Resonant Frequency, Q, and Voltage Gain

Modeling the Converter for Analysis

To understand the relationship between frequency and gain, we first need a simplified model of the converter. The primary side inverter creates a square wave, and the resonant tank filters it. However, the secondary side, with its transformer, rectifier diodes, and output capacitor, is a non-linear system. Analyzing this directly is complex.

A standard and powerful technique, which your maths and CS background will help you appreciate, is the First Harmonic Approximation (FHA). The core idea is that the resonant tank is a very effective low-pass filter. It strongly attenuates all the higher-order harmonics of the input square wave, meaning only the fundamental (first) harmonic contributes significantly to power transfer.

This allows us to make two key simplifications:

1. Replace the square-wave-generating switching stage with an equivalent sinusoidal AC voltage source.
2. Replace the entire non-linear rectifier and load on the secondary side with a single equivalent AC resistance ($R_{ac}$) on the primary side.

The following video provides an excellent conceptual walk-through of this modeling process and derives the final gain equation.

The Key Parameters Defining the Gain

As the video showed, the converter's gain depends on a few critical, interrelated parameters. Let's define them clearly.

Let's summarize those key parameters:

• Resonant Frequency ($f_r$): This is the natural resonant frequency of the series L-C pair, determined by $L_r$ and $C_r$. It serves as the central reference point for the converter's operation. $$f_r=\frac{1}{2\pi \sqrt{L_r{C}_r}}$$
• Normalized Switching Frequency ($F_x$): To make the analysis universal, we talk about the switching frequency $f_s$ in relation to the resonant frequency $f_r$. A normalized frequency of $F_x=1$ means the converter is switching exactly at resonance. $$F_x=\frac{f_s}{f_r}$$
• Quality Factor (Q): This is perhaps the most important parameter for understanding the gain curves. It's the ratio of the resonant tank's characteristic impedance to the equivalent AC load resistance $R_{ac}$. $$Q=\frac{\sqrt{L_r/C_r}}{R_{ac}}$$ Since $R_{ac}$ is the model for the output load, Q is a direct function of the load current. This leads to a crucial and somewhat counter-intuitive relationship:
  • Heavy Load $\to$ Low output resistance $\to$ Low $R_{ac}$ $\to$ High Q
  • Light Load $\to$ High output resistance $\to$ High $R_{ac}$ $\to$ Low Q
• Inductance Ratio (m): This is the ratio of the total primary inductance to the resonant inductance, $m=(L_r+L_m)/L_r$. It quantifies the influence of the magnetizing inductance $L_m$ on the tank's behavior.

Visualizing the Relationship: The Gain Curves

With these parameters defined, we can now plot the resonant tank's gain versus the normalized switching frequency for different load conditions (different Q values). This gives us the iconic LLC gain curves.

Let's interpret this fundamental graph:

1. Operation at Resonance ($F_x=1$): At this specific frequency, the gain of the resonant tank is 1, regardless of the load (Q). This is a highly efficient operating point, and designers often choose the transformer turns ratio so that the converter operates here under nominal input voltage and full load.

2. Operation Above Resonance ($F_x>1$): In this region, the tank's gain is less than 1. The converter acts like a "buck" or step-down converter. To decrease the output voltage (or to handle a higher input voltage), the controller increases the switching frequency.

3. Operation Below Resonance ($F_x<1$): Here, the tank's gain can be greater than 1. The converter acts like a "boost" or step-up converter. To increase the output voltage (or to compensate for a low input voltage), the controller decreases the switching frequency.

4. The Role of Q (Load): As the graph shows, the shape of the gain curve depends heavily on the load. At heavy loads (high Q), the curve is relatively flat. At light loads (low Q), the curve develops a high peak, offering a much larger potential gain.

Critically, the peak of each curve marks the boundary between two operating regions. As we discussed in previous lessons, ZVS is only guaranteed in the inductive region, which lies to the right of the gain peak. Operating to the left of the peak (the capacitive region) can lead to hard switching and component failure. Therefore, the controller must be designed to always keep the switching frequency higher than the frequency at the peak gain for a given load.

The Impact of the Inductance Ratio 'm'

The load (Q) is a dynamic variable, but the inductance ratio m ($=(L_r+L_m)/L_r$) is a fixed design parameter chosen by the engineer. This choice has a significant impact on the shape of the gain curves and represents a classic engineering trade-off.

As the reading shows, the trade-off is:

• Low 'm' value (e.g., m=3): This provides a higher maximum gain, which is useful for applications with a very wide input voltage range. The downside is that a smaller 'm' implies a smaller magnetizing inductor ($L_m$), which leads to higher magnetizing current. This "circulating" current doesn't go to the load but still causes conduction losses in the primary-side components, reducing efficiency.
• High 'm' value (e.g., m=12): This results in a lower maximum gain. The benefit is a larger magnetizing inductor, which means less circulating current and higher efficiency.

When you review a schematic, the ratio of $L_m$ to $L_r$ gives you a clue about the design priorities: was the design optimized for a very wide operating range (low 'm') or for peak efficiency under more nominal conditions (high 'm')?

Conclusion

In this lesson, we have demystified the control mechanism of the LLC converter. By modulating the switching frequency relative to the resonant frequency, a controller can navigate the converter's gain curves to regulate the output voltage against changes in both input voltage and output load.

Key Takeaways:

• LLC converters are controlled via frequency modulation.
• The relationship between frequency, load, and gain is captured by a set of gain curves.
• The key parameters that define these curves are the normalized frequency ($F_x$), the load-dependent quality factor (Q), and the design-dependent inductance ratio (m).
• The converter operates with a gain > 1 below resonance and a gain < 1 above resonance. At resonance, the gain is 1, independent of load.
• The choice of Q and m involves critical engineering trade-offs between the required gain range, operating frequency range, and overall efficiency. A heavy load corresponds to a high Q, and a light load to a low Q.

In our next lesson, we'll build on this by examining the operational challenges. Specifically, we'll discuss the trade-off between having a wide ZVS operational range and the penalty of circulating current losses that come with it.