How to explain voltage dependency effects using LTspice simulation?

The miniaturization of ceramic capacitors requires achieving higher capacitance values ​​in smaller and smaller spaces. To this end, materials with high dielectric constants (ε) and increasingly thin dielectric insulation layers are being realized, making it now possible to produce high-quality ceramic layers on an industrial scale.

Unfortunately, the dielectric constant ε _r = ƒ() is a function of the electric field strength, so the capacitance exhibits voltage dependence. Depending on the ceramic type and the layer thickness, this effect can be very significant. It is not uncommon for the capacitance to drop to less than 10% of the nominal value at the maximum permissible voltage.

In applications where a constant voltage is applied to an MLCC, such as a decoupling capacitor, it is easy to account for this effect. As long as the voltage remains constant, the remaining capacitance value can be obtained from the manufacturer's data sheet or online tools.

But what about cases where the voltage can vary? For example, in Figure 4, the input filter on the switching regulator is powered from a 5 V USB supply to a 24 V industrial supply. Another example is a 2-wire Ethernet PHY that is AC-coupled to a power supply of different voltage values ​​on the same line.

In such cases, circuit simulation using LTspice can provide useful insights. Some MLCC manufacturers have made the corresponding DC bias models available for download. In addition, LTspice provides methods and implementation tools to simulate voltage-dependent behavior. For this, the capacitance vs. voltage curve and one of the methods described in Figure 3 are useful.

LTspice provides a well-known capacitor model with constant capacitance as well as a nonlinear model. The nonlinear model is used to solve the charge equation. Solving the nonlinear capacitor model directly is not appropriate because the charge needs to be preserved. But this should not be a problem here because the capacitance is obtained by differentiating the charge with respect to the voltage. In turn, the voltage-dependent capacitance must be integrated. This has been done for the methods below, so no mathematical knowledge is required to use these models.

The first-order method uses a linear voltage dependence:

From this we can get by integration:

That’s the charge equation. Now we can plug it directly into LTspice terminology, substituting in the capacitance value of the capacitor:

Q=x*{c0V}-0.5*x**2*({c0V}-{cVmax})/{Vmax} .

However, the nearly constant initial capacitance of many MLCCs decreases rapidly even at moderate voltages and then remains nearly constant. In such cases, if only a linear model is used, the effective capacitance will be overestimated for a large range of capacitances. For such a wide distribution, a model based on the hyperbolic tangent (Tanh) can be used:

The parameters can be easily estimated without further aids.

Figure 1. Tanh approximation function and related parameters

The capacitance value can be replaced by the charge equation

Q=x*({C0+Csat})/2+({Csat-C0})/4*{Vtra}*ln(cosh((x-{Vth})*2/{Vtra})) .

Figure 2. 10 µF MLCC

To check the capacitor model in LTspice, apply

This creates a constant voltage ramp. The amount of current through the capacitor then corresponds exactly to the value of the capacitor because:

Figure 3 clearly shows the superiority of the proposed nonlinear model over the standard constant capacitance model. With this capacitance curve, the linear model is adequate for most applications.

Figure 3. Example of simulating a 10μF 6.3V 0805 MLCC with different capacitor models in LTspice.

Finally, it should be noted that only a single non-ideal effect was simulated here. Many other effects still exist with MLCCs, including aging, temperature dependence, frequency dependence, AC amplitude dependence, dielectric absorption, etc. For many applications, it is sufficient to consider the DC bias dependence as the only major effect. LTspice can be used as a practical tool to account for non-ideal characteristics such as DC bias before building the first prototype.

Figure 4. Interference current suppression of the input filter of the LT8619 buck regulator simulated from the converter side using the Tanh model for different supply voltages.