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Advanced semiconductor technology allows an increasing number of transistors to be integrated onto ever-smaller silicon chips. This increases the power density, however, and presents power-management challenges, especially in computing and telecommunications applications. First, the higher the integration, the higher the power required to operate the chip. In addition, with the transistors working at higher frequency, there is more dynamic load on the power supply. The output voltage can be decreased to avoid overheating the silicon, but the lower operating voltage tightens the voltage-regulation requirements.

In these applications, the most-widely adopted circuits are the multiphase buck converter shown in Figure 1, and its isolated versions, including full-bridge, half-bridge, and forward converters. To optimize the design, several factors, such as control and power devices, need to be considered. Another important factor is the output filter inductor, which influences several aspects of a power-supply's performance. With a larger inductance, there is smaller current ripple. This leads to higher efficiency, lower output voltage ripple, and lower EMI noise. Unfortunately, larger inductance also means that more energy is stored in the magnetic, which results in a larger footprint and slower transient response.

Figure 1. A two-phase buck converter

This article will discuss inductor design for switching converters, and a solution that uses coupled inductors. First, the fundamental limitations of the traditional inductor in low-voltage, high-current, fast-transient applications are described. Next, the operation and behavior of coupled inductors is explained. New problems are encountered when implementing coupled inductors, however, and several controller-based solutions are proposed. Lastly, a two-phase buck converter design is shown as an example.

Limitations of the traditional inductor solution
In computing and telecommunication applications, it is critical for power supplies to have fast transient response and high efficiency, but from the inductor design perspective this is a contradiction.

Inductor's influence on transient performance
Because of the inherent delays in power converters, the load current demand has a much higher slew rate than that of the inductor current. The difference between these two currents determines the charge that needs to be absorbed by the output capacitors, as shown in Figure 2. The overshoot due to the additional energy is calculated as:

where ΔI_o is the load current step, C is the output capacitance value, S_iL is the inductor current slew rate, and S_io is the load current slew rate. In order to reduce the overshoot with certain load current steps, it is necessary to increase C or S_iL . Because of the high cost and large footprint, a large output capacitor is undesirable. Therefore, high slew rate in the inductor is mandatory.

Figure 2. Difference in current slew rates determines output voltage overshoot

The slew rate of the inductor current is determined by the feedback control and the inductor value. The concept of critical inductance has been proposed to evaluate the relationship between the feedback control loop bandwidth, f_c , and the inductor value. With input voltage V_in and duty cycle D, the critical inductance is:

When the inductor value is less than L_critical , the current slew rate is determined by f_c . Otherwise, it is limited by the inductor value. Therefore, the maximum benefit for feedback control is obtained when the inductance is small.Inductor's influence on efficiency
On the other hand, a large inductance is desired for small current ripples, because:

A small inductance, yields a large current ripple. This causes extra conduction, switching losses, and reduced converter efficiency. In high-frequency switching converters, 40% of the power loss is from the turnoff of the top devices, which is proportional to the turnoff currents:

Transient versus efficiency
Small inductances improve the transient response but at the expense of efficiency. This tradeoff is illustrated in Figure 3. For example, with a 12-V-to-1.2-V 300-kHz 2-phase buck converter each phase has a maximum current of 25 A. With linear control of the switching power converters, the bandwidth, f_c , is designed to be about 20% of fs. With 60-kHz bandwidth, L_critical is 200 nH. With this inductance, the inductor current ripple is 18 A, or 70% of the full-load current! The 34-A turnoff current at full load leads to huge power losses. A practical inductor design of 300nH would result in 40—50% ripple. Although the ripple can be further reduced with a larger inductor, as shown in (4), there is diminishing return once the inductance is sufficiently large.

Figure 3. Inductor design tradeoff between power loss and output capacitor

For best system efficiency, a large inductor value is preferable, because it leads to small current ripple. On the other hand, a small inductor is desirable for good transient response due, because it can achieve fast energy transfer. This conflict inherently limits the performance of converters that use traditional inductors. It would be better if the inductors could have a nonlinear behavior, having large values at steady state and small values during the transients.Benefits of Using a Coupled Inductor
In multiphase interleaved applications, the coupled inductor solution shown in Figure 4 has this nonlinear performance. With self inductance L, and mutual inductance M=∝L, coupling the inductors from two phases gives:

Because of the interleaving operation of the paralleled phases, v₁ and v₂ have several different patterns during one switching cycle. The inductance determines the steady-state ripple for i_L1 and i_L2 as:

On the other hand, the duty cycle of the two phases follows the same command in the feedback control. Therefore, with a dynamic load, v₁ and v₂ have the same average transient response. In addition to L_ss , another inductance determines the transient performance:

Figure 5 illustrates the relationship between L_ss and L_tr with different duty cycles and coupling coefficients. L_ss is always larger than L_tr , which means that the coupled inductor is superior to the traditional inductor in its ability to provide both high efficiency and fast transients to a power supply.

Figure 4. A two-phase buck converter with coupled inductors

Figure 5. Ratio of Lss/Ltr as a function of duty cycle and coupling coefficient

Based on Figure 5, however, L_ss does not have to be higher than the self inductance. Therefore, the inductors need to be redesigned when introducing coupling. When working with a noncoupled inductor L_nc , it is possible to improve performance by keeping L_tr the same as L_nc . This implies that L_ss is larger than L_nc . This method maintains transient performance, reduces inductor current ripple, and can result in higher converter efficiency. An alternative solution is to design L_ss the same as L_nc . The inductor current ripple stays the same, the inductor current slew rate increases with small L_tr , and the transient performance improves.

When coupling inductors in the interleaved phases of a converter, two inductance parameters exist for each inductor, with one determining the steady-state current ripple and the other determining the transient performance. Proper design thus allows both high efficiency and fast transients to be achieved.Limitations in coupled inductors
Unfortunately, there is limited room for further improving efficiency based on today's noncoupled inductor designs. With 40″50% current ripple, the power loss due to the current ripple is relatively insignificant. The benefit of keeping L_tr and increasing L_ss is therefore not obvious. In addition, increasing L_ss leads to much higher self inductance, which normally results in much longer inductor winding traces and higher dc resistance (DCR). Consequently, the power loss on the DCR increases, which further reduces the increase in efficiency.

Therefore, most of coupled inductor designs try to reduce L_tr with similar L_ss and L_nc . This way, the transient performance increases, so that fewer output capacitors are needed. Even with this approach there are implementation limitations, however, First, there is a limit on improving the transient performance by reducing L_tr . As shown in Figure 3, when L_tr is smaller than L_critical , the inductor current slew rate is limited by the feedback loop bandwidth.

Meanwhile, maintaining the inductor current ripple does not imply the same output current ripple. As shown in Figure 6, with the noncoupled inductor, the ripple in the total current is smaller than the inductor current because of cancellation with the interleaving operation. However, with the coupled inductors, the two current ripples are added. Hence, with coupling, each phase's current ripple stays the same, but the total output current ripple increases, thus leading to higher output voltage ripple. Meanwhile, using a smaller L_tr reduces the output capacitor. With a reduced output capacitance, the output voltage ripple increases.

Figure 6a. Current ripples for interleaved buck converters with noncoupled inductors

Figure 6b. Current ripples for interleaved buck converters with coupled inductors

Two limitations are encountered when exploring the benefits of coupled inductors: limited bandwidth of the control loop, and higher output voltage ripple. To fully reap the benefits of coupled inductors, these barriers must be broken.

Design with coupled inductors
Fortunately, with proper controller designs, the performance with coupled inductors can improve.To overcome the limitation from the feedback loop bandwidth, it is necessary to use nonlinear control methods. One implementation is enhanced PWM (EPWM). With this method, steady-state operation maintains constant frequency and perfect interleaving. Meanwhile, during the load transient, all of the phases can be turned on or off to minimize undershoot and overshoot. With EPWM enabled, there is no longer a concept of bandwidth, and the inductor current slew rate is limited only by the inductor values. Therefore, smaller L_tr is always beneficial with EPWM.

The output voltage ripple is mainly determined by the parasitics of the output capacitors, including the equivalent series inductance (ESL) and equivalent series resistance (ESR). Therefore, selecting output capacitors with lower ESL and ESR can reduce the output voltage ripple. For example, a 100-�μF ceramic capacitor has 2-nH ESL and 1.4 m&Omega ESR, and a 22-μF ceramic capacitor has 1-nH ESL and 2-m&Omega ESR. Therefore, maintaining the same capacitance, while replacing 100-μF capacitors with 22-μF capacitors can significantly reduce the output ripple. Unfortunately, this method has its own limitation: using small-value capacitors leads to a larger number of capacitors and thus a larger footprint.

On the other hand, the ripple requirement is a part of the whole output voltage regulation specification. The ripple is important, because a larger ripple leaves less room for the voltage variation from the load transient, temperature, and other variations. Therefore, if the controller could minimize the error from the other aspects, there would be more room allowed for the output ripple. Hence, L_tr can be further reduced to reduce the output capacitor.

Therefore, a high accuracy controller with nonlinear control can maximize the benefit of coupled inductors. To illustrate this point, the ADP3192A controller from Analog Devices is evaluated. Figure 7 shows a 2-phase 600-kHz design with a 12-V input and a 1.2-V output. The coupled inductors in this case have L_tr of 60 nH and L_ss of 110 nH. With this design, only three 100-μF capacitors are needed in additional to eighteen 22-μF decoupling capacitors. With the load steps between 10 A and 60 A, perfect output voltage transient is obtained.

Figure 7. Transient performance

Conclusion
The coupled inductor is superior to the traditional inductor in its ability to provide both high efficiency and fast transients in a power supply. There are, however, design limitations to be considered. The solution lies in using a high-accuracy nonlinear controller that maximizes the benefit with coupled inductors.