Peak CurrentMode DCDC Converter Stability Analysis
Popularity and perceived advantages of currentmode control (CMC) have made it de rigueur as a loop control architecture with many power management IC manufacturers and power supply vendors. Also known as multiloop control, an external voltage loop and a widebandwidth inner current loop are standard. Peak, valley, average, hysteretic, constant on/offtime, and emulated currentmode techniques are realizable. Top of the agenda is usually peak currentmode control (PCMC) with slope compensation. Notable advantages of PCMC include automatic input line feedforward, inherent cyclebycycle overload protection, and current sharing capability in multiphase converters. Acute shortcomings are current loop noise sensitivity and switch minimum ontime limitations, particularly in high stepdown ratio, nonisolated converter applications. The emulated (sampleandhold) architecture^{[1,4]} largely alleviates these shortcomings, however, while preserving the benefits of PCMC.
The converter in Figure 1 represents a singlephase buck topology operating in continuous conduction mode (CCM), and whose duty cycle, D, is determined by recourse to the principles of PCMC. Note that the parasitic resistances of the filter inductor and output capacitor are denoted explicitly. Other buckderived power stage topologies  including isolated forward, fullbridge, voltagefed pushpull  could also be inserted here, while retaining a similar loop configuration (feedback isolation excepted).
In a peak currentmode architecture, the state of the inductor current is naturally sampled by the PWM comparator. The outer voltage loop employs a typeII voltage compensation circuit and a conventional operational transconductance error amplifier (EA) is shown with its inverting input, labeled the feedback (FB) node, connected to feedback resistors R_{fb1} and R_{fb2}. A compensated error signal appears at the EA output, labeled COMP, the outer voltage loop thus providing the reference command for the inner current loop. COMP effectively represents the programmed inductor current level. The current loop converts the inductor into a quasiideal voltagecontrolled current source, effectively removing the inductor from the outer loop dynamics, at least at DC and low frequencies.
The current sensing location in Figure 1 is shown schematically after the inductor. The implementation could be a discrete shunt resistor, or lossless using inductor DCR current sensing^{[5]} or measuring MOSFET onstate resistance^{[2]}. Alternatively, a current sense transformer can be exploited, but only if the current sense location is such that the current waveform is zero for part of the switching period to allow transformer reset, e.g. in series with the highside FET. In any event, the equivalent linear amplifying multiple is given by:
Where:
G_{i} = Current sense amplifier gain (if used)
R_{s} = Current sensor gain given by one of:
A perfect currentmode converter relies only on the dc or average value of inductor current. In practice, a peaktoaverage inductor current error exists in a PCMC implementation and this error can manifest itself as a subharmonic oscillation of the current loop in the time domain at duty cycles above 50%. Slope compensation is the wellknown technique of adding a ramp to the sensed inductor current to obviate the risk of this subharmonic oscillation. Figure 2 illustrates how a turnon command is activated when the clock edge sets the PWM latch. A turnoff command is imposed when the sensed inductor current peak plus slope compensation ramp reaches the COMP level and the PWM comparator resets the latch. This is known as trailing edge modulation. Se, earmarked in Figure 2, is the external slope compensation ramp slope and S_{n} and S_{f} are the ontime and offtime slopes of the sensed current signal, respectively. D' = 1D is the duty cycle complement.
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CMC SMALLSIGNAL REVIEW
The analytical nexus around which a smallsignal dynamic model can be derived is based either on Middlebrook and Cuk's famous state space averaging (SSA) technique or, more simply, Vorpérian's PWM switch model. An intuitive model of the smallsignal currentmode system is illustrated in the block diagram format of Figure 3[1].
Modulator gain block KM is the gain from the duty cycle to the switchnode voltage. Contingent upon the load characteristic, Ro represents the smallsignal AC load resistance given by:
Where:
R = DC operating point of the load.
The component Rdc in Figure 3 is the cumulative series resistance attributed to the inductor DCR, MOSFET onstate resistance, and PCB trace resistance
From Figure 3, the smallsignal ac variation of the switchnode and output voltages are written as
Thus, the controltooutput transfer function is:
This describes the smallsignal behavior of the modulator and power stage when the smallsignal input voltage variation is zero. The expressions for impedances Zo(s) and ZL(s) can be substituted into (4) to obtain the pole/zero form of the controltooutput transfer function as:
H(s), the highfrequency extension in the controltooutput transfer function to model the modulator sampling gain, is discussed in more detail in the next section. The relevant gain coefficients can be derived as:
The dominant filter pole and capacitor ESR zero frequencies are given respectively by:
A typical controltooutput transfer function frequency response is elucidated in Figure 4. The pole and zero locations are denoted by × and o symbols, respectively.
SAMPLING GAIN
A currentmode control system is a sampled system, the sampling frequency of which is equal to the switching frequency. A traditional lowfrequency averaged model can be modified to include the sampling effect in the current loop. Ridley[3] advised that the sampling action in the peak currentmode loop is an infinite order system but can be represented in the frequency range of interest by a pair of complex RHP zeros in the current feedback path, denoted using the gain block He(s) in Figure 3, where:
The closedcurrent feedback loop thus becomes unstable if it has enough gain leading to subharmonic oscillation. In the controltooutput transfer function, the sampling action is represented as a pair of complex RHP poles located at half switching frequency such that:
For any converter, the quality factor, Q, is:
with the slope compensation parameter defined as:
For singlecycle damping of an inductor current perturbation, the baseline requirement is that the slope compensation ramp should equal the inductor current downslope[1], i.e.
Accordingly, a perturbed inductor current will return to its original value in one switching cycle and the resultant Q factor, calculated from (12), is equal to 2/π or 0.637. Even though most PCMC implementations exploit a fixed slope compensation ramp amplitude, the ideal slope compensation level is proportional to output voltage.
Note that excessive slope compensation increases mc, decreases Q, reduces the current loop gain and bandwidth. This portends additional phase lag in the voltage loop and stymies the maximum attainable crossover frequency. The system becomes inherently tilted towards voltagemode control. Conversely, insufficient slope compensation decreases mc and increases Q, causing peaking in the current loop gain and ultimately voltage loop instability as duty cycles exceed 50%. A Q value in the range 0.5 to 1.0 is generally satisfactory.
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TRANSFER FUNCTION
A typeII compensator using an EA with transconductance, gm, is shown in Figure 5. The dominant pole of the EA openloop gain is set by the EA output resistance, REAout, and effective bandwidthlimiting capacitance, Cbw, as follows:
The influence of any EA high frequency poles, whether parasitic or included by design, is neglected in the above expression. The compensator transfer function from output voltage to COMP, including the gain contribution from the feedback resistor divider network, is given by:
With:
And the feedback attenuation factor is:
Evaluating (17) gives an expression of the form
As ωpEA1 and ωpEA2 are well separated in frequency, the low Q approximation applies and (19) becomes
Where:
Typically, Rcomp âª REAout, Ccomp â« (Chf + Cbw) and the approximations set forth in (21) are valid. The components creating the two compensator poles and one compensator zero are circled in Figure 5.
Here, the feedback attenuation is unity and the EA has openloop dc gain of 50dB and 5MHz singlepole gainbandwidth characteristic. Again, the poles and zeros are denoted with × and o symbols, respectively, and a + symbol indicates the EA bandwidth. Note that the 180° phase lag related to the EA in the inverting configuration is not included in the phase plots.
Compensator pole ωpEA1 shown in Figure 5 appears at very low frequency and can be superseded by an integrator term. The equation for the compensator transfer function thus can now be simplified to:
The integrator gain term, Ac, is given by
CMC COMPENSATOR DESIGN
A frequently employed, yet corrigible, compensation strategy is to equate the controltooutput transfer function to the compensator transfer function term by term to attain a singlepole 20dB/decade rolloff of the loop response. To demonstrate, consider:

One compensator pole, ωpEA1, positioned to provide high gain in the low frequency range, minimizing output steady state error for better load regulation;

One compensator zero located to offset the dominant load pole, ωzEA = ωp. Typically, the minimum load resistance (maximum load current) condition is used;

One compensator pole positioned to cancel the output capacitor ESR zero, ωpEA2 = ωesr;
The loop gain is expressed as the product of the controltooutput and compensator transfer functions. Substituting (5) and (22), the loop gain is
The crossover frequency, ωc = 2πfc, where the loop gain is 0dB, is usually selected between one tenth and one fifth of the switching frequency. If ωzEA = ωp and ωpEA = ωesr, the loop gain reduces to
Assuming a well designed current loop (Q = 0.637), the sampling gain contribution is insignificant at frequencies up to the crossover frequency. This assumption precludes the case where too much slope compensating ramp is added. The magnitude of the loop gain at the dominant pole frequency is:
Using basic bode plot principles, it is apparent that
Thus derived, a straightforward solution for the crossover frequency is
Finally, compensator component values can be calculated sequentially as
An initial value is selected for Rfb2 based on a practical minimum current level flowing in the divider chain. Note that the compensation zero frequency represents the dominant time constant in a load transient response characteristic. A large Ccomp capacitor is thus antithetical to a fast transient response settling time. Ccomp can be reduced, however, to tradeoff phase margin and settling time. A phase margin target of 50° to 60° is ideal. It is found that the compensator zero is optimally located above the load pole but below the power stage resonant frequency, ωLC ≈1/ √LCo, so that:
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Furthermore, a smaller compensation capacitance is advantageous when the transconductance EA has a low output drive current capability.
DESIGN EXAMPLE
The circuit operating conditions, key component values and control circuit parameters of a buck converter based on one channel of a LM5642X dualchannel synchronous buck
The relevant gains and power stage corner frequencies are calculated using expressions (6) through (9) as follows:
The compensation component values, assuming a desired crossover frequency of 60kHz, are found using (29) and (30) as:
Figure 6 shows a Mathcad derived loop gain and phase plot of the exemplified converter. The equivalent plots with an ideal EA are also shown. The phase margin, ÏM, is the difference between the loop phase and 180° (EA inversion phase lag notwithstanding).
SIMPLIS SIMULATION
Using a LM5642X PWM controller in a buck converter configuration (Table), a SIMPLIS switching model circuit simulation is used to substantiate the analysis. The SIMPLIS model is presented in Figure 7.
The loop gain T(s) of the system is measured by breaking the loop at the upper feedback divider resistor, injecting a variable frequency oscillator signal, and analyzing the frequency response. The element with reference designator X1 is the SIMPLIS clock edge trigger to find the circuit periodic operating point (POP) before running the ac analysis. POP analysis works on the full nonlinear switching time domain model of the circuit and enables subsequent ac or transient analyses. Figure 8 illustrates the bode plot simulation.
REFERENCES

R. Sheehan, National Semiconductor, ‘CurrentMode Modeling  Reference Guide’, www.national.com/analog/power/conference_paper_design_ideas

National Semiconductor LM5642 High Voltage, Dual Synchronous Buck Converter with Oscillator Synchronization from the PowerWise® Family, www.national.com/pf/LM/LM5642.html

R. B. Ridley, ‘A New, ContinuousTime Model for CurrentMode Control’, IEEE Transactions on Power Electronics, Vol. 6, No. 2, April 1991, pp. 271280.

National Semiconductor LM3000 Dual Synchronous Emulated CurrentMode Controller from PowerWise® Family, www.national.com/pf/LM/LM3000.html.

National Semiconductor LM27402 High Peerformance Buck Controller with DCR Current Sensing from PowerWise® Family, www.national.com/pf/LM/LM27402.html.