Electromagnetic noise in electric circuits: Ringing and resonance phenomena in the common mode

In modern life, we are surrounded by electromagnetic noise, the origin of which is not precisely known. This situation was caused by the fact that we did not have a theory to treat noise quantitatively. Very recently, however, a new multiconductor transmission line (MTL) theory was derived from Maxwell’s equations without approximation, where the radiation process was naturally incorporated into circuit theory.^1,2 The MTL theory is able to clarify the origin of noise theoretically and to provide solutions to reduce it by decoupling the normal mode from the common mode by arranging an electric circuit symmetrically.³ 

It is now important to know the noise in ordinary electrical circuits quantitatively using the new MTL theory. With this knowledge we are able to avoid noise in circuits and circumstance. There have been several attempts to understand noise through experiments and parametric modeling.^4–6 Specifically, however, we do not understand the origin of noise due to the common mode, which uses conducting materials around the circuit. The new MTL theory was able to include the surrounding conducting materials to handle the common mode noise explicitly. It is particularly important to understand the common mode noise in the field of accelerator, where noise is fatal for acceleration of particles up to the speed of light.⁷ In the discussion of Electro-Magnetic Compatibility (EMC), electromagnetic noise exists in the common mode, which is treated qualitatively by connecting a circuit through stray conductors with surrounding materials. Here, the surrounding materials are merely used as to return the common-mode current to the circuit. Instead, we must consider these materials explicitly as transmission objects, which interact electromagnetically with the circuit. Hence, in the discussion of noise, it is essential to treat the electric circuit and the surrounding conducting materials using the MTL theory.^3,8 In the present paper, we examine a three-conductor transmission-line circuit using the newly developed MTL theory. We observed an interesting phenomena whereby a standard circuit with grounding exhibited strong ringing patterns and resonance phenomena. Based on these findings, the proposed theory explains a priori recent experimental data obtained using various PCB arrangements.^4–6 Thus, we herein first introduce a theoretical framework to deal with MTL systems and define the normal and common modes for a three-line circuit. We then calculate the relation between normal and common modes and discuss couplings of these modes for various configurations. We compare our findings with recent experimental data obtained with PCB and demonstrate that experimental data can be explained by the newly developed MTL theory.

In order to understand the origin of noise, it is best to use a simple circuit that is grounded to earth or to metallic materials. To this end, we consider a three-conductor transmission-line circuit, as shown in Fig. 1. Here, the two lines, i = 1,  2, are the main lines (resistance is neglected), and are connected to a power supply PW and an electric load R_L. We add one more line, denoted as line 3, which represents a conductor in the environment. We cannot drop the third line, because conductors, such as metals, always exists around a circuit. Usually, we do not consider the earth or metallic materials in the environment as conducting objects, the voltages and currents of which change with position and time. However, conducting objects, through which electric signals propagate by interacting electromagnetically with the electric circuit, should be treated explicitly when discussing noise. The shape and material of conductors in the environment are arbitrary, but for simplicity, the conductors are considered to be wire-shaped. Line 3 can be longer than the circuit, but the characteristic impedance becomes much larger for lengths longer than the main lines, and almost all electric signals are reflected at the electric load.² Electromagnetic waves propagate in an approximately transverse electromagnetic mode (TEM) through conducting materials if at least two transmission lines are used. If line 3 is actually not a line but a plane, then only the region of the plane close to the two main lines contribute to the coefficients of inductance.^8,9

The two-conductor transmission-line circuit is connected to the third line through the ground. Potentials U_i(z, t) and currents I_i(z, t) are used for lines i = 1, 2, and 3, where these quantities depend on the direction denoted by the z-axis and the time t, as shown in Fig. 1 through.¹ We identify the normal mode as the differential mode of the two lines, the voltage and current of which are:

In the case without line 3, the normal-mode current is simply I_n = I₁. As for the common mode, we first treat the two lines as a whole and introduce the average potential $U 12 = 1 2 ( U 1 + U 2 )$ and the summed current I₁₂ = I₁ + I₂. We then define the common mode using two lines as a whole along with line 3 as the normal model for two lines:¹ 

With this definition, the common mode can be assigned to the noise mode, which uses line 3 for the propagation of the electric signal outside of the main lines. If the total current is zero, the common-mode current is the summed current, I_c = I₁ + I₂, which corresponds to the usual notion of the common mode in a two-line circuit. Using these quantities, we can calculate any quantities as the voltage of the third line measured from the second line: $U 3 ( z , t ) − U 2 ( z , t ) = 1 2 V n ( z , t ) − V c ( z , t )$⁠.

We write the multiconductor transmission-line (MTL) equations for the case of three lines in a general form as follows:^1,8

Note that, although the second equation (6) is usually written in terms of the coefficients of capacitance, the form of (6) makes rewriting the MTL equations in terms of the normal and common modes much easier.¹ In the case of thin wires of length l, we are able to calculate the generalized coefficients of inductance $L ij$ and of potential $P ij$ using Neumann’s formula:^1,9

where a_ij is the radius of wire i for the case in which i = j and is the distance between lines i and j for the case in which i ≠ j. In the case of plane, we have a much more complicated expression.⁸ The magnetic permeability μ and the permittivity ε in a homogeneous medium are related to the speed of light c in a vacuum as follows: $1/ ε μ =c$⁠. Thus, we are able to write coupled differential equations in terms of the normal and common modes with the coefficients of inductance for the normal L_n and common L_c modes and their coupling coefficient L_nc:¹ 

These coefficients appear in the modified MTL equation in terms of the normal and common modes:

and a similar expression for the common mode.

We can obtain the coefficients of potential for all of the normal and common modes and their coupling term. These coefficients are proportional to the coefficients of inductance, such as $P n = 1 μ ε L n$⁠.¹ The coefficients of capacitance are obtained by taking the inverse of the matrix of the coefficients of potential. Furthermore, we can obtain the coefficients of inductance and capacitance for the case in which the voltages are defined as V_i = U_i − U₃ and the currents are the currents of lines i.⁸ 

We simulated an electric circuit with grounding using a short pulse of 5 ns and a 100 V power supply V_PW. We inserted the resistance for the load R_L of the main lines in order to satisfy the impedance matching for the circuit, $R L = Z n = L n P n$⁠, so that the normal mode does not have a reflected wave at the load. We considered small and equal resistances near the power supply, r₁ = r₂ = 10 Ω. As the standard case, we considered a wire having a radius of 1 mm and wire separations of a₁₂ = 10 mm and a₂₃ = 10 mm. We varied D, which is the distance of line 3 from the center of the main lines, in order to investigate the coupling of the normal and common modes. The length of the wires were l = 1 m.

We calculated the voltages at the location of the load R_L and extracted the normal- and common-mode voltages V_n, V_c as functions of time, as shown in Fig. 2. The normal-mode voltage V_n has a slight voltage drop but a large input signal at T_l = l/c = 3.3 ns, which is the time required for the input signal to arrive at R_L. The large signal is followed by small square waves that oscillate for a long time, as is clear in the inset of Fig. 2. Without line 3, the small oscillating waves do not appear due to the impedance matching in the normal mode. In actuality, the existence of metallic materials, such as metallic cases or supports, plays the role of line 3, should generate the noise discussed herein. The common-mode voltage V_c exhibits a significant long-duration voltage oscillation with a cycle time of T_2l = 6.6 ns. This oscillatory behavior is caused by reflections at the load and source points. We are able to understand this behavior as a ringing phenomenon in the common mode.⁸ The ringing pattern observed in the normal mode is caused by coupling of the normal mode and the common mode at the reflection points. The voltage between lines 2 and 3 is $U 3 − U 2 = 1 2 V n − V c$⁠, which is a significantly large value. This means that, as a conductor, line 3 carries a significant amount of electric signals.

In order to study the effect of line 3, the separation distance D of line 3 from the center of lines 1 and 2 was then varied, where the magnitude of D is $a 23 + 1 2 a 12$⁠. For the sake of clarity, line 3 is shifted by 5  mm in the direction perpendicular to the plane formed by lines 1 and 2. The calculated results are shown in Fig. 3. The initial negative value of V_n at approximately t = 10 ns shown in Fig. 2 gradually decreases with D. As D approaches the center of lines 1 and 2, V_n decreases with D and becomes exactly zero when the third line intersects the center of lines 1 and 2. This is the case for the S3L arrangement, in which the two main lines are placed equidistant from the ground line,¹ which is shown as L_nc = 0 in Eq. (11). The normal mode completely decouples from the common mode for this symmetric configuration.³ The behavior of the common mode is shown as a function of D in Fig. 3(b). A similar oscillatory behavior is observed in the common mode. This large voltage is a result of the initial common-mode voltage V_c(z = 0, t = 0) being large and finite, which is achieved by running the grounding from line 2 to the ground. If we want to make the common-mode voltage small in order to reduce the noise, we should split the power supply into two separate power supplies and introduce the grounding from the midpoint of the twin-power supplies to line 3.^1,3 In this sense, the standard circuit maximizes electromagnetic noise by taking the grounding from one of the two main lines.

The long-duration ringing patterns indicate electric signals that are reflected numerous times at both ends in the normal and common modes. Therefore, we take the Fourier transforms of the voltages using a continuous square wave rather than a short pulse, where the square wave has a period of 10 ns and a duty rate of 0.5. The frequency spectra for both the normal and common modes are shown in Fig. 4(a). Several spikes are observed in the frequency spectra. The sharp peaks are caused by the Fourier transform of the square wave, whereas the peaks having wide bandwidths are caused by the resonance. The frequencies of the resonances are determined by the geometry of the circuit. The frequencies of the resonances correspond to the standing wave condition: $f n = ( 2 n − 1 ) c 4 l$ with n = 1, 2…The frequency spectrum in the common mode is shown using a sine-wave source with various frequencies in the right-hand panel. We can verify that numerical calculations using the MTL equations can be reproduced by simulations using the Simulation Program with Integrated Circuit Emphasis (SPICE).⁸ 

We have thus far discussed a simple electric circuit with grounding, where the ground is treated as the third conduction line, and have determined a number of other configurations using the concept of ringing and resonance phenomena in the common mode. Based on these findings, we were able to understand a series of interesting experimental results, obtained using a printed circuit board (PCB).^4–6 We next discuss two interesting cases: a case in which a signal line and a guard trace are arranged on one side of a test board and a return plane is arranged on the other side^4,5 and a case in which two signal lines are arranged on one side of a test board and a return plane is arranged on the other side.⁶ In the former case, the signal line was connected to a power supply on one end of the return plane of the PCB and an impedance-matched load on the other end. The guard trace was placed along the signal line and had vias connecting the guard trace and the return plane in order to reduce electromagnetic emission from the PCB system. Measurement of the radiation of the PCB system and the voltage of the guard trace for the case in which vias were placed at the terminal points of the guard trace revealed clear resonance signals. The resonance frequencies corresponded to the length l of the guard trace, $f n = n 2 l c ε eff$ for n = 1, 2,.., where l is the length of the signal and guard trace and ε_eff is the relative permittivity of the PCB. These frequencies correspond to standing waves in the common mode (guard trace) because impedance matching was imposed in the normal mode.

In the PCB experiment,⁴ the signal line corresponds to line 1 of the present study. The return plane corresponds to line 2, and the guard trace corresponds to line 3. Impedance matching was imposed for the normal mode in both cases. The voltage U₂(0, t) − U₃(0, t) was taken to be zero near the power supply in both cases, whereas U₂(z = l, t) − U₃(z = l, t) was made to be zero at the load in the PCB experiment. Moreover, lines 2 and 3 were not connected at the load in our theoretical study. Accordingly, the resonance condition was changed to that of the PCB experiment from that of the standard circuit of the present study. Since the resonance was generated in the common mode, impedance matching was introduced in the guard trace in order to remove the resonance behavior.⁵ As for the measurement of the resonance in the radiation, Watanabe et al. introduced an imbalance differential model to estimate the radiation. In a new circuit theory, the radiation process was introduced to the circuit theory using Maxwell’s equations, where this process verified that the presence of the common mode is accompanied by radiation from the system.² 

In the latter case, two signal lines were placed to one side of the PCB and a return plane was located on the other side. These two signal lines were arranged in several places at the edge of the test board and were placed at various distances from the edge toward the middle of the test board. Significant resonance behavior was observed for various configurations and its magnitude was found reduced as the two signal lines are placed closer to the middle of the test board.⁶ In the new theory we are able to show that the coupling of the normal mode to the common mode is reduced considerably as the case where the two signal lines are placed closer to the middle of the test board, as shown in Fig. 3.

We have investigated the performance of a standard circuit with grounding using the MTL theory. We observed ringing and resonance phenomena in the common mode and coupling of the common mode with the normal mode. We clarified the origin of the noise by properly defining the common mode in the MTL theory. The theoretical findings of the present study are supported by recent PCB experiments.