g m Trans-conductance amplifier I I ICIS 0 - i o /i i i Current amplifier LOW i i LOW v i HIGH LOW ~ ~ v v i v o r i i i v o VCVS ICVS i o i i i o v i HIGH g m v i HIGH LOW i v i HIGH VCIS ICIS EE 323 -Feedback and stability Page 8 of 27

VTGE-CONTLED VTGE SOUCE (VCVS) High input impedance +V CC Low output impedance Stiff voltage source v in + _ -V EE 2 v out Feedback fraction: v v out + 2 Closed loop gain: GainCL + Loop gain: + 2 CL + + 2 Error between ideal and exact values: %Error 00% + EE 323 -Feedback and stability Page 9 of 27

Impedances: Z in ( + ) in Z out + out Output voltage: v in v out v Negative feedback: Stabilizes voltage gain, Increases input impedance, Decreases output impedance, educes 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 is minimum. %Maximum error 00% + (min) EE 323 -Feedback and stability Page 0 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: Total harmonic voltage THD 00% Fundamental voltage EE 323 -Feedback and stability Page of 27

Negative feedback reduces harmonic distortion (closed-loop harmonic distortion): THD THD CL + Quantity + β 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 9-, 9-2, 9-3, 9-4 (page 667) EE 323 -Feedback and stability Page 2 of 27

CUENT-CONTLED VTGE SOUCE (ICVS) Low input impedance, Low output impedance. Stiff voltage source from a current input. Trans-resistance (r m ) (i.e., output voltage is proportional to the current by a resistance). _ 2 +V CC v out v out iin2 + i in 2 i in + -V EE 2 can be selected to have different conversion factors (trans-resistances). Input and output impedances: z in ( CL ) + 2 Z out ( CL ) + out Example: inverting amplifier, 9-5, 9-6 (page 674) EE 323 -Feedback and stability Page 3 of 27

VTGE-CONTLED CUENT SOUCE (VCIS) Transconductance, g m, (i.e., /) Both input and output impedances are high Stiff current source. v in + _ +V CC i out -V EE L 2 i i out out v in vin ( + + g m v in 2 ) where g m Input and output impedances: Z Z in(cl) out(cl) ( + ( + ) ) in Example 9-7 (page 677) EE 323 -Feedback and stability Page 4 of 27

CUENT-CONTLED CUENT SOUCE (ICIS) Low input impedance, High output impedance. Stiff current source. Current gain factor i. _ +V CC i in + i out -V EE L ( + + ) 2 2 i + L 2 Input and output impedances: Z in(cl) 2 where Z + + out(cl) ( + ) 2 Example 9-8 (page 678) EE 323 -Feedback and stability Page 5 of 27

BNDWIDTH 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 f 2(CL) f unity CL GBP Gain frequency Gain bandwidth product is constant f CL f CL f CL 2(CL) f unity GBP f unity constant for a given op-amp. Trade off gain to bandwidth in design. Less gain, more bandwidth (more gain, less bandwidth). Unity frequency determines GBP of the op-amp. EE 323 -Feedback and stability Page 6 of 27

Higher unity frequency op-amp may be needed for specific application (requires both high gain and high bandwidth.) Example 9-9, 9-0, 9-, 9-2, 9-3 (page 682) EE 323 -Feedback and stability Page 7 of 27

FEEDBCK LOOP STBILITY Stability of the negative feedback loop must be examined to verify that unwanted oscillations will not occur. Output of a linear system experiences a relative phase shift of 90 O if the driving frequency increases beyond one of the poles of the system function. System with three or more poles, a frequency will exist at which the phase shift exceeds 80 O. EE 323 -Feedback and stability Page 8 of 27

t some frequency, ω 80, the 80 O phase shift will change an otherwise negative feedback loop into a positive feedback loop. The response of the feedback loop at ω 80 : ( ) + ( ) EE 323 -Feedback and stability Page 9 of 27 v out 80 80

If 80 β, the denominator becomes 0 and the output becomes infinite (even v in 0). Such a condition is equivalent to an oscillation at the frequency ω 80. Less stringent inequality 80 β also leads to oscillation at ω 80. Note: in practice, the saturation limits of the op-amp limit the magnitude of oscillation. I. FEEDBCK LOOP COMPENSTION Use frequency compensation to prevent unwanted oscillations (at ω 80 ). lter the open loop response so that the stability condition is met: 80 β < Internal design (stability condition is met up to some maximum value of β, i.e., β). The LM74 is stable under all negative feedback conditions. The value of 80 of LM74 is less than unity. EE 323 -Feedback and stability Page 20 of 27

dding external components to the feedback loop: Evaluate the feedback loop for stability If the feedback loop is unstable, external compensation must be added. External compensation is sometimes preferred over internal compensation because the latter limits the GBP of the feedback loop. EE 323 -Feedback and stability Page 2 of 27

II. EVLUTION OF STBILITY CONDITION Gain margin and phase margin to determine stability: Gain Margin (j ) 80 *** Gain margin must be positive (i.e., 80 β < ). 80 Phase Margin arg(( j ) ) ( 80 ) 80 + arg( (PM) 0 0 (j ) (PM) ) j ) ( at ω PM. *** Negative phase margin: ( j ) is greater than unity at ω 80 (i.e., circuit unstable.) One margin passes the stability test, the other will also. Design a feedback loop with excess gain or phase margin to ensure stability. EE 323 -Feedback and stability Page 22 of 27

Example: Transfer function open-loop frequency response: where 0 0 6, ω 0rad/sec, ω 2 ω 3 0 6 rad/sec. (j ) ( + j )( + j 0 2 )( + j 3 ) Since ω << ω 2, the dominant pole of (jω) ω. a. How large can the feedback parameter β become before instability results? ssume β is not a function of frequency. b. Design a non-inverting amplifier that meets the stability conditions. Solution: Since β is not frequency dependent, ω 80 occurs at the frequency where the angle of the transfer function is at 80 O, that is where: arg( 80 80 80 0 ( j ) ) tan ( ) tan ( ) tan ( ) 80 2 3 solving for ω: ω 80 0 6 rad/sec EE 323 -Feedback and stability Page 23 of 27

The magnitude of (jω) at ω 80 as follows: 6 6 0 0 80 6 6 6 5 2 ( + 0 j 0 )( + 0 j 0 6 )( + 0 j 0 6 ) (0 )( 2) 5 Stability of the feedback loop requires: j ) 5 ( 0. 2 For the circuit to be stable, the closed-loop gain must therefore meet the minimum condition: v v out in 5 v in + _ +V CC v out -V EE 2 4 EE 323 -Feedback and stability Page 24 of 27

EXTENL COMPENSTION compensation network is added to an under-compensated negative feedback loop. Example: a. Show that the minimum allowed β that ensures stability is 0.00. b. To what closed-loop gain does β correspond? c. Design a compensation network that will stabilize the op-amp in a noninverting amplifier with a gain of 5 (i.e., 4dB). EE 323 -Feedback and stability Page 25 of 27

Solution: f 80 3.2*0 6 Hz (from the angle portion of the Bode plot.) 80 60dB000 ( j ) β < 0-3. non-inverting amplifier gain of 5 (β0.2) will be unstable. Compensation network adds a pole to the op-amp open-loop response at a frequency below f 80, The open-loop gain at f 80 will be reduced so that the stability condition can be met. If the pole is located well below f 90, it will add an additional 90 O phase shift at f 90, bringing the total phase shift at f 90 to 80 O. The original f 90 before compensation will become the new f 80 after compensation. 90 or 90 + 0 db db The uncompensated op-amp has a gain magnitude of 90dB at f 90. If an amplifier with closed-loop gain of 5 (i.e., 4dB) is to be made, then β/5 (i.e., - 4dB) and 90 must be reduced from 90dB to 4dB (i.e., 76dB). This can be achieved with a pole at f c 50.7Hz. How???????? EE 323 -Feedback and stability Page 26 of 27

o bove f c, the pole at f c will increase the roll-off by 20dB/decade. f f 90 c f c f 0 76dB 3.8decades 20dB / decade 90 3.8 5 3.2 0 Hz 50.7Hz 3 6.3 0 o The compensation pole at f c introduced by a simple C filter o C c c 2 f 39rad / sec EE 323 -Feedback and stability Page 27 of 27