Controlling and Preventing Core Saturation in Inductors – Technical Articles

In the previous article, we saw that a strong magnetic field can cause saturation in magnetic materials. In a saturated material, all of the magnetic domains in the core line up with the external field. Beyond that point, there are no additional domains to align, leading to a significant drop in the material’s permeability. Though there are applications that make use of core saturation, it’s mostly something to be avoided.

When trying to prevent saturation in an inductor, the number of turns is a particularly important design parameter. However, deciding whether we need to increase or decrease the number of turns can be a bit tricky. We’ll find out more about this topic after we review the core response model we’ll be using.

The Piecewise Linear Model of Core Response

The B-H characteristic of a magnetic material is highly nonlinear. To more easily analyze magnetic systems, we commonly model this curve using a piecewise linear function. Note that this piecewise model accounts only for saturation, not hysteresis.

The orange curve in Figure 1 illustrates the piecewise linear approximation of a hypothetical ferromagnetic material’s B-H curve. For purposes of comparison, the purple line shows the B-H curve of an air-core inductor.

Figure 1. Piecewise linear model of a magnetic material’s B-H curve (orange) and an air-core inductor’s B-H curve (purple).

For flux densities below the point of saturation (B < B_sat), the magnetic material’s response is approximated by a constant relative permeability. This portion of the B-H curve therefore has a slope of μ₀μ_r, where μ_r is the relative permeability and μ₀ is the permeability of free space. Keep in mind that this is only an approximation—the actual response of a magnetic material isn’t a straight line.

For B > B_sat, the relative permeability approaches unity and the material behaves like a non-magnetic medium. Its B-H curve, like that of the air core inductor, is approximated by a straight line with a slope of μ₀. This is the saturation effect.

The figure shows that a ferromagnetic material can have a very high permeability, but only if it isn’t saturated. When the material saturates, its permeability reduces to that of free space. However, practical designs usually set the maximum flux density lower than the saturation flux density. Since we can expect μ_r ≫ 1 in these designs, we can further simplify the model by approximating the saturation region with a horizontal line (Figure 2).

Figure 2. A simplified model of the B-H curve where the flux is assumed to be constant in the saturation region.

From Faraday’s law, we know that the voltage induced in a winding is proportional to the rate at which the magnetic flux changes over time. In saturation, however, the magnetic flux is almost constant. As a result, no voltage is induced across an inductor when its core saturates. Instead, the inductor behaves almost like a short circuit.

Saturation Limits the Maximum Magnetic Field Force

To avoid saturation, we need to limit the magnetic flux density (B) to below B_sat. We know that B is given by:

$$B~=~frac{phi}{A_c}$$

Equation 1.

where:

Φ is the magnetic flux

A_c is the cross-sectional area of the core.

We can therefore limit B by either increasing the core’s cross-sectional area or reducing Φ.

The product of the current and the number of turns (nI, or the magnetic field force) must also be limited to avoid core saturation. Increasing the number of turns for a given current moves the component toward saturation, as does increasing the current of an inductor that has a given number of turns.

For a solenoid with n turns and a length of l_m, the magnetic field intensity is H = nI/l_m. By noting that Φ = BA_c and nI = Hl_m, we can rescale the B-H curve to obtain the Φ-versus-nI curve of the core. This is shown in Figure 3.

Figure 3. The Φ-versus-nI curve of a magnetic core.

To avoid saturation, we should have:

$$nI ~leq~ H_{sat} l_m quad ~Rightarrow~ quad nI ~leq~ frac{B_{sat} l_m}{mu_0 mu_r}$$

Equation 2.

We obtain the right-hand half of Equation 2 by taking a flux density equation ((B~=~ mu_{0} mu_{r} H)) from our discussion of complex permeability, and rewriting it to solve for the field intensity ((H~=~ frac{B}{mu_{0} mu_{r}})).

Core Saturation and Inductor Voltage

In the above discussion, we assumed that the current flowing through the inductor (I) is known. When that’s the case, we can easily use Equation 2 to determine if a magnetic field force value leads to saturation or not.

However, we’re sometimes given the voltage across the inductor rather than the current through it, and need to verify whether the core saturates or not using that information. Also, if the core saturates, we need to determine which parameters can be changed to avoid saturation.

We know that the voltage and current of an inductor are related by:

$$V ~=~ L frac{dI}{dt}$$

Equation 3.

where:

L is the inductance

t is the time.

Using this relationship, we can calculate the inductor’s current from the voltage across it. Once we know the current, we can use Equation 2 to see whether or not the core saturates. A more straightforward method, however, is to directly use Faraday’s law:

$$V ~=~ N frac{d Phi}{dt} ~=~ NA_c frac{dB}{dt}$$

Equation 4.

By integrating the above equation, we obtain the flux density as a function of the voltage:

$$B(t) ~=~ frac{1}{NA_c} int_{0}^{t} V(t) dt ~+~ B(0)$$

Equation 5.

The above equation explicitly shows the time dependence of B and V. The integration also introduces an initial B term in the form of B(0), which represents the magnetic flux through the inductor at the initial time (t = 0).

Equation 5 has another important implication, though it might seem counterintuitive. It shows that for a given voltage waveform applied to the inductor, increasing the number of turns moves the inductor away from saturation. This is in contrast to applying a given current to the inductor, as we did in Equation 2. For a known current flowing through the inductor, increasing the number of turns pushes the device toward saturation.

The apparent contradiction can be explained by re-examining Equation 4. This equation shows that if we increase the number of turns (N), a relatively smaller change in the magnetic flux (Φ) is required to produce a given voltage (V). In other words, if the voltage waveform applied to the inductor is fixed, we can increase N to reduce the flux through the inductor, which moves the core away from the saturation region.

To better understand this, let’s examine the special case of a sinusoidal input voltage.

Core Saturation For a Sinusoidal Voltage

Assume that the voltage applied to the inductor is:

$$V ~=~ V_m sin(omega t)$$

Equation 6.

where V_m is the magnitude of the voltage (the amplitude of the sine wave).

This also produces a sinusoidal flux through the inductor. Applying Equation 5, we obtain:

$$B(t) ~=~ frac{1}{NA_c} Big (- frac{V_m}{omega} cos(omega t) Big ) bigg vert _0 ^t ~+~ B(0)$$

Equation 7.

which simplifies to:

$$B(t) ~=~ frac{V_m}{NA_c omega} Big (1~-~cos(omega t ) Big ) ~+~ B(0)$$

Equation 8.

We now find the peak-to-peak value of B(t). Noting that the input voltage is a sine wave, the input voltage is positive over the interval ⍵t = 0 to ⍵t = π. B(t), which is related to the integral of the input voltage, therefore reaches its maximum at ⍵t = π. Finding the difference between B(⍵t = π) and B(⍵t = 0) gives us the peak-to-peak value of B(t):

$$B_{pp}~=~ frac{2V_m}{NA_c omega}$$

Equation 9.

To avoid saturation, the amplitude of B(t) should be less than B_sat. Since the value of the amplitude is equal to half of the peak-to-peak value (B_p = B_pp/2), this leads to:

$$frac{V_m}{NA_c omega} ~leq~ B_{sat}$$

Equation 10.

As you can see, for a sinusoidal inductor voltage with amplitude V_m, we can increase N to avoid core saturation. Equation 10 also shows that reducing the frequency of the sine wave (⍵) can push the core toward saturation. To understand this, note that the flux through the core is proportional to the integral of the input voltage (Equation 5).

Reducing the input frequency means that the input voltage has longer positive and negative half-cycles. Over these longer half-cycles, the flux has time to increase to a larger positive or negative value. There is therefore a minimum frequency that the core can support without being saturated. To clarify these concepts further, let’s work through a couple of quick example problems.

Example 1

The saturation flux density of a core material is 0.2 T at a given temperature. Using this material, we build an inductor with a core cross-sectional area of 10^-4 m² and number of turns N = 10. If the voltage across the inductor is a sine wave with amplitude V_m = 10 V, what is the minimum frequency of operation needed to avoid core saturation?

Plugging the given values into Equation 10, we have:

$$omega ~geq~ frac{V_m}{NA_c B_{sat}}~=~frac{10}{10 ~times~ 10^{-4} ~times~ 0.2}~=~ 50,000 quad text{rad/s}$$

Equation 11.

Converted from rad/s, the minimum frequency of operation to avoid core saturation is f = 7.96 kHz.

Example 2

An inductor is designed to support a sinusoidal voltage of amplitude V₁ and frequency ⍵₁. The peak flux density for this input is B₁. If we double the number of turns, what is the lowest frequency that keeps the maximum flux density below B₁?

From Equation 9, we know that the amplitude of the flux density (B_p) is:

$$B_{p}~=~ frac{V_m}{NA_c omega}$$

Equation 12.

Since V_m and A_c are assumed to be constant, doubling N allows us to halve the frequency without exceeding B₁, the original flux density. Therefore, the new inductor can go as low as ⍵₁/2.

Wrapping Up

As we’ve now seen, the number of turns and the cross-sectional area of a core both affect the core saturation for a given input. By paying attention to these parameters, we can avoid saturation in our design.

Adding an air gap to the core is another popular way to prevent saturation. The air gap increases the saturation current at the expense of reducing inductance. If we choose the correct gap length and number of turns, however, we can still achieve the required inductance while avoiding saturation. We’ll discuss this further in the next article.

All images used courtesy of Steve Arar