Chapter 2

Fundamental Principles

2.1. Sources of noise: the switching cell and its control

The static conversion of electrical energy (switch-mode power supply, inverter, rectifier, etc.) is based on the principle of the switching cell: it is the connection of two switches, which enables the management of energy between an input source of voltage and an output source of current (Figure 2.1). The main switch is controlled by a periodic modulation function fm(t), with a binary value and a variable cyclic ratio (α = ton/Td), which regulates the transfer of power according to the value of this cyclic ratio α. The notion of source must be understood in the sense that it is capable of imposing a near-constant quantity (of voltage or current) at the time-scale of the switching period. This characteristic is generally due to the presence of reactive components (input capacitor Ce or output smoothing inductance).

Thus, it is noteworthy to see that the external parameters of the switching cell (E and Io) are constants, whereas the internal parameters (ie and vk) are variables, regulated by the function fm(t).

Figure 2.1. Switching cell and its corresponding wave shapes

It is possible to express the variable voltages and currents of the switching cell as a function of those that are constant:

[2.1]

[2.2]

From these relations, we deduce the law of converted power control:

[2.3]

Thus, by this principle, all static conversion functions can be undertaken. We sometimes add a transformer to the structure when a galvanic insulation is required.

2.1.1. Origin of conducted and radiated perturbations in static converters

The electrical quantities are very much variable in the switching cell. Indeed, in order to reduce losses during commutation (simultaneous presence of the voltage and the current in the switches), it is essential that the commutations be very quick. Currently, the size of switching gradients is on the scale of 100–1,000 A/μs for dI/dt and of 5–50 kV/μs for dV/dt.

To illustrate, Figure 2.2 outlines these phenomena in a chopper connected to a line impedance stabilization network (LISN): in the mesh surrounding the hatched area, the current Ie undergoes very quick high-frequency variations. The resulting loop connects to a magnetic radiative dipole: the input decoupling capacitor Ce, limited in its operation due to its imperfections (resistance and inductance in series lp), is generally not sufficient to prevent the propagation of an impulsive parasitic current Ip onto the network.

Figure 2.2. Origin and coupling mode of electromagnetic perturbations of a static converter

Moreover, the conductors shown in grey bold lines in Figure 2.2 endure the strong variations of the voltage VK. They constitute an electric radiative dipole and can transmit impulsive currents Imc to the earth via the parasitic capacitance denoted by Cp between the device and the earth.

2.2. Modeling

2.2.1. Simple model of the switching cell

We can now provide a model of the switching cell, representing the effects of perturbation [COS 93]. The input current of the cell is presented as a current generator creating the parasitic differential-mode current by means of coupling through a common impedance (input capacitor). The voltage of the switch is presented as a voltage generator generating the common-mode current via capacitive couplings. The switching cell can then be represented by one of the two models in Figure 2.3, where the sources of current Ie and voltage Vmc = VK appear.

Figure 2.3. Models of the switching cell for the depiction of conducted perturbations

These models can be connected with that of the electric environment of the converter (network, charge, control devices and connections to the earth) to determine and calculate the common- and differential-mode currents at the input while conforming to the standards as required.

The first model shows why it is important to see that the parasitic currents due to the voltage VK can flow along the two power supply lines, but it also includes the common-mode coupling capacitances. The source of voltage VK is in a floating reference. The second model is linked to the common reference voltage, which simplifies the analysis, and is more representative of the switching cell. These equivalent generators are, therefore, considered as equivalent sources of electromagnetic (EM) perturbations in the frequency domain. We complete the description of the model with the impedances linking the different potentials of the cells to the potential reference (earth, frame, etc.). These intangible impedances represent the effects of parasitic couplings between the converter and its electrical environment. All of the imperfections of the active and passive components can be included in this model (Figure 2.4). This representation, therefore, suggests that the effects are local, which is an acceptable hypothesis as long as the geometric dimensions of the devices remain small relative to the wavelengths of the quickest signals, which is generally the case.

Figure 2.4. Electromagnetic compatibility (EMC) models of the switching cell through equivalent sources and local couplings

By completing this model with the model incorporating the LISN, it is easier to carry out the calculation of the current Ip applied to the measuring impedance of the LISN, and representative of the contribution of each source to the disruptive signal. We can, therefore, express the current Ip that circulates through each impedance of the LISN as:

[2.4]

The functions C1(f) and C2(f) are representative of the source couplings Ie and VK in differential mode and common mode in the frequency domain. In fact, they are analogous to transfer functions that help to connect the internal perturbation sources to the quantities measured by the LISN. These functions can be calculated analytically by localizing all of the impedances of the system. They can also be obtained experimentally by directly measuring with the impedance analyzer. We can, therefore, see that the sources present in the converter contribute to the conducted perturbations. Nonetheless, certain coupling functions are dominant relative to each propagation mode; in particular, the function C1 is dominant in differential mode as is the function C2 in common mode. Two points, therefore, arise as determiners:

– the excitation sources Ie, VK, characterized in the frequency domain;

– their coupling functions C1(f) and C2(f).

A very simplified example of execution is illustrated in Figure 2.4. For this scenario, we have presented the coupling functions C1(f) and C2(f) (Figure 2.5). We can note the minimum of C1 due to the self-resonance of the capacitor Ce.

Figure 2.5. Evolution of the frequency of the coupling functions C1(f) and C2(f)

We can deduce the spectrum of conducted perturbations in the LISN through the application of expression [2.4]. Figure 2.6 shows a comparison between the calculated and measured spectra by using this modeling principle. The support is a chopper switching at 15 kHz, supplied at 50 V and providing an output current of 2 A. We will reintroduce it later on.

Figure 2.6. Comparison between a simulation based on the frequency model and measurements for a scenario including a chopper

This approach can also be applied to the radiated mode: the coupling functions are much more complex as they integrate the (generally three dimensional (3D)) geometric properties of the source circuits and the coordinates of the observation point of the radiation. Nevertheless, we can express the radiated fields as alternative equivalent forms, which means that the coupling functions are dependent on the nature of the observed field:

[2.5]

[2.6]

In every scenario, it is the electric parameters of the switching cell that must be considered as the noise sources of the converter.

2.2.2. More complex model of the switching cell

The model of the sources defined earlier do not take into account the presence of very high-frequency rays in the spectrum of parasite signals. Their origin resides in multiple parasitic components in the conversion structure (Figure 2.7), namely:

– the parasitic capacitances of the open-state switches;

– the parasitic inductances of the cabling or inherent components (bonding of active components, capacitor Ce, etc.);

– the capacitive and inductive couplings with the ground of the switching cell and its load.

Their effect is shown by the appearance of high frequency (HF) resonances in the spectrum of parasitic signals, which become greater the quicker the switch is controlled, as the switching losses are barely dampened in idle states. These systems can attain several tens of megahertz, and even more so for low-voltage high-current converters.

Figure 2.7. Model of the switching cell along with...