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A New Way to Model Current-Mode Control

By Robert Sheehan, Principal Applications Engineer, National Semiconductor, Santa Clara, Calif. Unified models using general gain parameters provide the solution for any peak- or valleyderived current-mode converter.

Part Two

I

n Part I of this article (Power Electronics Technology, May 2007), the basic operation of current-mode control was broken down into its component parts, allowing a greater intuitive understanding for the practical designer. A comparison of the modulator gain was made to voltage-mode operation, and a simple analogy showed how the optimal slope-compensation requirement could be obtained without any complicated equations. Now unified models using general gain parameters are introduced, along with simplified design equations, and an in-depth treatment of the analysis and theory is presented. This general modeling technique explains how previous models can complement each other on various aspects of the current-mode-control theory.

Modeling Continuous-Conduction Mode

This article provides models and solutions for fixed-

frequency, continuous-conduction-mode (CCM) operation. Reference [1] covers the theoretical background for this subject, providing an exhaustive analysis of the buck regulator with its associated models and results. To prevent duplication, the boost regulator of Fig. 1 forms the basis for the discussion here. A more rapid approach to using this information is to bypass reference [1] and follow the general guidelines for slope compensation described in the first part of this article. Then the simplified equations can be used to determine the frequency response. A current-mode switching regulator is a sampled-data system, the bandwidth of which is limited by the switching frequency. Beyond half the switching frequency, the response of the inductor current to a change in control voltage is not accurately reproduced. To quantify this effect for linear modeling, the continuous-time model of reference [2] successfully placed the sampling-gain term in the closed-current

Fig. 1. This switching model of a boost regulator topology provides an example for modeling and simulating continuous-conduction-mode operation. Power Electronics Technology June 2007

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CURRENT-MODE CONTROL

To place either sampling-gain term into the linear models for the buck, boost and buck-boost, the following relationships are applied: FM(s) = FM × HP(s) and GI(s) = GI × H(s). The accuracy limit for the sampling-gain term is identified by comparing Q to the modulator voltage gain KM and the feed-forward term K. Q is directly related to the slope-compensation requirement. The derivation starts with the ideal steady-state modulator gain, the physical reason being that at the switching frequency, the relative slopes are fixed with respect to the period T. A change in control voltage is then related to a change in average inductor current. Any transfer function that is solely dependent on KM in the forward dc-gain path will have excellent agreement to the switching model up to half the switching frequency. However, any transfer function that includes K in the forward dc-gain path will show some deviation at half the switching frequency.

Simplified Transfer Functions

No assumptions for simplification were made during the derivation of the transfer functions. The only initial assumptions are the ones generally accepted to be valid in a first-order analysis. Voltage sources, current sources and switches are ideal, with no delays in the control circuit. Amplifier inputs are high impedance, with no significant loading of the previous stage. Simplification of the results was made after the complete derivation, which included all terms. Reference [1] has examples for the buck regulator. To show the factored form, the simplified transfer functions assume that the poles are well separated by the currentloop gain. Expressions for the low-frequency model do not show the additional phase shift due to the sampling effect. The control-to-output transfer function with the samplinggain term accurately represents the circuit's behavior up to half the switching frequency. The line-to-output expressions for audio susceptibility are accurate at dc, but diverge from the actual response as frequency increases. The current-sense gain is defined as RI = GI × RS, where GI is the current-sense amplifier and RS is the sense resistor. For all transfer functions, K × RI 1 Z = and L = M . C OUT × R C L OUT To include the sampling-gain term in the control to output transfer function, the term s s2 s + 2 in the lowis replaced with 1 + N × Q N L frequency equations. This represents the closed-current-loop sampling-gain term. Inclusion of this term in the line-tooutput equations will not produce the same accuracy of results. For peak or valley current mode with a fixed slopecompensating ramp, L = Q × N . 1+

Fig. 2. For a buck regulator, sampling gain HP(s) is placed in the forward path (upper circuit), and sampling gain H(s) is placed in the closedcurrent feedback loop path (lower circuit).

feedback loop. This allows accurate modeling of the controlto-output transfer function using the term HE(s). To accurately model the current loop, the unified model of reference [3] placed the sampling-gain term in the forward path. For peak or valley current modes with a fixed slopecompensation ramp, this also accurately models the controlto-output transfer function using the term FM(s). To develop the theory for emulated current-mode control, reference [1] used a fresh approach, deriving general gain parameters, which are consistent with both models. In addition, a new representation of the sampling-gain term for the closed-current loop was developed, identifying limitations of the forward-path sampling-gain term. The upper circuit in Fig. 2 represents the unified form of the model, with K being the feed-forward term. In the lower circuit, KN is the dc audio susceptibility coefficient from the continuous-time model. The linear model sampling-gain terms, as shown in Fig. 2, are defined as:

H P ( s) = 1 Q 1+ s× N , H( s ) = 1 + ( s × K E ) + s2 and N = , 2 T N

where T is the switching period. The term KE is new and emerged from the derivation of the closed-loop expression for H(s). This derivation used slope-compensation terms other than the classic fixed ramp for peak or valley current 1 mode. KE can be expressed as , N × QE but this serves no purpose, because QE would need a value of infinity for the condition KE = 0. To date, no method has been found which successfully incorporates KE into the openloop expression for HP(s). Use of HP(s) is limited to peak or valley current mode with a fixed slope-compensating ramp, for which the value of KE = 0.

Power Electronics Technology June 2007

Sampling Gain Q

Using a value of Q = 0.637 will cause any tendency toward sub-harmonic oscillation to damp in one switching cycle. With respect to the closed-current-loop control-to-output 24

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CURRENT-MODE CONTROL

Mode Peak current mode Slope compensation Fixed slope VSL = SE × T SE = SE, SN VSL T VAP × D × R I D L VAP × D × K SL T VAP × D × R I D L VSL T mC, Q mC = 1 + SE SN KM = KM, K 1 T V (0.5 - D) × R I × + SL L VAP KE = 0 KE

SN = Peak current mode Proportional slope T K SL = R I × L For Q = 0.637 (single-cycle damping) Valley current mode Fixed slope VSL = SE × T

Q=

1 T D × (m C × D - 0.5 K = 0.5 × R I × × D × D D 5) L SE SN KM = 1 T (0.5 - D) × R I × + 2 × K SL × D L K E = -K SL × D × L RI

SE =

mC = 1 +

SN =

Q=

1 T 2 D × (m C × D - 0.5 K = 0.5 × R I × × D × D + K SL × D D 5) L SE SN 1 T V (D - 0.5) × R I × + SL L VAP T ×D×D D L K E = -K SL × D × L RI KE = 0

SE =

mC = 1 +

KM =

SN = Valley current mode Proportional slope T K SL = R I × L For Q = 0.637 (single-cycle damping) Emulated- Fixed slope peak VSL = SE × T current mode SE =

VAP × D × R I L VAP × D × K SL D T

Q=

1 × (m C × D - 0.5 5) SE SN

K = -0.5 × R I ×

mC = 1 +

KM =

1 T (D - 0.5) × R I × + 2 × K SL × D D L T × D × D - K SL × (D )2 D L 1 T V (D - 0.5) × R I × + SL L VAP T ×D×D D L 1 T (D - 0.5) × R I × + K SL L T × D × D + K SL × D D L

SN = SE =

VAP × D × R I L VSL T VAP × R I L VAP × K SL T VAP × R I L

Q=

1 × (m C × D - 0.5 5) SE SN

K = -0.5 × R I ×

mC =

KM =

K E = -D × T

SN = Emulated- Proportional peak slope current T K SL = R I × mode L For Q = 0.637 (single-cycle damping)

Q=

1 × (m C - 0.5 5) SE SN

K = -0.5 × R I ×

SE =

mC =

KM =

K E = -D × T

SN =

Q=

1 × (m C - 0.5 5)

K = -0.5 × R I ×

Table 1. Summary of general gain parameters.

function, the effective sampled-gain inductor pole is given by: 1 f L (Q) = × ( 1 + (4 × Q2 ) - 1) . 4×T×Q This is the frequency at which a 45-degree phase shift occurs because of the sampling gain. For Q = 0.637, fL(Q) occurs at 24% of the switching frequency. For Q = 1, fL(Q) occurs at 31% of the switching frequency. For second-order systems, the condition of Q = 1 is normally associated with best transient response. The criteria for critical damping

Power Electronics Technology June 2007

is Q = 0.5 ( = 1). Using Q = 1 may make an incremental difference for the buck, but is inconsequential for the boost and buck-boost with the associated right-half-plane zero of R. For the peak-current-mode buck with a fixed slope-compensating ramp, the effective sampled-gain inductor pole is only fixed in frequency with respect to changes in line voltage when Q = 0.637. Proportional slope-compensation methods will achieve this for other operating modes. To determine the effect of reducing the slope compensation to increase the voltage-loop bandwidth, an emulated26

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CURRENT-MODE CONTROL

for a stable voltage loop, at the expense of under-damping the current loop. With Q = 1, sub-harmonic oscillation is quite pronounced during transient response, but damps at steady state. The reader is encouraged to simulate and observe these effects directly. A simulation example for the boost is provided after the linear models and transfer functions are presented.

Linear Models

Fig. 3. The low-frequency linear model for this buck regulator was made using SIMetrix.

peak-current-mode buck with proportional slope-compensation switching circuit was implemented in SIMPLIS. A standard type-II 10 MHz error amplifier was used for frequency compensation. With T/L = (5 µs/5 µH) and RI = (0.1 V/A), the best performance was achieved with Q = 0.637 for a crossover frequency of 40 kHz and 45-degree V ×M K phase margin. By setting Q = 1, a crossover frequency of M = D, IC = AP and FM = M . R OUT VAP 50 kHz was achieved, DigiPwr 45-degree phase 1:13 PM PwrElec-Ventronics again with1/4p 5/9/07 margin butPage 1 The control-to-output simplified transfer function is: reduced gain margin. This appears to be the practical limit s 1+ v OUT R OUT Z = × , vC R1 × K D s s 1 + × 1 + P L

Simple, accurate and easy-to-use linear models have been developed for the buck, boost and buck-boost converter topologies. Each linear model has been verified using results from its corresponding switching model. In this manner, validation for any transfer function is possible, identifying the accuracy limit of the given linear model. General gain parameters are listed in Table 1. These parameters are independent of topology, and written in terms of the terminal voltage (VAP) and duty cycle (D). The coefficients for the linear model of the buck regulator shown in Fig. 3 are: V V - VOUT VAP = VIN , D = OUT , D =(1 - D)= IN , VIN VIN

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and the line-to-output simplified transfer function is: s 1+ v OUT R O × D × K N Z = × , RI × KD v IN s s 1 + × 1 + P L where KD = 1 + P = 1 C OUT OUT R OUT 1 K , KN = - and KM × RI KM D 1 1 × + . R OUT K M × R I O

The coefficients for the linear model of the current-mode boost regulator shown in Fig. 4 are: VOUT - VIIN V N VAP = VOUT , D = OUT , D = (1 - D) = IN , VOUT VOUT M= V ×M K 1 , IC = AP and FM = M . D R OUT VAP

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The control-to-output simplified transfer function is: s s 1 - × 1 + v OUT R OUT × D D R Z = × , RI × KD vC s s 1 + × 1 + P L 28

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Power Electronics Technology June 2007

CURRENT-MODE CONTROL

and the line-to-output simplified transfer function is: s 1+ v OUT R OUT × D × K N D Z , = × RI × KD v IN s s 1 + × 1 + P L where R ×D 2 1 D R1 K 1 K D = 2 + OUT × + , KN = + , RI K M R OUT × D 2 D K M D R × D2 R = OUT and L 2 1 D2 1 K P = × + × + . C out R OUT R I K M D The coefficients for the linear model of the current-mode buck-boost regulator shown in Fig. 5 are: VOUT VIN VAP = VIN + VOUT , D = , D = (1 - D) = , VIN + VOUT VIN + VOUT V ×M K D M = , IC = AP and FM = M . D R OUT VAP The control-to-output simplified transfer function is: s s 1 - × 1 + v OUT R OUT × D D R Z = × , RI × KD vC s s 1 + × 1 + P L and the line-to-output simplified transfer function is: v OUT v IN s s 1+ × 1+ R OUT × D × D × K N K Z = × , RI × KD s s 1 + × 1 + P L R OUT × D 2 1 D K × + , RI K M D

Fig. 4. The low-frequency linear model for this boost regulator was made using SIMetrix.

where KD = 1 + D + KN = K =

Fig. 5. The low-frequency linear model for this buck-boost regulator was made using SIMetrix.

R OUT × D 2 D RI × D 1 K - + , R = OUT , 2 K M D R OUT × D D L×D

R OUT × D 2 × K N OUT and L×K 1 + D D2 1 1 K P = × + × + . C OUT R OUT R I K M D T O

Boost Regulator Simulation Example

For the peak-current-mode boost converter example, comparisons of results from the switching circuit of Fig. 1 were made to the linear model of Fig. 4 using the samplinggain term HP(s). To use the forward-path sampling-gain term, slope compensation was implemented with a fixed ramp. The results will be slightly different if a proportional ramp is used, as this modifies the modulator gain term KM and feed-forward term K. For an actual boost-converter implementation with a fixed ramp, it is only possible to get

Power Electronics Technology June 2007

the optimal Q at one input voltage. The control-to-output gain plots in Fig. 6 show only a slight deviation between the two models at half the switching frequency, where fSW = 200 kHz. For the simulation, slope compensation was set for Q = 0.637. The choice of simulation program is important, since not all SPICE programs calculate parameters with the same degree of accuracy. For switching-model simulation, SIMPLIS is able to produce Bode plots directly from the switching model. This program was used to produce the switchingmodel simulation results. The low-frequency model was made with SIMetrix, which is the general-purpose simulator for the SIMetrix/SIMPLIS program. This simulator only handles Laplace equations for s in numerical form, where the numerator order must be equal to or less than the denominator order. PSpice is much better suited for linear models with Laplace functions in parameter form. It is more accurate than the SIMetrix/SIMPLIS program but cannot produce Bode plots directly from the switching model. PSpice or a program with similar capability may be used to obtain the simulation results for the linear model. 30

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CURRENT-MODE CONTROL

Fig. 6. This comparison of control-to-output transfer functions for a peak-current-mode boost converter using fixed-slope compensation reveals the switching and linear models behave similarly, with a slight discrepancy at 100 kHz, which is half the switching frequency.

Unified Modulator Modeling

In Part I of this article, the criteria for current-mode control was considered. This led to the linear model, with the gain terms being easily identified. The importance of the concept of KM as the modulator voltage gain cannot be overstated. Most linear models for current-mode control have allowed the math to define the model. In reference 1, an intuitive understanding of the modulator was used to drive the math. By algebraic manipulation, both the averaged

Topology Buck

iL iL = i i L = iG

LE LE = L E(s) = VOFF D

E(s)

Boost

LE =

L s×L E(s) = VOFF × 1 - (D )2 D D × (VOFF / ION OFF L VOFF s×L E(s) = OFF × 1 - (D )2 D D D × (VOFF / ION OFF

Buckboost

iL = i + iG

LE =

Notes: VOFF = VAP ; ION = IC Table 2. Corrections and clarifications for reference [3].

model and continuous-time model were redefined to fit the form of the unified model. Combining the unified-model gain blocks with the three-terminal PWM switch resulted in the linear models used here. A new closed-current-loop sampling-gain term has been defined that accommodates any fixed-frequency peak- or valley-derived operating mode. Limitation of the forwardsampling-gain term has been identified, providing direction for further development in linear modeling. PETech

References

1. Sheehan, Robert, "Emulated Current Mode Control for Buck Regulators Using Sample and Hold Technique," Power Electronics Technology Exhibition and Conference, PES02, October 2006. An updated version of this paper, which includes complete appendix material, is available from National Semiconductor Corp. 2. Ridley, R.B., "A New, Continuous-Time Model for Current Mode Control," IEEE Transactions on Power Electronics, Vol. 6, Issue 2, pp. 271-280, 1991. 3. Tan, F.D. and Middlebrook, R.D., "A Unified Model for Current-Programmed Converters," IEEE Transactions on Power Electronics, Vol. 10, Issue 4, pp. 397-408, 1995.

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