In Part I of this article (Power Electronics Technology, May 2007), the basic operation of currentmode 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 voltagemode operation, and a simple analogy showed how the optimal slopecompensation requirement could be obtained without any complicated equations.
Now unified models using general gain parameters are introduced, along with simplified design equations, and an indepth 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 currentmodecontrol theory.
Modeling ContinuousConduction Mode
This article provides models and solutions for fixedfrequency, continuousconductionmode (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 currentmode switching regulator is a sampleddata 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 continuoustime model of reference [2] successfully placed the samplinggain term in the closedcurrent feedback loop. This allows accurate modeling of the controltooutput transfer function using the term H_{E}(s).
To accurately model the current loop, the unified model of reference [3] placed the samplinggain term in the forward path. For peak or valley current modes with a fixed slopecompensation ramp, this also accurately models the controltooutput transfer function using the term F_{M}(s).
To develop the theory for emulated currentmode control, reference [1] used a fresh approach, deriving general gain parameters, which are consistent with both models. In addition, a new representation of the samplinggain term for the closedcurrent loop was developed, identifying limitations of the forwardpath samplinggain term.
The upper circuit in Fig. 2 represents the unified form of the model, with K being the feedforward term. In the lower circuit, K_{N} is the dc audio susceptibility coefficient from the continuoustime model. The linear model samplinggain terms, as shown in Fig. 2, are defined as:
where T is the switching period. The term K_{E} is new and emerged from the derivation of the closedloop expression for H(s). This derivation used slopecompensation terms other than the classic fixed ramp for peak or valley current mode. K_{E} can be expressed as 1/ω_{N}×Q_{E}
but this serves no purpose, because Q_{E} would need a value of infinity for the condition K_{E} = 0. To date, no method has been found which successfully incorporates K_{E} into the openloop expression for H_{P}(s). Use of H_{P}(s) is limited to peak or valley current mode with a fixed slopecompensating ramp, for which the value of K_{E} = 0.
To place either samplinggain term into the linear models for the buck, boost and buckboost, the following relationships are applied: F_{M}(s) = F_{M} × H_{P}(s) and G_{I}(s) = G_{I} × H(s). The accuracy limit for the samplinggain term is identified by comparing Q to the modulator voltage gain K_{M} and the feedforward term K. Q is directly related to the slopecompensation requirement. The derivation starts with the ideal steadystate 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 K_{M} in the forward dcgain 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 dcgain 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 firstorder 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 lowfrequency model do not show the additional phase shift due to the sampling effect. The controltooutput transfer function with the samplinggain term accurately represents the circuit's behavior up to half the switching frequency. The linetooutput expressions for audio susceptibility are accurate at dc, but diverge from the actual response as frequency increases.
The currentsense gain is defined as R_{I} = G_{I} × R_{S}, where G_{I} is the currentsense amplifier and R_{S} is the sense resistor. For all transfer functions,
To include the samplinggain term in the control to output transfer function, the term
1+5/ω_{L} is replaced with 1+5/ω_{N}×Q+5^{2}/ω_{N}^{2} in the lowfrequency equations. This represents the closedcurrentloop samplinggain term. Inclusion of this term in the linetooutput equations will not produce the same accuracy of results. For peak or valley current mode with a fixed slopecompensating ramp, ω_{L}=Q×ω_{N}.
Sampling Gain Q
Using a value of Q = 0.637 will cause any tendency toward subharmonic oscillation to damp in one switching cycle. With respect to the closedcurrentloop controltooutput function, the effective sampledgain inductor pole is given by:
This is the frequency at which a 45degree phase shift occurs because of the sampling gain. For Q = 0.637, f_{L}(Q) occurs at 24% of the switching frequency. For Q = 1, f_{L}(Q) occurs at 31% of the switching frequency. For secondorder systems, the condition of Q = 1 is normally associated with best transient response. The criteria for critical damping is Q = 0.5 (δ = 1). Using Q = 1 may make an incremental difference for the buck, but is inconsequential for the boost and buckboost with the associated righthalfplane zero of ω_{R}. For the peakcurrentmode buck with a fixed slopecompensating ramp, the effective sampledgain inductor pole is only fixed in frequency with respect to changes in line voltage when Q = 0.637. Proportional slopecompensation methods will achieve this for other operating modes.
To determine the effect of reducing the slope compensation to increase the voltageloop bandwidth, an emulatedpeakcurrentmode buck with proportional slopecompensation switching circuit was implemented in SIMPLIS. A standard typeII 10 MHz error amplifier was used for frequency compensation. With T/L = (5 µs/5 µH) and R_{I} = (0.1 V/A), the best performance was achieved with Q = 0.637 for a crossover frequency of 40 kHz and 45degree phase margin. By setting Q = 1, a crossover frequency of 50 kHz was achieved, again with 45degree phase margin but reduced gain margin. This appears to be the practical limit for a stable voltage loop, at the expense of underdamping the current loop. With Q = 1, subharmonic 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
Simple, accurate and easytouse linear models have been developed for the buck, boost and buckboost 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 (V_{AP}) and duty cycle (D).
The coefficients for the linear model of the buck regulator shown in Fig. 3 are:
The controltooutput simplified transfer function is:
and the linetooutput simplified transfer function is:
where
The coefficients for the linear model of the currentmode boost regulator shown in Fig. 4 are:
The controltooutput simplified transfer function is:
and the linetooutput simplified transfer function is:
where
The coefficients for the linear model of the currentmode buckboost regulator shown in Fig. 5 are:
The controltooutput simplified transfer function is:
and the linetooutput simplified transfer function is:
where
Boost Regulator Simulation Example
For the peakcurrentmode 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 H_{P}(s). To use the forwardpath samplinggain 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 K_{M} and feedforward term K. For an actual boostconverter implementation with a fixed ramp, it is only possible to get the optimal Q at one input voltage. The controltooutput gain plots in Fig. 6 show only a slight deviation between the two models at half the switching frequency, where f_{SW} = 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 switchingmodel simulation, SIMPLIS is able to produce Bode plots directly from the switching model. This program was used to produce the switchingmodel simulation results. The lowfrequency model was made with SIMetrix, which is the generalpurpose 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.
Unified Modulator Modeling
In Part I of this article, the criteria for currentmode control was considered. This led to the linear model, with the gain terms being easily identified. The importance of the concept of K_{M} as the modulator voltage gain cannot be overstated. Most linear models for currentmode 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 model and continuoustime model were redefined to fit the form of the unified model. Combining the unifiedmodel gain blocks with the threeterminal PWM switch resulted in the linear models used here.
A new closedcurrentloop samplinggain term has been defined that accommodates any fixedfrequency peak or valleyderived operating mode. Limitation of the forwardsamplinggain term has been identified, providing direction for further development in linear modeling.
References

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.

Ridley, R.B., “A New, ContinuousTime Model for Current Mode Control,” IEEE Transactions on Power Electronics, Vol. 6, Issue 2, pp. 271280, 1991.

Tan, F.D. and Middlebrook, R.D., “A Unified Model for CurrentProgrammed Converters,” IEEE Transactions on Power Electronics, Vol. 10, Issue 4, pp. 397408, 1995.