Two resonant frequencies

There’s a substantial trend in offline converters operating at a few hundred watts of output power to use an LLC half bridge for the dc-dc converter stage. This topology provides favorable efficiency and lower electromagnetic interference (EMI) signatures due to resonant operation. Also, it doesn’t require an output inductor or any of the resultant circulating currents and losses in the secondary windings of the transformer.

As a longtime fan and veteran of fixed-frequency pulse-width modulation (PWM), I’ve found that the design approach to the LLC half-bridge converter is notably different from that which I’m used to. The LLC converter is a resonant half-bridge converter operated at 50/50 duty cycle with a little deadtime between the half-bridge switches. Two resonant frequencies in this converter separate the operation from most any other LC resonant converter in use.

The first is a lower-frequency, load-dependent resonance that varies so the maximum Q occurs at light load (*see the figure*). This frequency is determined by the sum of the leakage inductance (Llk) and the magnetizing inductance (Lm) of the transformer primary winding and the resonating capacitor.

The second is a stable series LC resonance that exhibits little to no load dependency determined by the Llk of the primary winding and the resonating capacitance (Cr). Most of these converters are operated above resonance so that at light-load conditions the switching frequency is higher than the resonant design points (region 1). As load increases, the switching frequency droops. At maximum load, the converter operates slightly below the LlkCr resonant point in region 2.

As it relates to line voltage, the frequency is higher at high-line conditions, and the operating frequency drops as line voltage drops. The lowest-frequency point is at low line and max load. The current waveforms in this converter are sinusoidal when operated at or around resonance. The excitation voltage to the transformer primary (and hence the secondary voltage waveforms) are square-wave signals.

Transformer Discussion

The first thing to consider in the design of the LLC half bridge is the transformer design. When considering the delta B flux swing in the primary of the transformer, we need to consider the minimum operating frequency and the 50/50 square-wave excitation at the primary. For the turns ratio, we need to understand the power transfer mechanism in the LLC converter.

We basically have a circuit with two resonant frequencies: an upper designed resonant frequency and a resultant high-Q load-dependent resonance somewhere below this point. It is important to note that the inductances that form these resonant circuits are formed by the leakage inductance and the sum of the leakage and magnetizing inductance of the transformer respectively.

Having spent a lot of time designing transformers for fixed-frequency PWM converters, I’ve found that a transformer of this nature usually requires tightly coupled windings and minimal leakage inductance. Limitations on delta B flux excursions and losses from primary circulating current generally call for considerably high magnetizing inductance in these fixed-frequency structures. This is not the case at all for the LLC transformer!

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There needs to be enough circulating current in the magnetizing inductance to resonate the switch node up or down for zero voltage switching (ZVS) switching conditions at light-load or no-load conditions during the deadtime of the half-bridge switches. Too much magnetizing inductance will drive lower circulating currents in the primary and hard switched conditions. The leakage inductance of the primary winding needs to be fairly small for good regulation, clearly smaller than the magnetizing inductance.

But how small can it be? To answer this we need to understand the transfer function of the LLC network. As Lm and Llk get closer together, the peak of the (Lm + Llk)Cr resonance gets higher and higher. This peak shows up as voltage stress on the resonating capacitor. To maintain a cost-effective capacitor, we don’t want this voltage peak going too high. As good design practice, it is advisable to start out with Lm/Llk equal to 5.

What this means for the transformer houses is that the primary and secondary can be wound with a little less attention to all of those fixed-frequency PWM details like interleaving, magneto-motive force (MMF) stackup, and proximity effect. Most LLC transformer bobbins are partitioned so the primary is wound on one partition and the secondary is wound on the other.

This simplification tends to hold the leakage inductance higher than conventional interleaving techniques and more constant from unit to unit and shift to shift at the transformer manufacturer while satisfying the prescribed Lmag/Llk ratio. (I remember the days of interleaving and carefully considering proximity effects, MMF stackup, and eddy current losses when Lmag/Llk was 100 or higher. We don’t need that for LLC half-bridge circuits!)

So we now have a feel for the role of leakage and magnetizing inductance and the volt*time products involved, but we haven’t discussed turns ratio yet. The worst-case operating condition for the LLC resonant converter is where the transfer function shows the lowest V_{Out}/V_{In} value. This is at high-line and light-load conditions where the operating frequency is above LlkCr resonance and the low-pass portion of the transfer function envelops the gain.

This is where we need to understand the upper boundary of the switching frequency of the LLC converter. We don’t want to approach harmonic modes of operation with any of the resonant components and the lumped parasitic capacitances of the MOSFETs (drain to source), so it’s usually advisable to keep the maximum frequency around twice the LlkCr resonant frequency.

LLC Circuit Design

With this in mind, if we look at the transfer function of the converter under no-load conditions, we will see that V_{Out}/V_{In} is slightly below unity. This is a great time for a simulation. If I excite the primary with my high-line voltage condition, perhaps 400 V from a power factor correction (PFC) preregulator, my half bridge will then apply a square-wave signal varying between 0 and 400 V to the Cr/transformer/secondary network. Considering V_{Out}/V_{In}, the transformer primary will see something slightly less, in most cases about 87% of this voltage or ±174 V. This is the lowest primary voltage.

The secondary turns ratio needs to be sized from here. For example, for a 24-V output, I might include a 2-V lumped diode/dc resistance/printed-circuit board (PCB) trace drop and set up the secondary to deliver 26 V from the center tap to either leg at 174-V primary excitation (Np/Ns = 174/26 or 6.692). A good practical turns ratio from this may be Np/Ns = 40/6 or 6.67. This would mean that either leg of the secondary would have six turns to the center tap or 12 turns across the legs.

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To determine the leakage inductance values, we need to go back to the simulation and look at the transfer curves. The K value, or Lm/Llk, determines the peak of the transfer curve and (Lm + Llk)Cr resonance. Initially you might want to start with reasonable values like Llk = 50 µH and Lmag = 250 µH and set up Cr so LlkCr resonance occurs at your nominal switching frequency (Fr1 in AN1160). The reflected load resistance, Rac, is placed across Lm. The resistor value is sized so it dissipates the full output power of the converter.

From this point, we can graphically examine the maximum value of the transfer function of the LLC network. This value is referred to as Mmax in AN1160. The frequency at which this occurs is our minimum operating frequency, Fmin. It usually comes out to 60% to 70% of the nominal switching frequency. Next we use Mmax and K to calculate Qmax, the full load Q of the LLC circuit:

Once we know this value we can calculate Llk as:

Llk = (Qmax*Rac)/(2*π*Fr1)

Once we know Llk, we can calculate Lm as K*Llk. Cr is calculated to resonate with Llk to form Fr1:

Cr = 1/(Llk*4*π^2*Fr1^2)

The composition and voltage rating of Cr needs to be carefully considered. I strongly recommend a good polypropylene capacitor. We can back out the capacitor voltage over the operating conditions in the simulation. Clearly the capacitor needs to block the dc component, pass the high-frequency ac current, and handle any resultant high-frequency ac voltage drop. In most cases a 1000-V dc rated polypropylene capacitor will work, but this needs to be verified in circuit and against the manufacturers’ high-frequency derating.

The IRS27951 and the IRS27952 LLC control solutions from International Rectifier will accommodate the minimum and maximum frequency settings with small signal resistors around the IC. AN1160 at www.irf.com presents a much more detailed description of the LLC half-bridge design. As for the feedback loop, we can write the transfer function for the LLC circuit and see that there are two zeros at the origin and four poles at the breakpoints of the LLC(C_{Out}) resonances. Compensating for this typically requires a type II compensation network, usually placed around a secondary side error amplifier into an optoisolator for hot/cold isolation.

Conclusions

While markedly different from a fixed-frequency PWM converter, the LLC resonant half bridge is an excellent solution to offline medium-power requirements. The converter requires careful consideration of the leakage inductance and magnetizing inductance of the primary winding of the transformer and the transfer function of the LLC circuit. I’ve made an attempt at presenting a condensed, nuts and bolts version of how the circuit works and how to design it. However, much more detailed information is available at www.irf.com under AN1160.