Amathematical model is always helpful in determining the optimal compensation components for a particular design. However, compensating the loop of a WLED currentregulating boost converter is a bit different than compensating the same converter configured to regulate voltage. Measuring the control loop with traditional methods is cumbersome because of low impedance at the feedback (FB) pin and the lack of a topside FB resistor. In Reference 1, Ray Ridley has presented a simplified, smallsignal controlloop model for a boost converter with currentmode control. The following mathematical model explains how to modify Ridley's model so that it fits a WLED currentregulating boost converter; it also explains how to measure the boost converter's control loop.
LOOP COMPONENTS
As shown in Fig. 1, any adjustable DC/DC converter can be modified to provide a higher or lower regulated output voltage from an input voltage. In this configuration, if we assume ROUT is a purely resistive load, then V _{OUT} = I_{OUT} × R_{OUT}. When used to power LEDs, a DC/DC converter actually controls the current through the LEDs by regulating the voltage across the lowside FB resistor as shown in Fig. 2. Because the load itself (the LEDs) replaces the upper FB resistor, the traditional smallsignal controlloop equations no longer apply. The DC load resistance is:
R_{EQ} = V_{OUT}/I_{LED}
(1)
V_{OUT} = n × V_{FWD} + V_{FB}
(2)
V_{FWD}, taken either from the diodes' datasheet or from measurements, is the forward voltage at I_{LED}; and n is the number of LEDs in the string.
However, from a smallsignal standpoint, the load resistance consists of R_{EQ} as well as the dynamic resistances of the LEDs, r_{D}, at the I_{LED}. While some LED manufacturers provide typical values of r_{D} at various current levels, the best way to determine r_{D} is to extract it from the typical LED IV curve, which all manufacturers provide. Fig. 3 shows an example IV curve of an OSRAM LW W5SM highpower LED. Being a dynamic (or smallsignal) quantity, r _{D} is defined as the change in voltage divided by the change in current, or r_{D} =πV_{FWD}/πI_{LED}. To extract r_{D} from Fig. 3, we simply drive a straight tangent line from the V _{FWD} and I_{LED} for the application and compute the slope. For example, using the dotted tangent line in Fig. 3, we get r _{D} = (3.5  2.0 V)/(1.000  0.010 A) = 1.51W at I_{LED} = 350 mA.
SMALLSIGNAL MODEL
As an example of a smallsignal model, the TPS61165 peak currentmode converter driving three series OSRAM LW W5SM parts will be used. Fig. 4a shows an equivalent smallsignal model of a currentregulating boost converter, while Fig. 4b shows an even more simplified model. Equation 3 shows a frequencybased (sdomain) model for computing DC gain in both the currentregulating and the voltageregulating boost converters:
where the common variables are:
and
The duty cycle, D, and the modified values for V_{OUT} and R_{EQ} are computed the same way for both circuits. S_{n} and S_{e} are the natural inductor and compensation slopes, respectively, for the boost converter; and f_{SW} is the switching frequency. The only real differences between the smallsignal model for the voltageregulating boost converter and the model for a currentregulating boost converter is the resistance K_{R} — which multiplies by the transconductance term, (1D/R_{i}) — and the dominant pole, ω_{P}. These differences are summarized in Table 1 ^{[1]}.
Since the value of R_{SENSE} is typically much lower than that of R_{OUT} in a converter configured to regulate voltage, the gain for a currentregulating converter, where R_{OUT} =R_{EQ}, will almost always be lower than the gain for a voltageregulating converter.
MEASURING THE LOOP
To measure the control loop gain and phase of a voltageregulating converter, a network or dedicated loopgain/phase analyzer typically uses a 1:1 transformer to inject a small signal into the loop via a small resistance (R_{INJ}). The analyzer then measures and compares, over frequency, the injected signal at point A to the returned signal at point R and reports the ratio in terms of amplitude difference (gain) and time delay (phase). This resistance can be inserted anywhere in the loop as long as point A has relatively much lower impedance than point R; otherwise, the injected signal will be too large and disturb the converterís operating point. As shown in Fig. 5, the highimpedance node where the FB resistors sense the output voltage at the output capacitor (lowimpedance node) is the typical place for such a resistor.
In a currentregulating configuration, with the load itself being the upper FB resistor, the injection resistor cannot be inserted in series with the LEDs. The converter's operating point must first be changed so the resistor can be inserted between the FB pin and the sense resistor as shown in Fig. 6. In some cases, a noninverting, unitygain buffer amplifier may be necessary to lower the impedance at the injection point and reduce measurement noise.
With the measurement setup in Fig. 6 but without the amplifier, and with R _{INJ} = 51.1W, a Venable loop analyzer was used to measure the loop. The model of a currentregulating converter was constructed in Mathcad® using the datasheet design parameters of the TPS61170, which has the same core as the TPS61165. With V_{IN} = 5 V and I_{LED} set to 350 mA, the model gives the predicted loop response for the TPS61165EVM as shown in Fig. 7, which provides an easy comparison with measured data.
We can easily explain the differences between the measured and simulated gain by observing variations in the WLED dynamic resistance and using the typical LED IV curve as well as chiptochip variations in the IC's amplifier gain.
CONCLUSION
While not exact, the mathematical model described here gives the designer a good starting point for designing the compensation circuit for a WLED currentregulating boost converter. In addition, the designer can measure the control loop with one of the alternate methods.
REFERENCES

Ray Ridley. (2006). Designer's Series, Part V: CurrentMode Control Modeling. Switching Power Magazine[Online]. Available: http://www.switchingpowermagazine.com/downloads/5%20Mode%20Control%20Modeling.pdf