**EMC Resonance**

Part I: Non-Ideal Passive Components

his article is part of a two-article series devoted to the concept of resonance in EMC. In Part I the fundamental circuit background is presented and illustrated by the resonance phenomenon in the non-ideal models of passive circuit components: capacitors, ferrite beads, resistors, and inductors. Part II (to appear in the next issue) describes the resonance in the decoupling capacitor circuits.

*2*-order circuits, series and parallel

^{nd}*RLC*configurations. These circuits, shown in Figure 1, contain a single lumped capacitor and a single lumped inductor connected either “purely” in series or “purely” in parallel.

Actual circuits differ from these classical configurations; in addition to the intentional discrete reactive components, they contain distributed parasitic inductances and capacitances. Nevertheless, the study of these basic *RLC* configurations provides an insight into the more complex topologies and their behavior. Let’s begin with a series *RLC* circuit.

*RLC*resonant circuit shown in Figure 2.

Since the study of resonance is performed in the sinusoidal steady-state, the voltage and current, in Figure 2, are shown in the phasor forms, and the component values are replaced by their impedances [1].

In order to introduce the concept of resonance, let’s calculate the input impedance to the circuit.

_{0}, the input impedance is purely real, it follows that

_{0}, the voltage and current are in phase! We have arrived at the definition of the resonant frequency:

*The resonant frequency, ω _{r} , is a frequency at which the voltage and current phasors are in phase (with respect to the same two terminals of the circuit).*

*RLC*circuit, the resonant frequency is the same as the undamped natural frequency.

*Note:*1) Not every

*RLC*circuit is resonant. 2) When the circuit is resonant, its resonant frequency, in general, is different from ω

_{0}. 3) The “classical” series and parallel circuit configurations are always resonant, and their resonant frequency is the same as ω

_{0}.

At resonant frequency, the magnitude of the input impedance is minimum. Let’s illustrate this using a circuit model of a nonideal capacitor [2] and plotting its input impedance. This is shown in Figure 3.

*RLC*resonant circuit shown in Figure 4.

Let’s calculate the input impedance to this circuit.

*RLC*resonant circuit, shown in Figure 6.

This circuit corresponds to a non-ideal model of a resistor [2]. Let’s calculate the input impedance to the circuit.

Note: the simulation model in Figure 7 is used only to show that the hybrid series *RLC* circuit *with the component values shown* is indeed resonant. This simple model is not valid beyond 2 GHz frequency, and thus, beyond 2 GHz, the simulated impedance plot and the value of the resonant frequency do not reflect the results that would be obtained from the laboratory measurements.

*RLC*resonant circuit, shown in Figure 8.

This circuit corresponds to a non-ideal model of an inductor [2]. Let’s calculate the input admittance to the circuit.

- Bogdan Adamczyk,
*Foundations of Electromagnetic Compatibility with Practical Applications*, Wiley, 2017. - Adamczyk, B., Teune, J., “Impedance of the Four Passive Circuit Components: R, L, C, and a PCB Trace,”
*In Compliance Magazine*, January 2019. - https://www.analog.com/en/analog-dialogue/articles/ferrite-beads-demystified.html