The value of the optimum snubber resistor is mathematically derived using the analysis in the frequency domain, leading to a simple—yet rigorous—equation.
« Previously: Frequency-domain analysis for optimal snubber design
For now, assume that the value of the snubber capacitor Csnb is already known. The method to identify this value will be described in the next paragraph.
By grouping the components shown previously and describing them with the Laplace transform, the circuit can be simplified as shown in Figure 1a, where:
Finally, Figure 1b shows the last simplification obtained by calculating the parallel of Z2 and Zsnb:
Figure 1: Model of the electrical circuit including the parasitic components and the snubber circuit in the Laplace domain.
The gain of the circuit, defined as the ratio between the voltage on VX node and the input voltage, can then be calculated for both of the cases, with and without the snubber network.
Plotting the |Gsnb(jω)| for two extreme values of Rsnb (Figure 2) provides a clear picture of the impact that the value of this component has on the system response.
Figure 2: |Gsnb(jω)| for extreme values of Rsnb.
For Rsnb = infinite (red curve), i.e. no snubber, the gain response peaks at the resonant frequency between L1 and C1 (f01), which is about 183MHz considering the standard values mentioned earlier.
On the contrary, replacing Rsnb with a 0Ω resistor, i.e. making the snubber as strong as possible, pushes the resonant frequency all the way down to f02 ≈ 92MHz since the snubber capacitor in parallel to C1 increases the overall capacitance.
Evaluating Eq(18) for different values of Rsnb (Figure 3) shows what happens for intermediate values of the snubber resistor. Two are the key concepts. The first is that, while increasing Rsnb, the peak of the black curve slowly decreases until, eventually, it turns into growth of the red peak. The optimal Rsnb is at the edge of this transition, which provides the lowest overall gain. The second key point is that there is one specific frequency (f03) where all the curves intersect. At f03 the gain is constant regardless of Rsnb and corresponds to the lowest overall gain when the optimal Rsnb is adopted.
Figure 3: |Gsnb(s)| for different values of Rsnb.
The information collected so far tells us that there is an optimal snubber resistor that minimises the overall amplitude of the gain response, but its value is not yet clear. To get to it there are a few other steps to take.
Since the mathematical analysis is quite complex, let’s try to simplify the circuit where possible. The impedance Z1 (Eq. (13)) is dominated by R1 at low frequency, which becomes negligible, compared to the inductor impedance, at high frequency. This means that, for angular frequencies much higher than the frequency of the zero determined by R1 and L1, the impact of R1 is negligible.
As explained in Figure 2, even with the strongest snubber (Rsnb = 0Ω), the minimum resonant frequency will be f02 (Eq. (20)). By replacing ω02 in the previous equation, R1 can be ignored for
This is the case in most applications requiring a snubber circuit. For example, considering Csnb = 3*C1 = 1.5nF, R1 can be neglected.
From the approximation, Gsnb(s) from Eq.(18) becomes:
We also know that |Gsnb(jω)| at ω03 is constant regardless of Rsnb. Which means:
Replacing Eq.(28) into Eq.(26) gives:
The optimum value of Rsnb is the one associated with the |Gsnb(jω)|curve whose maximum happens at ω03.
With L1 = 1.5nH, C1 = 0.5nF and Csnb = 0.5nF the optimum snubber resistor is:
Rsnb_opt = 3Ω
The simulation results reported in Figure 4 (conditions in the caption) confirm that using Rsnb_opt = 3Ω provides the lowest overall ringing.
Figure 4: Simulations confirm that Rsnbopt provides the lowest Vx ringing (R1 = 10mΩ, L1 = 1.5nH, C1 = 0.5nF, Csnb = 0.5nF).
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