Technical Article

# Demystifying Resonance

June 15, 2020 by Michael Doktor

## This article discusses resonant power and give some insight to scale a documented and closely modeled kernel to a desired power level.

Resonant power is tricky—to state it mildly! The devil is not in the details! Not initially at least. One can easily mistake the forest for the trees, getting truly lost, and not even realize it. We need a soft transition from the relative comfort of our familiar world of classical (“PWM”-based) power, into the yet-nonintuitive world of resonant power, including wireless power transfer (WPT). Preferably with a stopover at the “LLC topology”.

WPT and LLC are very much the same topology, or should have been, but most WPT solutions today use the “double resonator” shown as the second from top circuit in Figure 1, (shaded red), which actually creates multiple resonant peaks (lower half of Figure 2). This strange and wholly anticipated gain profile is unfortunately not even amenable to any proper control or tracking/optimization algorithm. We need to disband it quickly. The upper half of the figure, is what we get with only primary-side resonance, which is strongly recommended even for creating proper correct WPT, as also commonly used in LLC. It is amenable to smart, but possibly proprietary control algorithms for enhancing “user experience” and handling variable coupling, the key distinguisher between LLC and WPT.

## Defining the “Kernel”

We can take any kernel and scale it to any power, frequency, and input voltage range.

As a primer, here are the scaling law patterns that we will be using. To double the power, halve the inductance and double the capacitance

• To double the frequency, halve the inductance and halve the capacitance
• To quadruple the power, double the input voltage

Instead of “double” or “halve”, we can use other scaling factors to generalize, quite obviously. The trend is clear. Here, just for historical continuity, we have used the same values of LP and CP (57.2µH and 225.8nF) that we had arrived at for the PoE application in Switching Power Supply Design and Optimization, Second Edition. Since we are scaling anyway, we could actually use any initial kernel. We then inputted these LP and CP values into a general Mathcad spreadsheet, to generate the graphical aids presented herein.

##### Figure 2: Plotting Gain curves for the PoE converter, with and without CS

In Figure 2, just to throw more light on the gain curves that we typically get in an LLC topology, as an intermediate step, we have used our Mathcad spreadsheet to suggest the “recommended” load resistor of 20.6 ohms, for the PoE application. Then we have used multiples, or fractions, of that 20.6 ohms “recommended” resistor, to generate the stacked gain plots, to reveal the general trend. Note that “20.6 ohms” is the equivalent AC load resistor, not the DC load in the final DC-DC converter. It is the one meant to be applied to the equivalent AC-AC, non-isolated model using the First Harmonic Approximation (FHA).

In Figure 3, we see that the resonant peak shifts dramatically as a function of load. In fact, it is also a function of coupling K, because the right extreme, fHI, is simply the resonance of the leakage with the resonant capacitor. fLO is the resonant peak at very light loading.

##### Figure 3: Varying load in an LLC, causes a frequency shift, depending on coupling

Note that simple control algorithms, as being currently used, just can’t suffice. Frequency modulation techniques, based on trying to raise the gain by simply lowering the frequency, are in obvious danger of rolling “over the top”, as the peak flattens and shifts to the right as load increases. We can easily land up on the wrong side of the resonant peak, to its left, where power suddenly collapses as we lower the frequency futher. Now, expect a “mysterious” turn-OFF (and an Amazon return!). Other simplistic control methods, try to use modulation of duty cycle, or phase or input voltage, while keeping a “fixed frequency”. They also easily land up on the wrong side of the resonant peak, and not only is the wrong side “lossy”, but EMI increases dramatically due to hard-switching, negating the very purpose of “fixed-frequency” methods. Those actually have no place in resonant power, as the wireless electric vehicle charging (WEVC) teams around the world are realizing. These mistakes are magnified at higher power levels. And we haven’t even yet considered the effects of the “double resonator” flaw, which introduces another capacitor into the picture, thus causing two moving peaks, not one. Because now we have an LLCC, not an LLC!

For example, if any current method is based on an intrinsic assumption that the resonant peak is fixed at 100kHz, and has thus described a “control algorithm” which simply states: “head to 115kHz to increase power”, a good question to also ask is: what if the resonant peak has not just moved to “100 /√0.5 = 141.4kHz”? That would be true for K=0.5, and single resonant peak case, including LLC. For a higher coupling, say K=0.7 it would shift to 100 / √0.3 = 182.6 kHz. For K=0.8, it would be 100 / √0.2 = 223.6kHz. It gets much worse in the case of double peaking. But most likely, the current method has also mandated that the maximum upper frequency of operation is fixed at say, 200kHz (a plausible/laudable goal, to simplify power circuitry). But then it means, we will perpetually be on the wrong (left) side of the peak. We can never get to the right side, literally. How can we even ever hope to regulate? Such a charging pad may “sort-of work” for some random receiver, or odd placement (coupling), but the interoperability question looms large.

Let us move on, without CS here-ever-after!

## Practical Design Example of a Wide-Input 900W LLC/WPTconverter

With the above kernel, we have generated two key design curves, shown in Figure 4, and Figure 5. Keep in mind that the values used here, were LP = 57.2µH and CP = 225.8nF. So, this is the challenge:

We want to deliver 900W into a 48V output. Select the best L and C values for the converter for an estimated coupling of K = 0.5. For EMI compliance reasons, we want to stay below 150kHz guaranteed. What are the best-operating frequencies to carry out meaningful simulations, to test its performance? We are assuming an input DC varying from 200 to 400V.

On account of the high-power, the obvious choice is a full bridge, not a half-bridge. At 400VDC therefore, the equivalent AC wave applied to the input of the resonant network is as per the first harmonic approximation (FHA):

This is the equivalent amplitude of the sine-wave applied as per FHA. (For a half-bridge we would have divided this by 2).

##### Figure 4: Resistance lookup aid based on PoE kernel

From Figure 4, for a coupling of 0.5 and a VINMAX/VINMIN ratio (“gain factor”) of 2 (to allow us to reach 200VDC from the original 400VDC), we see that the recommended resistor to be placed on the output of our non-isolated AC-AC equivalent circuit is 14.7 ohms.

This load resistor is actually valid, irrespective of the applied/assumed input actually, because it produces just the right amount of “bulge” in the gain profile curve, in conjunction with the selected C and L (the critical “C/L” ratio in effect, which determines the power capability of a resonant network, in relationship to a given load, R), to allow for a gain of 2, occurring somewhere between fHI and fLO, before the gain curve rolls off! If we achieve that, there is no hint of overdesign either! No excess costly resonant capacitors. Just the right “bulge” in the resonance curve.

Now, had we applied an input of 400VDC through a full bridge, or an equivalent AC voltage of 509.3 Volts, the power at the peak of the AC sine wave input would have been

$$Peak\_AC\_Power = \frac{V^{2}}{R} = \frac{509.3^2}{14.7} = 17650 Watts$$

Note that the average of any Sine-squared function is known to be half. In other words, the average (DC) power we got from the kernel, with the desired maximum input applied to it, is

$$Average\_Kernel\_Power = \frac{Peak\_AC\_Power}{2} = \frac{17650}{2} = 8825 Watts$$

But we want only 900W from our proposed LLC converter, So the Power Scaling Factor we need to apply is less than 1 in our case:

$$Power\_Scaling = \frac{Desired\_Power}{Average\_Kernel\_Power} = \frac{900}{8825} = 0.102$$

Coming to the frequency scaling factor, the fHI of our kernel (for the desired K=0.5) is

$$fhi\_Kernel = \frac{1}{2 \pi \sqrt{(1-K)L_p C_p}} = \frac{1}{2 \pi \sqrt{(1-0.5) \times 57.2 \mu \times 225.8n}} = 62.63kHz$$

To stay below 150kHz, we would like to set fHI of the new converter exactly at 145kHz. So, the desired frequency scaling factor is

$$Freq\_Scaling = \frac{Desired\_fhi}{fhi\_Kernel} = \frac{145k}{62.63k} = 2.315$$

Final step, the recommended LP and CP values for our 900W converter are thus

$$C_p = C_{p\_KERNEL} \times \frac{Power\_Scaling}{Freq\_Scaling} = 225.8nF \times \frac{0.102}{2.315} = 9.95nF$$

$$L_p = \frac{L_{p\_KERNEL}}{Freq\_Scaling \times Power\_Scaling} = \frac{57.2 \mu H}{2.315 \times 0.102} = 242.24 \mu H$$

These are the values to try out in a simulation. They establish the desired power capability of the resonant network in effect, at the desired input voltage, allowing for the desired input variation too, and based on out estimate of K.

In the simulation, the calculated LP will need to be split up as follows

$$L_{LKG} = (1-K) \times L_P = (1-0.5) \times 242.24 \mu H = 121.12 \mu H$$

$$L_{MAG} = K \times L_P = 0.5 \times 242.24 \mu H = 121.12 \mu H$$

Now, if we had applied 400VDC at the input, then that would have given us exactly 400VDC output too, for the gain target of 1, if we introduced a 1:1 transformer and a diode bridge and output smoothing capacitors. In other words, we calculated it all for a VOR = 400V (or slightly higher as recommended earlier). So, with a 48VDC output instead, to appear as 400VDC in the equivalent DC-DC transformerless circuit, the desired turns ratio is

$$N = \frac{V_{INMAX}}{Vo} = \frac{400}{48} = 8.33$$

That is exactly how voltages scale through the transformer, and we are just using the same principle to reflect 400V into the Primary, from 48V at the Secondary. Note: when we actually wind the transformer, we will be using full (or maybe half) integral turns, so we will not get exactly 8.33 as recommended above, anyway. Further, we want to aim for a slightly higher VOR than VINMAX, i.e. to set a gain target slightly more than unity, to assure that we can regulate fully between fLO and fHI, never needing to go higher than fHI, for any load. So, if we manage to set the actual turns ratio NP/NS slightly greater than the calculated value above, it would work better. We may have to wind a few iterations of the transformer, to ensure that we stay within fLO and fHI always, with no “mysterious” dropouts.

Simulators usually use the inverse of this as their “turns ratio”. So, for them we may need to input the following turns ratio into the transformer used in our simulations (connected across the magnetizing/coupled inductance only)

$$N_{SIM} = \frac{V^2o}{V_{INMAX}} = \frac{48}{400} = 2.56$$

Rload 2.56 Power 900 We also realize that in the simulator, to test it for 900W (at the desired = = = output voltage), we simply need to put a resistive load of value

$$Rload = \frac{V^2o}{Power} = \frac{48^2}{900}=2.56 Ohms$$

Finally, looking at Figure 5, we see that the gain factor = 2 curve intersects with the K=0.5 to give us a frequency ratio of 0.75. That is the ratio of the location of the resonant peak of the applicable resonant curve with respect to fHI. Now, this ratio will remain unchanged through the scaling exercise, though the fHI has changed, and is now at 145kHz. So, assuming the same ratio, the location of the resonant peak, at which we should be able to get 900W at 200VDC input is 0.75 × 145kHz = 108.75kHz. That is the frequency we need to use at the lowest input voltage, to confirm maximum RMS current too.

##### Figure 5: The location of the resonant peak

Note: There is no point doing a simulation at maximum input and fHI, i.e. 145kHz to “validate” our design. That simulation point is meaningless. We will definitely get the desired output voltage because we are assuming that at least the correct turns ratio has been set in the transformer. But we also get almost any power we want really! We can keep reducing the load resistor, and we can get 1800W, or 3000W, you name it! Because looking at Figure 2, we realize that at exactly 145kHz (“fHI”), we essentially have a converter that can deliver as much power as we want, assuming ideal conditions (parasitics not included, or too small to matter).

Every LC network has an inherent power capability, which in practice affects the input range we want it to handle. Knowing that fact, we can arrive at an optimal design, instead of “just make the copper as thick as possible”. If a converter is inadvertently designed for more power by an incorrect choice of L-C components, then of course, if we understand basic power scaling, that has come about because we have too large a C value, and too small an L value (but far thicker copper of course). Because to double the power for example, we always halve the inductance (but not the size of the inductor though, which depends on LI2!), and double the capacitance! Keep in mind that resonant capacitors are far more expensive than coils or windings! Overdesign will cost us money.

Think of the money already saved by removing the receiver side resonant capacitor (CS) too!

With that we conclude this article. We have learned to scale a documented and closely modeled kernel to any power level, any frequency, and capable of being the “front-end” converter too, since it is tolerant to quantifiably wide input variations too. No overdesign either. The ability to change coupling too, to almost any desired value, means we can use it even for relatively loosely coupled wireless power transfer systems.