Power Management Circuits and Systems Basic Concepts: Amplifiers and Feedback Jose Silva-Martinez Department of Electrical & Computer Engineering Texas A&M University January 2, 209
Non-Inverting Amplifier Feedback Properties: Transfer function and sensitivity function The following non-inverted amplifier will be used as a testbed to verify the properties of passive feedback systems. R F H(s) = +A V ( A V ) +R F Where A V = V 0 V + V is the v i - + v o amplifier s gain. The term A V ( ) corresponds to the system s loop transfer function. +R F Factor +R F represents the feedback factor. 2
Feedback Properties H(s) = ( +R F ) ( + R A V ( in ) +R F ) ( +R F ) ( A V ( ) +R F ) In case the loop gain A V ( ), the system can safely approximated by the first +R F factor, then we called this term as the ideal system transfer function H ideal (s) = ( +R F ) = + R F This is a very desirable result since the transfer function is a ratio of passive elements connected through the feedback systems; in fact the gain becomes equal to /. The overall (closed loop) transfer function is then insensitive to amplifier s gain (AV) variations. 3
Practical amplifier gain is usually modeled as Feedback Properties A V = ( A V0 + s ω P ); A V0 is amplifier DC gain and ω P is Magnitude LG H(s) the amplifier s dominant pole. The gain error is determined by H signal =H(s)/(+LG) Frequency ( s ) Loop Gain s P AV 0 R in R in R F. It is important to recognize that the error function is equal to /(loop gain). Error function monotonically increases after the first pole of loop gain. 4
Error Function High frequency signals do not benefit as much as in-band (low frequency) signals s P ( s ) AV 0 R in R in R F 20 log 0 ( A V (s) ) 06 db -20 db/decade Error (db). The error function is inversely proportional to amplifier s frequency response. 0 db 20 db/decade unity-gain frequency f (Hz) 5
To maintain good accuracy, loop gain must be high enough until the maximum frequency of interest. For a closed loop system with a Error Function targeted error 0, we must satisfy the 0 db Error (db) following condition: f f max P in F ( s ) 0 AV 0 R in 2 R R -00 db 20 db/decade unity-gain f max frequency f P f (Hz) max is the maximum frequency of interest, often defined as system bandwidth. 6
Error Function For the case max /P >0, the condition for limited overall gain error can be simplified to +RF/Rin ωmax/ ωp Error magnitude 0 0.0 ~ 0-4 f f max R R P in F ( s ) 0 A V 0 R in If AV0 = 0 5 V/V (considered very large for CMOS solutions), the error measured at different frequencies is listed in this Table.. 0 ~.4 0-4 0 0 ~0-3 0 00 ~0-2 0 000 ~0-00 0 ~0-2 000 0 ~0-0000 0 ~0 0 7
Inverting Amplifiers Inverting Amplifier. The case of the inverting amplifier is a bit more complex than the case of the non-inverting amplifier. Assume that the amplifier s gain includes the effect of the feedback network. R F The voltage at the amplifier s inverting terminal (ERROR VOLTAGE) is: v i - v o R F V = ( ) V +R i + ( ) V F +R 0 F + Output voltage R F V 0 = A V V = A V ( ) V +R i A V ( ) V F +R 0 F The first term of the right most term is the so-called direct path gain The second right hand most term is obtained in this case by breaking the loop and grounding Vi terminal. 8
Inverting Amplifiers It can be found that signal transfer function R F is determined as v i - v o V 0 = ( R F ) ( V i + R A V ( in ) +R F ) 20 log 0 ( A V (s) ) + Error (db) 06 db -20 db/decade Error factor is the same for both inverting and non-inverting configurations 20 db/decade 0 db unity-gain frequency f (Hz) 9
Inverting Amplifier The ERROR VOLTAGE (inverting R F terminal of the amplifier) is determined by R F V = ( ) V +R i + ( ) V F +R 0 F v i - + v o If the amplifier gain is given by AV and it is not affected by RF, then the error voltage is computed as Magnitude LG Frequency V = ( +(A V )( R F ) +R F ) +R F V i ( R F A V ) V i H error =/(+LG) The larger the loop gain, the smaller the value of the error voltage is. 0
Transconductance Amplifiers Let us go a bit deeper: Systematic and Intuitive Analysis Transconductance amplifiers with floating elements in feedback Network Typically the CMOS amplifier does not have a low-impedance output stage: these are the most practical cases since buffers present voltage headroom issues. The analysis of this type of amplifiers is cumbersome, and often we get lost on the algebra. The results of typical analysis are not Z in v i Z i Z F v o Z L evident and hard to be properly interpreted; results are not intuitive! We would like to get insights and so guidelines to optimize our architecture and design procedures. + -
Transconductance Amplifiers Let us first consider the case of the inverting amplifier including amplifier s input and output impedance. The transimpedance amplifier s small signal model employing a voltage controlled current source is used here. i o Z F Using KCL at both nodes v x and v o we find v i Z in v x V X ( Z in + Z + Z F ) V 0 Z F = V i Z in i i i Z v + g m v x Z L v o V X (g m Z F ) + V 0 ( Z L + Z F ) = 0 Solving these equations result in V 0 V i = Y in Y F (+( Y in +Y +Y F )( Y L +Y F )) Y F gm Y F Notice that loop gain is then given by LG = ( ) ( g m Y F Y in +Y +Y F Y F Y L +Y F ) 2
Transconductance Amplifiers LG = ( Z in Z ) [(g Z in Z + Z m ) (Z F Z L Z F )] F The first term is the feedback factor, The 2 nd factor is the amplifier s gain. Notice that amplifier s transconductance gain is affected by the feedback element! A right hand plane zero arises when Z F is capacitive; that may hurt your phase margin. When considering capacitors or more complex networks it Z F is not evident where the poles and zeros are located. The culprit for these v i Z in v x Z v + g m v x Z L v o complications is the floating element Z F. 3
Floating Passives Modeling Bi-Directional Floating Impedances A major issue is the mapping of circuits that have bi-directional elements connecting different nodes. i ab i ab v a v b i ab v a v b Z ab v b Z ab v a Z ab Z ab Z ab The representation of the floating element ZF through the four components depicted above is consistent and accurate. The 4-element representation uses unidirectional (without floating passives) Resulting passive components are grounded (unidirectional) This model can be easily incorporated into the block diagram s representation and then solved by using Mason s rule. 4
Floating Passives Design Example: Inverting Transconductance Amplifier Converting the floating element into grounded (unidirectional) passive elements and unidirectional voltage controlled current sources Using the 4-element equivalent, and combining elements, the amplifier s circuit using unidirectional elements results in Z in v v x i Z v + Z F v o g m v x Z L vin Z in Z Z F Z F v 0 v X g m Z F v X Z Z L F vo 5
Floating Passives Feedforward gain and loop Gain can be easily found as vin Z in Z Z F Z F v 0 v X g m Z F v X Z Z L F vo FF = ( Z Z F ) ((g Z in +Z Z m ) Z F Z L Z F ) ; F LG = ( Z in Z Z F Z F ) ((g m Z F ) Z L Z F ) The effect of the feedback impedance on the amplifier s transconductance is evident in this model. Exact location of poles and zeros are now captured, and exact computation of both loop gain and closed loop transfer function is straightforward. 6
Mason s Rule Unidirectional Block Diagrams and Mason Rule An elegant yet more insightful solution for unidirectional networks employs the Mason rule. Unidirectional building blocks means that the output is driven by the input, but variations at the output does not affect at all the block s input. Examples of these blocks are a) Voltage controlled voltage sources, voltage controlled current sources, current controlled voltage sources and current controlled current sources. b) Grounded passives (resistors, capacitors and inductors) c) Examples of non-directional elements are the transformers, and floating impedances The transfer function of a given linear system represented by unidirectional building blocks can always be obtained by identifying loops and direct trajectories. 7
Feedback Properties Once the system is represented by unidirectional elements, we must: Identify the direct trajectories(paths) from the input(s) to output KyAn(s) from Vy to output Vy AmAn from Vx to output Ky The loops must also be identified Am(s) Vx S Am(s) S An(s) Vo Loops that are not touched by certain direct paths; e.g. The loop is not touched by the direct path KyAn The loops that are not sharing elements or nodes (un-touched loops) have to be identified. In this case we do not find any of these loops. 8
Mason s Rule For the schematic, the following paths and loop can be identified: i) Direct path from Vy: KyAnVy ii) Direct path 2 from Vx: Am(s) An(s)Vx iii) Loop: Am(s) iv) Loop2: An(s) v) Loop3: Am(s)An(s) vi) Notice that Am(s) and An(s) do not have any element in common. This is an example of un-touched loops MASON Rule: If the system is linear, then every single input generates an output component that can be computed according to the following rule: Vy Vx S Ky Am(s) S An(s) Vo vo vi direct paths loops product of direct paths and untouched loops product of untouched loops...... 9
In the case of the following block diagram composed by unidirectional blocks we can obtain the following transfer functions: H ox = V 0 = A m(s)a n (s) V x A m (s) H oy = V 0 V y = K ya n (s) A m (s) Once all transfer functions are identified, the overall output voltage is computed as follows: V 0 = H oy V y + H ox V x For the case of the second schematic, the transfer functions can be computed as H ox = A m (s)a n (s) A m (s) A n (s) A n (s)a m (s) A n (s)a m (s) The denominator is the result of the single loop Am(s), the single loop An(s), the loop involving Am(s)An(s) and last term is due to the product of the two non-touching loops An(s)Am(s). Mason s Rule Vy Vx Vy Vy Vx Vx Ky + Am(s) + loop is not touched by Vy path Ky Ky + Am(s) + S Am(s) S Untouched loops An(s) An(s) An(s) Vo 20
Mason s Rule The transfer function for the second input Vy can be computed as follows: H oy = V 0 V y = K y A n (s) {K y A n (s)}{a m (s)} A m (s) A n (s) A n (s)a m (s)+ A n (s)a m (s) Vy Ky Vx S Am(s) S An(s) Vo The direct path does not touch the loop determined by Am(s), then the term {K y A n (s)}{a m (s)} arises in the numerator. 2
Properties of Sensitivity Function Sensitivity Function When dealing with multivariable functions and some of the parameters may have significant variations then it is useful to compute the sensitivity functions to quantify the individual effects on system performance. The sensitivity of a multivariable function H(x, y, z, ) as function of the parameter x is defined as H x = ( x ) H (dh) = ( dx or dh = H ( dx ) H x X dh dx ) H x The sensitivity function represents the first order variation of H as function of the parameter x, normalized by the factor H/x. 22
Properties of Sensitivity Function To get more insight, let us consider the following approximation: H x = dh H dx x H H x x The sensitivity function then measures the variation of the normalized transfer function H H (percentage). x (percentage) as function of the variation of normalized parameter If the sensitivity function is computed as 0, then % variation in parameter x will produce a variation of 0% in the overall transfer function H. Then, it is highly desirable to maintaining the sensitivity function of H with respect to critical parameters lesser than. For the case of large parameter variations such as gain and bandwidth of the operational amplifier, it is highly desirable to keep the sensitivity functions well below unity. x 23
Be careful with circuit enhancing devices such as the ones that make use of negative resistors to boost amplifier s gain! Example: Typical Amplifier s gain: A VT = g m G L Enhanced voltage gain: A VE = Enhancing gain factor = Properties of Sensitivity Function g m G L G C = A VT ( G C G L ) = A VT ( G C G L ; for more gain boosting, GC GL X ) Sensitivity function: A VE X = X X = GC G L G C G L = G L G C The design trade-off is evident! The larger the gain boosting is, the larger the amplifier sensitivity to GC, GL tolerances is. Sweet Spots result in most of the cases in very sensitive systems! Corner and Montecarlo simulations are mandatory in these cases. 24
Main Concepts behind Feedback Non-Inverting Amplifier. Let us consider the case of the non-inverting amplifier: H(s) = ( +R F ) ( + R A V ( in +R ) F ) = ( +R F ) ( +ξ ) The computation of the sensitivity of H with respect to the error function ξyields, H ξ = ( ξ ) ( d ( +R F )( +ξ ) dξ (Rin+RF ) ( +ξ )) = ξ +ξ In case ξ, the sensitivity function can be approximated as H ξ ξ The error function is computed as ξ = = Loop Gain A V ( ) +R F If, we expect variations of ξ in the range of 00% but we do want the effects on H(s) to be no more than %, then ξ must be maintained under 0.0; e.g. loop gain > 40dB. 25
Better Linearity: Intuitively, when loop gain is large the error function decreases, thus The ERROR VOLTAGE is determined by Properties of Feedback: Linearity R F V = R F ( ) + R F + (A V ) ( ) + R F V i v i - + v o ( R F A V ) V i If loop gain is large, then amplifier s (true) input signal swing reduces at the same time! Non-linearities due to amplifier s gain A V are drastically attenuated! 26
General Remarks: Properties of Feedback: Linearity Entire loop transfer function is important. Be careful when breaking the loop; be sure the loading effects are included in your model. Pay special attention to the error voltage Verror V error = ( ) V +β H(s) in = ( ) V +LG in; Large loop gain reduces the error voltage If V error 0; then V in β V out ; then β becomes the most sensitive block of the closed loop system. Accuracy on the absolute value of β is critical. Noise and non-linearities in the feedback network have direct impact on closed loop system performance. Since V error is very small in the range where the gain of the high-gain network is high, then its linearity is quite good. V in S V error + - V out High Gain Network Feedback Network V out 27
Properties of Feedback: Linearity Linearity improves if and only if LG is large. If LG is small, then V error = V in then the high-gain network has to manage the entire input signal. H(s) V out = ( ) V +β H(s) in Since this is a closed loop system LG = β H(s) is critical for loop stability. Phase margin is relevant to avoid excessive closed loop peaking. Root Locus provides more information on system stability. Root locus show how openloop poles move when the loop is closed and β changes How the poles move when feedback factor changes? How stability is affected if poles and zeros move with process variations? 28
Properties of Feedback: Linearity Intuitive explanation on the linearity issue: Let us consider 3 cases for different frequencies V in S V error + - High Gain Network V out Magnitude V out Feedback Network H signal =H(s)/(+LG) Frequency V error = ( +LG ) V in = (H error (s))v in H(s) V out = ( ) V +β H(s) in = (H signal (s)) V in f f 2 H error =/(+LG) f 3 Case : f=f in the graph LG is large then error transfer function is quite small Closed loop system linearity is superior! 0.5 0-0.5 V error V in - 29
Case 2: f=f2 in the graph is such that loop gain is 0 (20dB) LG is not quite large then error transfer function is small and error voltage increases Properties of Feedback: Linearity V in 0.5 0 V error -0.5 Case 3: f=f3 in the graph is such that loop gain is close to unity (0dB) LG gain is modest, then error voltage is comparable to Vin. Remark: error voltage is the input of the non-linear amplifier! HD3 is proportional to Verror 2 - V in 0.5 0 V error -0.5-30
Properties of Feedback: Linearity Unfortunately, Root locus is not covered in this course due to lack of time, but it is highly recommended to be knowledgeable on this topic. Recommended book by Melsa and Schultz: Very enjoyable chapters on Stability Analysis and also on Root Locus Methodologies. 3