Analysis and Design of Analog Integrated Circuits Lecture 12. Feedback

Open Loop Versus Closed Loop Amplifier Topologies Open Loop Closed Loop Z f Source Source V src Z src Amp V src Z src Amp / / (db) (db) w bw w 1 w 2 w bw w 1 w 2 Open loop want all bandwidth limiting poles to be as high in frequency as possible Closed loop want one pole to be dominant and all other parasitic poles to be as high in frequency as possible 2

Consider an Open Loop Integrator w unity s / (db) w unity Parameterize integrator in terms of its unity gain frequency, w unity rad/s - Define = w unity /s - Note that H(w unity ) = 1 3

Now Surround the Integrator with a Feedback Path Z 2 20log /( -V x ) (db) Z 1 V x w unity The feedforward path is = w unity /s The feedback path is formed by Z 1 and Z 2 Derivation of closed loop transfer function: V x Z 2 = V x 0 Z 1 and = ( V x ) = 1+Z 1 /(Z 1 + Z 2 ) 4

Observations of Impact of Feedback 20log / Z 2 Z 1 V x 20log(1/β) w bw w unity Define = Z 1 /(Z 1 +Z 2 ) and rewrite transfer function as: = 1+Z 1 /(Z 1 + Z 2 ) = 1+β At low frequencies: At high frequencies: = s 0 1+β = 1 β = s (since β 1) 5

General View of Feedback 20log / Z 2 Z 1 V x 20log(1/β) β w bw w unity Closed loop transfer function: = V - in 1+β This is called Black s formula At low frequencies: At high frequencies: = 1 = s 0 β s 6

General Observations of Feedback 20log / β 20log(1/β) = 1+β w bw w unity The feedback path sets the closed loop gain at low frequencies - Assumes the open loop gain is large at low frequencies - Implies that accurate closed loop gain can be achieved at low frequencies despite variations in open loop gain The feedback path also influences the closed loop bandwidth 7

Gain Bandwidth Product for Closed Loop Systems 20log / = β 1+β The low frequency gain is: 20log(1/β) w bw s 0 = 1 β w unity The bandwidth roughly corresponds to: H(w bw ) 1 β w unity For = w unity w bw 1 β w unity = 1 β w bw s Closed loop systems exhibit constant gain-bandwidth product set by the unity gain frequency of the open loop amplifier 8

Example: Unity Gain Amplifier 20log / β = 1 = 1+ The low frequency gain is: s 0 =1 The bandwidth roughly corresponds to: w unity H(w bw ) w unity Gain bandwidth product is w unity : 1 w unity = w unity Unity gain closed loop amplifiers maximize the closed loop bandwidth assuming closed loop gain 1 9

Issue: Open Loop Amplifiers have Finite DC Gain 20log / 20log(K) β = 1 = 1+ Let us now model as: = w dominant w unity To first order, the closed loop bandwidth and gain are relatively unchanged K 1+s/w dominant What is the impact of having finite, open loop, DC gain? 10

Further Examination of Finite, Open Loop, DC Gain 20log / 20log(K) = K 1+s/w dominant β = 1 = 1+ w dominant w unity For unity gain configuration of closed loop amplifier: = = K s 0 1+ s 0 1+K = 1 1+1/K We see that finite open loop DC gain leads to a slight reduction of the closed loop DC gain - We want K >> 1 for the unity gain closed loop amplifier 11

More General View of Finite, Open Loop, DC Gain 20log / 20log(K) = K 1+s/w dominant β 20log(1/β) = 1+β w bw w unity For general configuration of closed loop amplifier: µ K 1 = s 0 1+β K = 1 β 1+(1/β)/K Finite open loop DC gain still leads to reduction of closed loop DC gain - We want K >> 1/ in this case - We will see implications of this issue later in the class 12

The Issue of Parasitic Open Loop Poles 20log / 20log(K) β 20log(1/β) = 1+β w bw w p Practical amplifiers have non-dominant poles, too: µ µ K 1 = 1+s/w dominant 1+s/w p - Of course, there can be multiple parasitic poles and also zeros A key issue of such parasitic poles is their influence on the stability of the closed loop amplifier 13

Key Tool for Assessing Stability: Open Loop Response 20log / 20log(K) We define the open loop response, A(s), as: A(s) =β Note that the unity gain frequency, w 0, of A(w) is approximately the same as the closed loop bandwidth, w bw = β 1+β 20log(1/β) A(w - 0 ) =1 β H(w 0 ) =1 H(w 0 ) =1/β Looking at the plot above, we can see that the intersection of and 1/ corresponds to the closed loop bandwidth, w bw w bw w p 14

Stability Analysis Based on Phase Margin of A(w) Phase margin is a key metric when examining the stability of a system - Phase margin is defined as 180 + phase{a(w 0 )} w 0 corresponds to the unity gain frequency of the open loop response (i.e., A(w 0 ) = 1) w 0 is approximately the same as the closed loop bandwidth, w - bw Phase margin must be greater than 0 degrees for the closed loop system to be stable Typically want phase margin to be greater than 45 Key skill: you must be able to plot Bode plots in both magnitude and phase! 15

Review of Bode Plot Basics Example: A(w) = - Log of magnitude (db): 1+jw/w z (1 + jw/w p1 )(1 + jw/w p2 ) 20 log A(w) =20log 1+jw/w z 20 log 1+jw/w p1 20 log 1+jw/w p2 Taking the log allows the poles and zeros to be plotted separately and then added together - Phase: 6 A(w) = 6 (1 + jw/w z ) 6 (1 + jw/w p1 ) 6 (1 + jw/w p2 ) Phase of poles and zeros can also be plotted separately and then added together 16

Review: Plotting the Magnitude of Poles Plot the magnitude response of pole w p1 20 log A p1 (w) =20log 1 1+jw/w p1 = 20 log 1+jw/w p1 - For w << w p1 : - For w >> w p1 : 20 log A p1 (w) 20 log 1 =0 20 log A p1 (w) 20 log w/w p1 20log A p1 (ω) ω ω p1-2/decade 17

Plotting the Phase of Poles Plot the phase response of pole w p1 6 A p1 (w) = 6 (1 + jw/w p1 )= arctan (w/w p1 ) - For w << w p1 : - For w = w p1 : - For w >> w p1 : 6 A p1 (w) arctan (0) = 0 6 A p1 (w) arctan (1) = 45 6 A p1 (w) arctan ( ) = 90 A p1 (ω) ω ω 0 o p1 /10 p1-45 o ω p1 10 ω -90 o 18

Review: Plotting the Magnitude of Zeros Plot the magnitude response of zero w z 20 log A z (w) =20log 1+jw/w z - For w << w z : - For w >> w z : 20 log A z (w) 20 log 1 =0 20 log A z (w) 20 log w/w z 20log A z (ω) 2/decade ω ω z 19

Plotting the Phase of Zeros Plot the phase response of zero w z 6 A z (w) =6 (1 + jw/w z )=arctan(w/w z ) - For w << w z : - For w = w z : - For w >> w z : 6 A z (w) arctan (0) = 0 6 A z (w) arctan (1) = 45 6 A z (w) arctan ( ) =90 90 o A z (ω) 45 o 0 o ω z /10 ω z ω z 10 ω 20

Example of Closed Loop Stability Evaluation 20log / β 20log(1/β) = 1+β Consider the case where: µ µ K 1 = s 1+s/w p w bw w p - This implies that: A(s) =β µ K s µ 1 1+s/w p 21

Phase Margin Versus Open Loop Gain Open loop gain increased 20log A(f) Evaluation of Phase Margin Dominant pole pair Closed Loop Pole Locations C B Im{s} -90 o angle(a(f)) f p C B A f A 0 Re{s} A -120 o PM = 59 o for A -150 o PM = 45 o for B PM = 33 o for C B -180 o C Note the closed loop pole locations versus open loop gain - Is the closed loop system unstable for any case above? 22

Corresponding Closed Loop Behavior Closed Loop Frequency Response Closed Loop Step Response 5 db -5 db A B C 1.4 1 C B A 0.6 Frequency response sees more peaking with higher open loop gain - How does this relate to the movement of the closed loop pole locations? Step response see more ringing with higher open loop gain f p f - How does this relate to the closed loop frequency response? 0 t 23

Some Key Observations 20log / β 20log(1/β) = 1+β A(s) =β w bw w p We have seen that increasing the open loop gain of A(w) leads to higher closed loop bandwidth - How is this consistent with the statement that increasing closed loop gain leads to lower closed loop bandwidth? As an exercise, consider the impact of the following: - Keep unchanged and increase the open loop gain of - Keep unchanged and increase 24

Example 2 of Closed Loop Stability Evaluation 20log / = β 1+β 20log(1/β) Consider the case where: µ µ K = s - This implies that: µ µ K A(s) =β s w bw 1 1+s/w p1 1 1+s/w p2 1 1+s/w p3 w p1, w p2, w p3 1 1+s/w p1 1 1+s/w p2 1 1+s/w p3 25

Phase Margin Versus Open Loop Gain Open loop gain increased -90 o -165 o -180 o 20log A(f) angle(a(f)) Evaluation of Phase Margin f p1 f p2 f p3 C B A f PM = 72 o for A PM = 51 o for B PM = -12 o for C Closed Loop Pole Locations Non-dominant poles Dominant pole pair A A B B Im{s} C Re{s} 0-240 o -315 o C Note the closed loop pole locations versus open loop gain - Is the closed loop system unstable for any case above? 26

Corresponding Closed Loop Behavior Closed Loop Frequency Response Closed Loop Step Response C A B C 1 B A Frequency Time Frequency response again sees more peaking with higher open loop gain - How does this relate to the movement of the closed loop pole locations? Step response ringing grows for high open loop gain - How does this relate to the closed loop pole locations? 27

Open Loop Versus Closed Loop Amplifier Topologies Open Loop Closed Loop Z f Source Source V src Z src Amp V src Z src Amp / / (db) (db) w bw w 1 w 2 w bw w 1 w 2 Now that we understand the phase margin criterion, can you explain why amplifiers designed to be within a closed loop system should have one dominant pole that is much lower in frequency than the parasitic poles? 28

Summary Feedback systems offer the benefit of accurate gain at low frequencies - Assumes accurate feedback and high open loop DC gain - Gain-bandwidth product of the closed loop system equals w unity of the open loop amplifier Accuracy of the closed loop DC gain is reduced with lower open loop DC gain - Want the open loop DC gain to be much higher than the desired closed loop DC gain for reasonable accuracy Stability of the closed loop system is often evaluated using the phase margin criterion - Examines the phase at unity gain frequency of the open loop response, A(w 0 ) = H(w 0 ), where A(w 0 ) = 1 w 0 is approximately the same as the closed loop bandwidth, w bw 29