Lecture notes 1: ECEN 489

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1 Lecture notes : ECEN 489 Power Management Circuits and Systems Department of Electrical & Computer Engeerg Texas A&M University Jose Silva-Martez January 207 Copyright Texas A&M University. All rights reserved. No part of this manual may be reproduced, any form or by any means, without permission writg from Texas A&M University.

2 Power Management Circuits and Systems Department of Electrical & Computer Engeerg Texas A&M University I. Feedback Properties: Transfer function and sensitivity function In the first part of this section, the followg non-verted amplifier will be used as a testbed to verify the properties of passive feedback systems. Please verify that amplifier transfer function is given by the followg expression H(s) = A V +A V ( ) +R F () R F v i - + v o Fig.. Resistive feedback non-vertg amplifier configuration. Where A V = V 0 V + V is the amplifier s ga. Notice that if we break the loop at amplifier vertg termal, the term A V ( R ) corresponds to the system s loop transfer function and the factor represents +R F +R F the feedback factor. The previous equation can be rewritten as H(s) = ( +R F ) ( + R A V ( ) +R F ) (2) Although this expression looks more complex than the previous one, it is more practical sce it is a bit more tuitive when designg systems. Notice that case the loop ga A V ( ), the system can +R F safely approximated by the first factor sce the second factor eqn 2 is approximately unity, then we called this term as the ideal system transfer function H ideal (s) = ( +R F ) (3) This is a very desirable case sce the transfer function is function of the passive elements connected 2

3 through the feedback systems; fact the ga becomes equal to /. The overall (closed loop) transfer function is then sensitive to amplifier ga (A V ) variations. Unfortunately this amplifiers with fite ga over the entire bandwidth of terest are not available; practical amplifier ga is usually modeled as A V = ( A V0 + s ω P ) (4) where A V0 is the amplifier DC ga and ω P is the amplifier s pole. For a sgle domant pole system (other poles and zeros are placed far beyond this frequency), this pole defes the amplifier bandwidth. In equation 4, s (=j) is complex frequency variable. For example, the -3dB frequency of the μa74 is only the 0 Hz range while the open-loop DC ga is around V/V. The product of the open-loop DC ga and the bandwidth is defed as the OPAMP s ga-bandwidth product (GBW). For the OPAMP μa74, the typical value of GBW is around.5 MHz. These parameters and the transfer function of the μa74 are illustrated the followg Figure. 20 log 0 ( A V (s) ) 06 db -20 db/decade f (Hz) Fig. 2. Typical OPAMP open-loop magnitude response showg a sgle low-frequency pole and a roll-off ga of -20 db/decade at medium and high frequencies. Let us defe the transfer function error troduced if the open-loop ga of the amplifier is fite. Accordg to (2) and (3), the ga error is determed by Eq. 5. If A v0 is frequency-dependent, then by makg use of Eq. 4, we can obta the general form for the error function cludg the effects of the fite OPAMP bandwidth: s P R RF ( s ). (5) AV 0 R It is important to recognize that the error function is equal to /(loop ga). Remember that loop ga is composed by the product of amplifier s ga and feedback factor. 3

4 20 log 0 ( A V (s) ) 06 db -20 db/decade Error (db) 0 db 20 db/decade unity-ga frequency f (Hz) Fig. 3. OPAMP open-loop magnitude response and its effect on the error response for an vertg amplifier. Based on the above expression, the pole of the OPAMP leads to a zero the error transfer function. Fig. 3 illustrates the relationship between error function and OPAMP frequency response. The low-frequency error can be estimated from Eq. 5 (where s = 0). At the frequency of the OPAMP s pole ω P, the voltage ga decreases due to the presence of the pole. As a result, the error function creases as predicted by Eq. 5. In a first-order approximation, the error function creases by a factor of 0 when the frequency creases by the same factor after the location of the pole. The higher the OPAMP bandwidth ( P ), the smaller the high frequency error is. Both limited DC ga and fite bandwidth of the amplifier crease the error function. For the case of the monotonic amplifier s transfer function (eqn 4), the maximum error occurs at the edge of targeted closed loop transfer function. The rule of thumb is that for a closed loop system with a targeted error, due to fite amplifier s parameters, smaller than 0, we must satisfy the followg condition: 2 max R R. (6) P F ( s ) 0 AV 0 R where max is the maximum frequency of terest, often defed as system bandwidth. For the case max / P >0, the condition for limited overall ga error can be simplified to 4

5 max R R. (7) P F ( s ) 0 A V 0 R Let us consider some practical cases to get more sight on these issues due to limited performance the operational amplifier. If A V0 = 0 5 V/V and R F / = 9, the error measured and different frequencies is listed Table. Notice the first 5 rows that the error creases monotonically with frequency. This is a result of the limited bandwidth of the OPAMP. We have to remember that the open-loop voltage ga reduces approximately proportionally to frequency after P. The desired amplifier ga also affects the error magnitude function; these effects are visible when comparg the results rows with /p=0 and different desirable voltage ga. Table 3.2 Closed-loop ga magnitude error: example amplifier for different ideal ga factors +R F / and ω/ω P ratios. +R F / ω / ω P Error magnitude ~ ~ ~ ~ ~ ~ ~ ~0 0 Sensitivity Function 5

6 When dealg with multivariable functions and some of the parameters may have significant variations then it is useful to compute the sensitivity functions to quantify the dividual effects of system performance. The sensitivity function of H(x, y, z, ) as function of the parameter x id defed as H x = ( x ) H (dh) = ( dx dh dx ) H x The sensitivity function is formally obtaed represents the first order variation of H as function of the parameter x, normalized by the first order factor H/x. To get more sight on the relevance of this function, let us consider the followg approximation: (8a) H x = dh H dx x H H x x where the first derivative is expressed by a discrete approximation H/x. The sensitivity function then (8b) measures the normalized variation of the transfer function H H as function of the normalized variation of parameter x. For stance if the value of the sensitivity function 8 is 0, then % variation parameter x x will produce a variation of 0% the overall transfer function H. It is highly desirable to matag the sensitivity function of H with respect to critical parameters lesser than. For the case of large parameter variations such as ga and bandwidth of the operational amplifier it is highly desirable to keep the sensitivity functions well below. Ma concept behd Feedback. Non-Invertg Amplifier. Let us consider the case of the equation 2, rewritten as follows: H(s) = ( +R F ) ( + R A V ( ) +R F ) = ( +R F ) ( +ξ ) (2b) The computation of the sensitivity of H with respect to ξyields, H ξ = ( ξ ) ( d ( +R F )( +ξ ) dξ (R+RF ) ( )) = ( ξ +ξ +ξ ) ( d dξ ( +ξ )) = (ξ( + ξ)) ( ( +ξ )2 ) = ξ +ξ If, for stance, we expect to see variations of ξ the range of 00% but we do want to reduce those effects on H(s) to be no more than %, then ξ must be mataed under 0.0. Some properties of the sensitivity (9) 6

7 function are cluded appendix A to facilitate its computation. In case ξ, equation 9 can be approximated as H ξ ξ (9) max where P R A V 0 amplifier. R R F max GBW R R R F ; GBW is the ga-bandwidth product of the Invertg Amplifier. The case of the vertg amplifier is, surprisgly, a bit more complex than the case of the non-vertg amplifier. There are two fundamental reasons for this complications: the voltage at the amplifier s vertg termal is determed by the combation of vi and vo, leadg to the followg results: R F V = ( ) V +R i + ( ) V F +R 0 (0) F 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 open loop ga and can be obtaed breakg the feedback factor at the output and groundg the right most termal of R F. A voltage divider results cascade with the vertg amplifier ga. The second right hand most term is our old friend: the (open) loop ga which is obtaed this case by breakg the loop and groundg Vi termal. Solvg equation leads to the desired transfer function; after some mathematical manipulations we can fd that V 0 = ( R F ) ( + R A V ( ) +R F ) (2) Fortunately the error function is the same as for the case of the non-vertg amplifier and we do not have to repeat the analysis. R F v i - + v o 7

8 Fig. 4. Resistive feedback vertg amplifier configuration. Voltage Controlled Current Amplifiers with floatg elements feedback. The analysis of this type of amplifiers is cumbersome and often we get lost on the algebra. The results are not evident and hard to be properly terpreted; more relevant is that we get very little sight and so guideles to optimize the system design process. Let us first consider the case of the vertg amplifier cludg amplifier s put and output impedance as depicted Fig. xx. The amplifier s small signal model employg a voltage controlled current source is depicted Fig. xxb. i o v i Z Z i - + Z L v o v i i Z i i v x Z i v - v + g m v x Z L v o (a) (b) Fig. 3.35a) Invertg amplifier with OPAMP put impedance Z and load impedance Z L, and b) its smallsignal equivalent circuit where R o = 0. Usg KCL at both nodes we fd V X ( Z + Z + ) V 0 = V i Z V X (g m ) + V 0 ( Z L + ) = 0 () () A more elegant yet more sightful solution consists on the solution of the network employg the Mason rule. Although it is not demonstrated here, it is true that the transfer function of a given system represented by unidirectional buildg blocks can always be obtaed by identifyg i) Unidirectional buildg blocks means that the output is driven by the put, but variations at the output does not affect at all the block s put. a. Examples of these blocks are the voltage controlled voltage sources, voltage controlled current sources, current controlled voltage sources and current controlled current sources. b. Examples of non-directional elements are the transformers, and floatg impedances 8

9 Once the system is represented by unidirectional elements, we must: ii) Identify the direct trajectories(paths) from the put(s) to output iii) The loops must also be identified iv) Loops that are not touched by certa direct paths v) The loops that are not sharg elements or nodes (un-touched loops) Some of the defitions used above are highlighted the followg examples. Assumg that the buidg blocks are unidirectional, we can identify the followg direct paths: i) KyAn(s)Vy ii) Am(s)An(s)Vx We can also identify one loop iii) Am(s) Vy Ky Vx + Am(s) + An(s) loop is not touched by Vy path For the schematic, the followg paths and loop can be identified: Vy Ky Vx + Am(s) + An(s) Untouched loops i) Direct path from Vy: KyAnVy 9

10 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 show any element common. MASON Rule: Once the direct paths and loops are identify, system transfer function can be easily obtaed accordg to Mason s rule as follows: If the system is lear, then every sgle put generates an output component that can be computed accordg to the followg rule vo vi direct paths product of direct paths and untouched loops... loops product of untouched loops... In the case of the first diagram we can obta the followg transfer functions: H ox = V 0 = A m(s)a n (s) V x A m (s) and H oy = V 0 = K ya n (s) {K y A n (s)}{ A m (s)} V y A m (s) (2) (2) (2) Notice that VyAn(s) does not touch the loop Am(s). Once computation of the overall output voltage is computed as follows: these transfer functions are obtaed, the V 0 = H oy V y + H ox V x (2) For the case of the second schematic, the dividual transfer functions can be computed as H ox = V 0 V x = 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 denomator is the result of the sgle loop Am(s), the sgle loop An(s), the big loop volvg Am(s)An(s) and the last term is due to the product of the two non-touchg loops An(s)Am(s). (2) 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) The overall voltage ga is given by equation xx. Other examples will be discussed class. (2) 0

11 Floatg Impedances. A major issue is the mappg of circuits that have bi-directional elements connectg different nodes. Let us consider the followg generic diagram volvg a transfer function H(s) and an admittance feedback; see fig. xa. It is not difficult to demonstrate through nodal equations that the representation of the floatg element through the four components depicted fig. xb. The importance of this representation is that the properties of the passive feedback became apparent this equivalent diagram. i) The put impedance of the architecture is obtaed as Y = Y F + [ V 0 Y V F = Y F ( + [ V 0 ) V ]Q ]Q ii) The amplifier output impedance can be computed as follows Y out = Y F + [ V 0 Y V F = Y F ( + [ V 0 ) V ]Q ]Q iii) Will contue!