# Non-linear AC circuits

Non-linear AC circuits are AC circuits containing non-linear components. Nonlinear components can be resistive, capacitive, and inductive.

Non-linear resistors can be inertial, and non-inertial. Inertial non-linear resistors are characterised with the resistance that changes with temperature. Non-inertial, non-linear resistor resistance does not depend on thermal conditions.

Non-linear inductances are characterised with inductance that is dependent on the current through the inductance. They can be made with different shapes of cores. The core material of the inductor plays an important role, and it is usually made with ferromagnetic material, that creates magnetic flux through the inductance coil winding.

These type of inductances can be controlled and uncontrolled. There is always some losses at the core of the inductance – and the losses can be significant and negligible. To make the losses negligible, the core material and its geometrical characteristics should be varied.

The AC current through the inductor induces the alternating magnetic flux, and that induces the eddy currents. Losses in the inductor due to eddy currents are proportional to the square of EMF and are reverse proportional to the inductance resistance. The losses are also proportional to the geometrical characteristics of the inductor core.

Some inductance cores are manufactured with ferromagnetic material, which makes a big difference on the device’s characteristics. Ferromagnetic materials are characterised by the magnetic hysteresis. Magnetic hysteresis is the gap between the change of magnetic induction B and the change of magnetic field strength H.

The square of the hysteresis loop (BH) is characterised by the energy emitted by the inductor core. The losses are proportional to the core volume, and the current frequency. If the magnetic field of the inductor changes quickly, the hysteresis loops are dynamic. It depends on the ferromagnetic material characteristics. Non-linear inductance can be replaced with the circuit depicted in Figure 1.

Figure 1. Replacing a circuit for non-ideal inductance (here ${R}_{h+e}$ is the losses due to the hysteresis and eddy currents, R_w is the losses due to the inductor winding).

As mentioned above the losses in the inductor depend on the core material. In some cases the losses due to the eddy currents and hysteresis can be negligible.

Non-linear capacitors are the capacitors where the dielectric permittivity of the material between the capacitor plates is the function of electric field strength. In this case the capacitor charge is the non-linear function of potential difference of the capacitor plates.

Normally capacitors that are used in the electronics are linear, but linearity is still dependent on the material between the capacitor plates. Non-linear capacitors contain the ferroelectric material between the plates. The non-linear capacitor is also characterised with the hysteresis (the gap between D and E), and it also provokes losses in the component. Similarly to the non-linear inductor, a non-linear capacitor can also be represented with the equivalent scheme (Figure 2), that takes into account the losses of the non-linear capacitor.

Figure 2. The equivalent scheme of non-ideal capacitor.

Non-linear devices, for example, two- or four-terminals can be very useful for making signals transformations, like current transformation, multiplication, regulation and another functions.

These are AC circuits that contain a few non-linear elements. AC circuits can contain resistive, capacitive or inductive elements. The controlled non-linear elements usually contain controlled terminals.

Non-controlled non-linear resistive elements are diodes, thyristors and others. Resistors are the inertia components. The resistors, whose resistance does not depend on the thermal processes, are non-inertia.

Non-linear inductive components are inductors whose cores are looped and consist of ferromagnetic material. Their inductive resistance depends on the AC value. These inductors can be controlled and uncontrolled. Cores of non-linear inductors can usually be packet and spiral. The core plates are usually covered with isolating material. When the work frequency is high, the inductor core is made with soft ferrite.

Let’s imagine that there is an AC current through the inductor core. The alternate magnetic field is generated in the inductor, and that provokes the eddy currents creation. The power losses are proportional to the square of EMF and reverse proportional to the resistance of the inductor. So the eddy current losses are proportional to the induction, are square of the frequency and square of the core width.

As we know ferromagnetic materials are characterised with the hysteresis phenomena. Hysteresis losses depend on the core volume, frequency and hysteresis squared. The hysteresis can be dynamic and static. Dynamic hysteresis is changing much faster than the static. It also depends on the geometrical characteristics of the core and its physical characteristics.

Non-linear inductance can be replace with the circuit, with the ideal inductance and parallel resistance, that describes the hysteresis losses and eddy current losses.

Non-linear capacitor components are those with a non-linear q(u) formula.

If we will turn on the AC current through the non-linear component, the output current will also be non-linear.

Let’s consider the four-terminals that contain a few non-linear components. This will be called non-linear four-terminals. These structures have the following practical significance:

• To transform the AC current to DC current;
• To transform non-linear current to linear current;
• To make the frequency multiplication;
• To make the frequency derivation;
• To stabilise the current and voltage;
• To make a modulation and many others.
Tags: