Inductors Allow Loss-Less Current Sensing in Multiphase DC-DC Converters

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High performance microprocessors require a low-voltage, high-current power supply with fast transient response. Therefore, interleaved multiphase synchronous buck converters have gained popularity as the VRM for these microprocessors. Multiphase converters must share load current to balance thermal stress on each phase, so it's important to address current sensing and sharing. You can use the current signal in adaptive voltage positioning to reduce the output voltage deviation during transients, which requires accurate current information.

Current sense resistors have high accuracy, less than ±5%, and their temperature coefficient (TC) is small. However, the sense resistor connection is in the power path, which generates additional power loss and lowers efficiency — especially in high output current applications ^[1]. Another disadvantage of the sense resistor is its relatively high cost. Current sensing through MOSFET R_DS(on) is a loss-less method, but this on-resistance has a large tolerance from lot to lot even for a given manufacturer and changes with the gate-drive voltage of MOSFET ^[1]. The TC tolerance of R_DS(on) isn't well defined, and is not always available.

You can use a dc resistance of an inductor to sense current. The inductor dc resistance changes with temperature, but the TC of copper wire is well defined, and independent of the inductor design of the manufacturer. A resistor-capacitor circuit in parallel with the inductor supplies the current information, which is then amplified by a current sense amplifier. The use of the amplified current signal is not only for the PWM ramp and adaptive voltage positioning, but also for current protection.

Fig. 1, on page 60, shows a buck converter with the inductor current sensing circuit, where L is the inductance and R_L is the inductor's dc resistance. In parallel with the inductor is a filter consisting of low-power resistor R_S and small ceramic capacitor C_S^[2]. Current sense amplifier, CSA, amplifies the voltage across C_S, which represents dc and ac components of inductor current. One reference cited a similar scheme ^[4] to obtain dc current information, mainly for overcurrent protection. It received a patent in 1995, despite a prior art in 1981 ^[5] and a relevant paper in 1994 ^[6]. There are other current sensing schemes that utilize inductor dc resistance. Instead of the simple resistor-capacitor filter across the inductor, you can use the differential active filter ^[7] or inductor current estimator circuit ^[8].

As shown in Fig. 1, if the current through the inductor is i_L, the voltage across the capacitor C_S is:

Therefore, the transfer function of the current sensing circuit is:

Fig. 2 shows the transfer function of Equation (2) under three different conditions. If R_SC_S = L/R_L, the R_S-C_S pole cancels the L-R_L zero, and the voltage v_c is proportional to the inductor current i_L as if a sense resistor R_L was used. If the two time constants are mismatched, the gain of the current sensing circuit is frequency dependent. The gain is R_L at low frequency, and equals to L/R_SC_S at high frequency, which is higher or lower than the dc gain depending on the relative magnitude of the two time constants.

It's difficult to match the time constants in the real applications, because the dc resistance varies with temperature, whereas the inductance changes with its bias current. When used in current protection, R_SC_S≥L/R_L normally improves noise immunity of the current sensing circuit. In multiphase buck converters using peak current mode control or enhanced V² control ^{[2] [3]}, you can employ the current signal for current limit as a major part of the ramp signal of PWM comparator. Use of proper ramp magnitudes optimizing converter performance determines the selection of R_S and C_S values, rather than on the match of time constraints. The flexibility to choose R_S and C_S values makes it possible to remove the restriction in selecting the inductance and inductor dc resistance and to use a fixed gain current sense amplifier integrated in the IC circuit.

Accuracy

Accuracy of the current sensing scheme depends on variation of dc resistance with temperature, manufacturing tolerance of dc resistance and the inductance, change of inductance with bias current, and tolerance of components R_S and C_S^[4]. Although R_L variation affects the dc gain of the current sensing circuit, the changes in L, R_S and C_S influence the high frequency ac gain. Table 1, on page 58, shows the gain errors due to these factors, and classifies them as dc gain error and high frequency ac gain error.

Inductor dc resistance changes with temperature because copper has a positive TC of 0.385%/°C. Assuming a maximum temperature of 90°C and an ambient temperature of 25°C, the gain error would be 25%, or ±12.5%.

The manufacturing error of the dc resistance depends on wire sectional area tolerance and wire length variation. For inductors using large gauge magnet wire, the dc resistance variation with wire sectional area is ±2%. The length of the wire is on the order of 10 cm and its error is less than ±1%. Manufacturing error of inductance is mainly from core permeability variations, and is about ±8%. The error from mechanical tolerance is about ±2%.

Inductance changes with its bias current, and obtains the relationship from the core-magnetizing characteristic. The variation of inductance depends on the core material, ranging from -1% to -30% at certain current levels. For the typical cores used in the VRM applications, the inductance change is -10% at rated current.

R_S and C_S component values only affect the high frequency ac gain. If the resistor and capacitor have ±1% and ±5% tolerance, respectively, their worst-case error would be ±6%.

The dc and high frequency ac gain errors of the current sensing circuit in the worst case are the sum of all the above errors, respectively. The total current sensing error, however, is not the sum of the dc and ac gain errors. It depends on the relative magnitude of the dc and ripple current, and is the larger of the two gain errors. The worst case is when the current consists only of ripple and the dc current is zero, and then the total error is equal to the ac gain error, i.e.-26% to +16%. The current sensing error affects the pulse-by-pulse current limit and adaptive voltage positioning of the converter.

As shown in bold letters in Table 1, on page 58, the current sharing errors are smaller than the current sensing errors, because the inductor temperature and current of the multiple phases are close. If the temperature difference among inductors is within ±5°C, the temperature influence on the difference of the inductor dc resistance and therefore the dc gain is ±2%. The influence of bias current on inductance difference is also smaller in current sharing. ±5% difference in inductor current causes 1% of inductance difference for the typical cores used.

For multiphase dc-dc converters, load current sharing balances the thermal stress on the components of each phase. If you use the peak current mode control (PCMC) for current sharing, the error in average current is half of the error of ripple current. Therefore, the ac current sharing error is half of the high frequency ac gain error, and the total current sharing error is the larger of the dc and ac gain error. If you use the average current mode control (ACMC), the high frequency ac gain error doesn't affect the total current sharing accuracy. The above errors are for the worst case, and matching the inductors and RC components can reduce current sharing error. It's possible to achieve ±5% of system current sharing error by the current sensing scheme, which is acceptable for the VRM application.

Besides the advantages of low cost and less power loss, the current sensing scheme has self-balance function in current sharing. If one phase supplies more current, the higher temperature of its inductor causes an increase of the dc resistance. Therefore, the dc sensing gain becomes higher, which tends to reduce the current of this phase.

Current Signal Compensation

Because of the low and tight output voltage requirement of microprocessors, adaptive voltage positioning is adopted to reduce the output voltage deviation during transients. At light load, the output voltage is set higher than the nominal value, and the output voltage becomes lower when load current increases. Fig. 3, on page 64, shows the 3-phase buck converter with the current sensing scheme. The current sensing filters consist of Rs1-Cs1, Rs2-Cs2 and Rs3-Cs3, and the current signals of the three phases connected to the current sense amplifiers CSA1-CSA3 respectively. You can use the amplified current signals CO1-CO3 not only as the ramp of the PWM comparators, but also summed together with the reference voltage VREF to form the droop signal VDRP. Assuming the identical component values in the three phases, the droop voltage change is proportional to the inductor current of one phase as follows, where A_i is the gain of the current sense amplifier.

The droop signal is combined with feedback voltage at the input of the error amplifier, VFB, and it changes the output voltage deviation, which is related to the droop voltage and the inductor current as:

Time constants of inductors and R_S-C_S circuits are mismatched in the application. If R3, C3, and C4 are not used in Fig. 3, the droop signal is different from the actual inductor current because of the mismatch. The overshoot of the droop voltage causes more output voltage drop and the output voltage is out of the spec, as shown in Fig. 4(a), on page 65. If R3=R_S, C3=C_S, and R3×(C3+C4)3=L/R_L, it cancels the poles and zeroes in Equation (5). The mismatch of the time constants is compensated and the droop signal is proportional to the inductor current, as shown in Fig. 4(b), on page 65. The output voltage is nearly a square wave when the current jumps between no load and full load.

Application

Fig. 5, on page 66, shows the current sensing scheme applied in the current sensing and sharing of a 3-phase synchronous buck converter. The converter is designed for the next generation microprocessor, requiring output voltage of 1.45V and an output current of 60A with a slew rate of 350A/µs. The switching frequency is 200 kHz.

Inductors L1-L3 have inherent dc resistance of 0.9mΩ and their currents are sensed through the resistors and capacitors Rs1-Cs1, Rs2-Cs2, and Rs3-Cs3 respectively. Looking at Fig. 6, you can see that the system current sharing accuracy of the converter is smaller than ±3.5% at the full load. The reason for the small error is that the current sensing circuit has reasonable accuracy. Above all, the IC integrates the three current sense amplifiers and PWM comparators, ensuring matching accuracy.

References:

M. George, and G. Baker, “Universal Current Sharing Issues for VRMs,” in Intel Technology Symposium 2000, Seattle, Sept. 2000.
ON Semiconductor, “CS5301 Three-Phase Buck Controller with Integrated Gate Drivers and Power-Good,” Data Sheet, July 2000.
W. Huang, “A New Control for Multiphase Buck Converter with Fast Transient Response,” in APEC Proc., Anaheim, Calif., March 2001, pp.273-279.
I. Cohen, “Overload Protection of Switch Mode Converters,” US Patent, 5465201, Nov. 1995.
H. Kubach, and H. Rupp, “Controlled D-C Current Supply System, with Controlled Current Flow through a Choke,” US Patent, 4293812, Oct. 1981.
B. Carsten, “Designing High Frequency Current Shunts and Current Transformers,” in Proc. of HFPC, San Jose, Calif., April 1994, pp.539-550.
E. Dallago, M. Passoni, G. Sassone, “Lossless Current Sensing in Low-Voltage High-Current DC/DC Modular Supplies,” IEEE Trans. Ind. Electron., vol. 47, pp.1249-1252, Dec. 2000.
B Arbetter, and D Maksimovic, “DC-DC Converter with Fast Transient Response and High Efficiency for Low-Voltage Microprocessor Loads,” in APEC Proc., Feb. 1998, pp.156-162.