LLC Resonant Converter Design and Calculation

This article based on Monolithic Power Systems article explains what is an LLC resonant converter, its basic principles and characteristics.

Introduction

LLC resonant converter is a type of DC-to-DC power converter that is widely used in electronic applications for efficient power conversion. It uses a resonant tank circuit composed of an inductor (L) and two capacitors (C) to convert an input voltage to a different output voltage.

The resonant operation allows for zero-voltage switching (ZVS) or zero-current switching (ZCS), reducing switching losses and improving efficiency, especially under varying load conditions.

LLC resonant converters have gained significant attention in power electronics due to their ability to meet the demanding performance requirements of modern power supply designs. Among the diverse family of resonant converter topologies, LLC stands out as one of the most prominent.

Resonant tanks, the foundation of LLC converters, are circuits composed of inductors and capacitors that oscillate at a specific frequency known as the resonant frequency. This unique characteristic of resonant tanks enables LLC converters to achieve higher switching frequencies (fSW) and minimize switching losses.

In high-power, high-efficiency applications, switch-mode DC/DC power converters with LLC resonant converters are particularly advantageous. They are ideal for power supply systems with delicate components (e.g., high-end consumer electronics) or power-demanding operations (e.g., charging electric vehicles).

An LLC converter comprises four essential blocks: power switches, resonant tank, transformer, and diode rectifier (as depicted in Figure 1). The process begins with the MOSFET power switches converting the input DC voltage into a high-frequency square wave. This square wave then enters the resonant tank, where it undergoes a process of harmonic elimination, resulting in the generation of a sine wave with the fundamental frequency.

The sine wave is subsequently transmitted to the secondary side of the converter through a high-frequency transformer. This transformer plays a crucial role in scaling the voltage, ensuring that it meets the specific requirements of the application. Finally, the diode rectifier converts the sine wave into a stable DC output.

The remarkable ability of LLC converters to maintain high efficiency even at extremely high power levels stems from their resonant nature. This resonant characteristic enables soft switching in both the primary and secondary sides of the converter, leading to increased efficiency by reducing switching losses.

In addition to saving board space, an LLC topology eliminates the need for an output inductor, allowing all inductors to be integrated into a single magnetic structure, reducing area and cost. This integration also enhances electromagnetic compatibility by simplifying and reducing the cost of shielding a single structure compared to three.

Power Switches

Power switches can be implemented in full-bridge or half-bridge topologies, each resulting in a unique output waveform (Figure 2).

The primary distinction between these topologies lies in the generation of the output waveform. Full-bridge topologies produce a square wave with no DC offset and an amplitude equal to the input voltage (VIN). Conversely, half-bridge topologies generate a square wave with a DC offset of (VIN / 2), resulting in a half the amplitude of the full bridge wave.

Each topology presents its own set of advantages and disadvantages. Full-bridge topologies require more transistors, making their implementation more expensive. Additionally, the increased number of transistors introduces higher series resistance (RDS(ON)), potentially leading to increased conduction loss. On the other hand, a full-bridge implementation reduces the necessary transformer turn ratio (N) by half, thereby minimizing copper loss in the transformer.

In contrast, a half-bridge topology is more cost-effective to implement and offers the benefit of reducing the RMS current across the capacitor by approximately 15%. However, it also introduces higher switching loss.

Considering these trade-offs, it is recommended to use a half-bridge power switch topology for applications with power below 1kW, while a full-bridge topology is preferred for applications requiring higher power.

Resonant Tank

The resonant tank, composed of a resonant capacitor (CR) and two inductors—the resonant inductor (LR) in series with the capacitor and transformer, and the magnetizing inductor (LM) in parallel—filters out the square wave’s harmonics, outputting a sine wave of the fundamental switching frequency to the transformer’s input.

The resonant tank’s gain varies with frequency and load applied to the secondary side (Figure 4). Designers must tune these parameters to ensure the converter’s efficient operation across a wide range of loads by designing the tank’s gain to exceed 1 for all load values.

The LLC converter’s wide operation range and high efficiency stem from the resonant tank’s dual inductors. Let’s examine how this works by considering the tank’s response to heavy and light loads, depending on the inductor.

Figure 5 illustrates the resonant tank’s gain for various loads if it were solely composed of the resonant capacitor and the magnetizing inductor. At light loads, a distinct peak in the resonant tank’s gain is evident. However, the gain for heavy loads doesn’t peak—instead, it exhibits a dampened response and only achieves unity gain at extremely high frequencies.

If the resonant tank were solely composed of the resonant inductor (LR) in series with the resonant capacitor, the behavior would differ. The gain doesn’t exceed 1, but when the load is heaviest, the tank achieves unity gain much faster compared to the parallel inductor.

By implementing both inductors in the resonant tank, the resulting frequency gain response ensures that the converter can adequately respond to a much larger range of loads — in addition, it can enable stable control for the entire load range (see Figure 4). The resulting LLC tank has two resonant frequencies (f_R and f_M), calculated with Equation (1) and Equation (2), respectively.

$$f_R=\frac{1}{2\pi \sqrt{L_R\times <!--\; \times \; -->C_R}}$$ $$f_M=\frac{1}{2\pi \sqrt{(L_M+L_R)\times <!--\; \times \; -->C_R}}$$

The tank’s gain response is dependent on three parameters: the load, normalised inductor, and normalised frequency.

The load is expressed through the quality factor (Q), which is dependent on the load connected to the output. However, using the value of the load is not accurate, since there is a transformer and a rectifier between the output of the resonant tank and the load (see Figure 1). Therefore, we must use a primary-referenced value for the load, called RAC. RAC and Q can be estimated with Equation (3) and Equation (4), respectively:

$$R_{AC}=\frac{8x{n}^2}{\pi }\times <!--\; \times \; -->R_O$$ $$Q=\frac{\sqrt{L_R/C_R}}{R_{AC}}$$

The normalized frequency (fN) is the ratio between the MOSFET’s switching frequency (fSW) and the tank’s resonant frequency (fR). It can be calculated using Equation (5):

$$f_N=\frac{f_{SW}}{f_R}$$

The normalized inductance (LN) is expressed as the relationship between the resonant and magnetizing inductors, estimated using Equation (6):

$$L_N=\frac{L_M}{L_R}$$

With these parameters, we can calculate the converter’s gain response using Equation (7):

$$M_G(Q,Ln,Fn)=\frac{V_{OUT[AC]}}{V_{IN[AC]}}=\frac{f_N^2\times <!--\; \times \; -->(L_N-<!--\; -\; -->1)}{(f_N^2-<!--\; -\; -->1{)}^2+f_N^{2}x(f_N^2-<!--\; -\; -->1)\times <!--\; \times \; -->(L_N-<!--\; -\; -->1{)}^2\times <!--\; \times \; -->Q^2}$$

Note that these calculations are based on first harmonic analysis (FHA). This approach is suitable because we assume that the LLC operates within the resonant frequency (fR). By applying Fourier analysis, we can represent the resonant tank’s input as a square wave composed of multiple sine waves with varying amplitudes and frequencies. Since the resonant tank filters out all sine waves with frequencies different from the fundamental fSW, we can disregard all waves except the fundamental sine wave, significantly simplifying our analysis.

Soft Switching

A notable feature of LLC converters is their ability to perform soft switching.

Soft switching aims to minimize switching losses by synchronizing the electronic switches with the natural rise and fall of current and voltage within the circuit. This ensures that the switches turn on and off at the most optimal points. If switching occurs when the current is close to zero, it’s known as zero-current switching (ZCS). If switching happens at low voltages, it’s called zero-voltage switching (ZVS). LLC converters possess the capability to perform both ZVS and ZCS due to their resonant nature.

Figure 7 illustrates the four fundamental operating modes of an LLC converter. Modes 1 and 3 represent the conventional LLC operation, which we’ve already discussed. In Mode 1, the current flows from the source to the resonant tank and the secondary of the transformer (Q1 is on, while Q2 is off). Conversely, in Mode 3, the remaining stored power in the resonant tank is transferred to the secondary of the transformer with the current flowing in the opposite direction compared to Mode 1 (Q1 is off, and Q2 is on). ZVS occurs during Modes 2 and 4, when both switches are turned off. During these periods, current passes through the body diode of the transistor (e.g., Q2 in Mode 2 or Q1 in Mode 4), which is also known as freewheeling.

Freewheeling results in a drop in the voltage across the transistor (VDS) until it approaches zero, limited by the minimal voltage drop of the body diode. Since this happens when both gate signals are low, by the time the circuit transitions from Mode 2 to Mode 3 or Mode 4 to Mode 1, the voltage across the transistor is close to zero, thereby minimizing switching losses.

Conclusion

In conclusion, comprehending how an LLC resonant tank functions is essential for designing an LLC converter. The tank’s resonant characteristics make the LLC converter highly sought after, as it can maintain efficient and stable operation across a wide range of loads and power levels. However, this resonance also necessitates meticulous attention to circuit parameter design, as the tank’s gain response is influenced by various factors, including the load and the converter’s operating point (refer to Equation (7)).

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