Core saturation is a major concern when designing magnetic components. Most applications seek to avoid it. As we discussed in the preceding article, it’s possible to limit a core’s magnetic flux density to below saturation level by reducing the inductor’s number of turns. However, this also lowers the inductance.
A different, more useful technique is to add an air gap to the core while increasing the number of turns by an appropriate factor. This method allows us to control both the inductance and saturation current parameters. Adding an air gap also increases the inductor’s energy storage capacity and makes it less susceptible to changes in the core’s magnetic properties.
We’ll discuss each of these advantages at length over the course of this article. Before we dive in, however, let’s answer a basic question: why do inductors and transformers use magnetic cores?
Air-Core vs. Magnetic-Core Inductors
An air-core inductor acts as an antenna. It emits unwanted electromagnetic radiation to the nearby circuits and receives electromagnetic signals from the environment that can interfere with the circuit operation.
A magnetic core’s high permeability, on the other hand, allows it to concentrate the field in a predefined region of space. This lets us increase the magnetic coupling between windings. In this way, using a magnetic core enables the creation of inductors with large inductances and transformers with high coupling.
Figure 1 shows three different magnetic core geometries. These can be divided into two types: closed-loop and open-loop.
Figure 1. Left and center: Closed-loop magnetic cores. Right: An open-loop magnetic core.
The magnetic circuits at the left and center of Figure 1 are closed, causing the magnetic flux to be mostly confined inside the core. When we examine the rod core on the far right, however, the field lines close their paths through the surrounding air. The core therefore forms an open magnetic circuit. In an inductor, a toroidal or other type of closed-loop core is usually chosen to maximize the magnetic field in the core and limit the flux leakage to the core’s outside.
To summarize, magnetic cores enable compact, high-value inductors and minimize electromagnetic interference, especially when closed-loop cores are used. Despite these advantages, however, magnetic materials have two main non-idealities:
To achieve a balance between the advantages and drawbacks of ferromagnetic cores, it’s common to add an air gap within the core loop.
What is a Gapped Core?
Figure 2 shows an inductor built using a core with an air gap.
Figure 2. An inductor wound around a core with an air gap.
Air is a linear material and doesn’t exhibit hysteresis. The air gap therefore improves the linearity and reduces the hysteresis effect. However, these improvements are achieved at the cost of lower overall inductance, as we’ll discuss shortly.
Counterintuitive though it might seem, a gapped core can also store a relatively greater amount of energy in the air gap. This energy storage capability can be very helpful in power supply design applications, where we need to output a large amount of power at the lowest material cost, size, and weight.
Analysis of the Gapped Core
Let’s analyze the gapped core in Figure 2 to see how the air gap influences different inductor parameters. Assume that:
- The core has relative permeability μc and mean length lc.
- The gap has a relative permeability of unity and a length of lg.
- The cross-sectional areas (A) of the core and the air gap are equal.
Figure 3 shows the equivalent magnetic circuit of this gapped core.
Figure 3. Equivalent magnetic circuit of a gapped core.
In the above model:
n is the number of turns in the inductor
i is the inductor current
ℛmc is the reluctance of the core
ℛmg is the reluctance of the air gap.
Reluctance quantifies how much the magnetic circuit resists the magnetic field flow and is measured in At/Wb. The core and air gap reluctances can be found, respectively, by Equations 1 and 2:
$$mathcal{R}_{mc} ~=~ frac{l_c}{ mu_c mu_0 A}$$
Equation 1.
$$mathcal{R}_{mg} ~=~ frac{l_g}{ mu_0 A}$$
Equation 2.
As mentioned above, the core and the air gap are assumed to have an equal cross-sectional area (A). This is a reasonable assumption when lg is small compared to the dimensions of the cross-section.
From the circuit model in Figure 3, we have:
$$ni ~=~ Phi (mathcal{R}_{mc}~+~mathcal{R}_{mg})$$
Equation 3.
This equation relates the flux through the core (Φ) to the applied magnetomotive force.
Effective Permeability of a Gapped Core
If the permeability of the core is much larger than unity (μc ≫ 1), a gapped core has an effective relative permeability of:
$$mu_{eff} ~approx~ frac{l_c}{l_g}$$
Equation 4.
where:
lc is the mean length of the core
lg is the gap length.
When lc = 100lg, for example, the gapped core has an effective relative permeability of 100. The important takeaway here is that the gap dominates the core behavior as long as μc ≫ 1.
The Air Gap Reduces Inductance
Since the gap lowers the effective relative permeability of the core, it’s no surprise that adding a gap also lowers the inductance of the structure. Another way to reach the same result is by applying the definition of inductance. We know that inductance is defined as:
$$L~=~frac{n Phi}{i}$$
Equation 5.
By combining Equations 3 and 5, we find the inductance of the gapped core to be:
$$L~=~frac{n^2}{mathcal{R}_{mc}~+~mathcal{R}_{mg}}$$
Equation 6.
The air gap increases the total reluctance and decreases the inductance. Despite this apparent decrease, gapped cores offer three important advantages:
- They reduce sensitivity to material permeability.
- They increase the saturation current.
- They increase the stored energy.
Let’s go over each of these advantages.
The Air Gap Reduces Sensitivity to Material Permeability
Without an air gap, the inductance is directly proportional to the core material permeability, which changes with temperature and is a nonlinear function of the applied magnetic field intensity. This makes it hard to precisely control the inductance.
Now consider a gapped core. Since the reluctance of the air gap is much larger than that of the core material, Equation 6 can be rewritten as:
$$L ~approx~ frac{n^2}{mathcal{R}_{mg}} ~=~ frac{n^2}{(frac{l_g}{mu_0 A})} ~=~ frac{n^2 mu_0 A}{l_g}$$
Equation 7.
From the above, we can see that a gapped core’s inductance depends mainly on the gap properties (A and lg). Since the permeability of air (μ0) is constant, it’s possible to adjust the gap length to build well-controlled inductances that are less sensitive to variations in permeability.
Figure 4 compares the B-H curve of the core material with that of the gapped core.
Figure 4. B-H comparison of the gapped core with the un-gapped core.
As we see in the figure above, introducing the air gap reduces the slope of the curve—or, equivalently, the inductance—but also produces a more linear response. Recall that the gap dominates the core behavior as long as the relative permeability of the core is much greater than unity (μc ≫ 1).
The Air Gap Increases the Saturation Current
Figure 4 clearly shows that the air gap increases the saturation field intensity (or, correspondingly, the saturation current). With no gap, the reluctance that the flux experiences is small. Therefore, a relatively small current can drive the core into saturation.
When a gap is introduced into the core, the effective reluctance increases. A larger current is thus required to saturate the core. Let’s calculate the maximum current that the inductor can handle without reaching saturation.
From Equation 3, the B value of a gapped inductor is given by:
$$B~=~frac{ni}{(mathcal{R}_{mc}~+~R_{mg})A_c}$$
Equation 8.
where Ac is the cross-sectional area of the core. The current at the onset of saturation, therefore, is:
$$I_{sat} ~=~ frac{B_{sat}A_c}{n}(mathcal{R}_{mc}+mathcal{R}_{mg})$$
Equation 9.
where Bsat is the saturation flux density. The air gap increases the effective reluctance, and hence the saturation current, of the core.
The Air Gap Increases the Stored Energy
We know that magnetic fields store energy. The energy per unit volume stored in a magnetic field (wm) is the integral of the field intensity (H) over the range of the flux density variation:
$$w_m ~=~ int_{B_1}^{B_2} H ; dB$$
Equation 10.
This is the same equation we derived when analyzing the hysteresis loss earlier in this article series.
Figure 4 showed that introducing the air gap reduces the slope of the B-H curve. This enlarges the area to the left of the B-H curve, indicating that the inductor can store a larger amount of energy.
Figure 5 compares the energy that a gapped core can store with that of an ungapped core. The green hatched area (A1) corresponds to the power density of the ungapped core. The blue hatched area (A2) shows the power density of the gapped core.
Figure 5. The green and blue shaded areas show the power density of an ungapped core and a gapped core, respectively.
A quick visual comparison of A1 with A2 makes it clear that the gapped core can store more energy than the ungapped core. If we increase the length of the gap, the slope of the B-H curve reduces further, leading to an even greater energy storage capacity. Most of the energy in a gapped inductor is actually stored in the air gap.
Choosing the Gap Length and Number of Turns
We saw that the air gap increases the saturation current but lowers the inductance. To compensate for the loss of inductance due to the air gap, we can increase the coil’s number of turns (n). This increases the magnetic field generated by the coil, restoring the inductance to the desired value.
Assuming that the gap reluctance is much greater than that of the core, Equations 6 and 8 simplify to:
$$L ~approx~ frac{n^2}{mathcal{R}_{mg}}$$
Equation 11.
and:
$$B ~approx~ frac{ni}{mathcal{R}_{mg}A_c}$$
Equation 12.
Increasing the value of n causes both the inductance (L) and the magnetic flux density (B) to increase as well. However, L is proportional to n2 and B is proportional to n. The inductance therefore grows faster than the flux density when n is increased.
If we increase ℛmg as well as n, it’s possible to reduce the flux density without changing the inductance. For example, say that k is an arbitrary value greater than unity. If we increase ℛmg by a factor of k, and n by a factor of (sqrt{k}), then L remains the same but B reduces by a factor of (sqrt{k}).
When to Use Transformers With Gapped Cores
In many applications, we use transformers to transfer an AC signal from the source to the load. When that’s the case, we usually use ungapped cores. Adding an air gap reduces the achievable inductance for any given form factor and leads to a less ideal transformer.
Other transformers, such as ignition coils and flyback transformers, are used to store energy and then transfer it to a secondary winding. These applications may use gapped cores because they store a larger amount of energy and significantly reduce hysteresis loss.
For example, an ignition coil used in a typical gasoline-powered car has a primary coil of about 250 turns and a secondary coil with some 25,000 turns. The primary is connected to the car’s battery and stores energy by producing a strong magnetic field. To fire a spark plug, the primary coil current is cut. This makes the magnetic field collapse, inducing a high-voltage electromotive force in the secondary coil.
In this way, the magnetic field energy becomes a strong current pulse in the secondary coil, which sparks the plug and ignites the fuel-air mixture in the engine’s cylinders. Using a gapped core in the ignition coil helps ensure that sufficient energy reaches the secondary winding.
Wrapping Up
In this article, we learned that a gapped core can handle larger currents without saturation. Introducing the air gap also makes the inductor more stable against changes in the core’s magnetic properties and enhances its energy storage capability.
All of these advantages are achieved at the expense of a smaller inductance. However, by appropriately choosing the gap length and the number of turns, we can restore the desired inductance while avoiding core saturation. This technique is commonly used to design inductors for power electronics applications.
This article concludes our discussion of magnetic cores for the moment. The articles in this series are listed below in order of publication:
- Key Concepts of Magnetic Materials
- Diamagnetic, Paramagnetic, and Ferromagnetic Materials Explained
- Using Complex Permeability to Characterize Magnetic Core Losses
- Understanding the Effect of Eddy Currents on the High-Frequency Behavior of Magnetic Cores
- Understanding Dimensional Resonance in High-Frequency Magnetic Cores
- Understanding Magnetic Field Energy and Hysteresis Loss in Magnetic Cores
- Hysteresis Loss: Estimation, Modeling, and the Steinmetz Equation
- Understanding How Laminated Cores Reduce Eddy Current Loss
- Using Permeability to Understand Magnetic Core Saturation
- Controlling and Preventing Core Saturation in Inductors
- Understanding Inductors with Gapped Cores
All images used courtesy of Steve Arar
