Circuit model of transient cross-coupling

Using a signal generator and an oscilloscope, measurements were made of the differentialmode current and the common-mode current in a twin-conductor cable installed on a test rig, over a range of frequencies which included half-wave resonances. Test data was used to assign component values to a Triple-T circuit model of the assembly. This model was then transformed into a transient coupling model. The signal generator was reset to generate square waves, and photographs were taken of the waveforms of the input voltage and the common-mode current. Close correlation was achieved between these waveforms and those of the transient coupling model. This demonstrates that the technique of circuit modelling and bench testing is reliable and accurate, in both the frequency domain and the time domain. The technique allows potential hazards due to transient interference to be analysed, tested, and quantified during the design process. Electromagnetic interference can be analysed without recourse to the mathematics of full-field modelling. It is also shown that, from the point of view of Electromagnetic Compatibility, the concepts of the single-point ground and the equipotential ground are both misleading and counter-productive.

The best way of illustrating the technique is to build a test rig which involves coupling between the differential-mode loop and the common-mode loop of a representative signal link. The construction of such a rig is described, as is the circuitry of the interface at the near end. The terminals at the far end are short-circuited. The test equipment is extremely simple; a signal generator, an oscilloscope, and a current transformer.
A circuit model which simulates coupling in the frequency domain is described, and component values assigned using data on the geometry of the assembly. A test is carried out to record the variation of differential-mode current with frequency. A similar test is used to record the variation of the amplitude of the common-mode current. An essential feature of these tests is that it covers the frequencies of half-wave resonance of both circuit loops.
A comparison was carried out of the responses of the model and that of the test results. Initially there was significant deviation. But this was corrected by using a process of successive approximation of the components of the model. The resulting model provides good correlation between theoretical and actual responses in the frequency domain. A general circuit model which simulates coupling in the time domain is described. Component values are derived from those measured by the frequency response tests. Equations are derived which allow the amplitude of reflections at each end to be computed, and a Mathcad program described which simulates the propagation of energy back and forth along the assembly. The mechanism is analysed in terms of the propagation of discrete packets of charge.
The signal generator was then set to generate square-wave pulses and the oscilloscope used to monitor the waveform of the input voltage to the test rig and the waveform of the differentialmode current. Good correlation was achieved between the actual responses of the hardware and the simulation of the model. The same was done for the common-mode loop.
The model was used to simulate the response of the setup to a step input voltage and the steady-state values of the currents computed. These proved to be exactly the same as those derived from DC analysis.
The results of this set of tests demonstrate the modelling technique is highly reliable, indicating that it can be used during the circuit design process of any electrical system mounted on a conducting structure.
The fallacy in the concept of the 'single point ground' is identified, and it is shown that there is no such thing as an equipotential conductor.
The naming of variables and constants follows normal practice in computer programing and assigns several characters to each parameter. For example, RL1, RL2 and RL3 represent the values of the three resistors of the load circuitry. As far as possible, subscripts are used to identify the elements of vectors and matrices.

Initialisation
Basically, the rig consisted of a copper pipe installed round the walls of a room. A wooden batten was fixed along the length of the pipe and a twin-conductor cable routed along the surface of the batten. This construction allowed a reasonably constant spacing between cable and pipe to be maintained along the length of the assembly. This configuration can be correlated with any signal link on any vehicle, aircraft, spacecraft, or ship. The copper pipe represents the electrical properties of the conducting structure. In this context, the terms 'conducting structure', 'ground' and 'earth' are synonymous.
Resistances of the three conductors were assumed to be: Page 4 of 25

Test setup
A schematic diagram of the test setup is shown in Figure 1a. This includes the interface module which links the test rig to the test equipment. The signal generator provides a voltage Vin across the 10 ohm resistor via a 41 ohm resistor. The voltage across the 10 ohm resistor is monitored by channel 1 of the oscilloscope. This configuration allows the impedance presented to the generator to be 50 ohm and the source resistance Rs1 applied to the cable input terminals to be: This resistance is significantly less than the impedance presented by the cable assembly over the range of frequencies involved in obtaining test data. So the alternating voltage applied to the assembly-under-review is reasonably constant.
A current transformer was clamped round the send conductor to allow the differentialmode current to be monitored. There were 10 turns on the secondary winding, to ensure that the impedance presented to the cable was very low; about 0.24 ohm. Details of the construction and calibration of this transformer are provided by reference [2].
The three terminals at the near end of the test rig were connected to the interface module as illustrated in Figure 1a and the terminals at the far end were shorted together.

Test Procedure
The signal generator was used to apply a sinusoidal signal to the cable at a set of spot frequencies, over the range 100 kHz to 12 MHz. Measurements were taken of the peak-topeak voltages Vch1 and Vch2 displayed by the oscilloscope. Since Vch1 is proportional to Vin and Vch2 is proportional to the current Idiff in the differential-mode loop, it was possible to calculate the ratio Yt1 at each spot frequency.

Idiff Yt1
Vin Yt1 is a measure of the current which would flow in the differential-mode loop when the input voltage was 1 volt. The blue circles on Figure 2a show the variation of this parameter with frequency.
The current transformer was then clamped round both cable conductors and the test repeated. The blue circles on Figure 2b are a record of the variation of the parameter Yt2 with frequency, where

Icm Yt2
Vin  and Icm is the common-mode current. Page 5 of 25

Creating the model
The differential-mode current is defined here as that which flows along the send conductor and back along the return conductor. The common-mode current is defined as that which flows along the return conductor and back along the ground.
The part of the model which simulates the test rig can be described as a Triple-T network, with each T-network representing the properties of one conductor. In order to provide an accurate simulation of the frequency response of the assembly-under-test, it was necessary to transform the impedance value of each branch of the network into a distributed parameter.
For the send conductor, the transformations are: The impedance Zh1 replaces the impedances of the horizontal branches representing conductor 1, whilst Zv1 replaces the impedance of the vertical branch. The same transformations apply to the other two conductors. [3]

Computations
A Mathcad worksheet was compiled to convert the impedance of each branch of the network to a distributed parameter at each frequency and then to calculate the values of the currents Ic1, Ic2, Ic3, and Ic4. The admittance Ym1 is the ratio of the differential-mode current Ic1 to the voltage Vin, and the variation of Ym1 with frequency is illustrated by the solid red curve in Figure 2a. The transfer admittance Ym2 in Figure 2b illustrates the variation of the common-mode current Ic2 with frequency when 1V Vin  In the worksheet, the initial values of the lumped parameters were those defined by equations (1) to (4). When the frequency response of the model was first compared with that of the test results, there was a significant deviation between the solid red curves and the blue circles of Figures 2a and 2b. However, a few iterations of the program, with different values assigned to these lumped parameters, yielded the response illustrated. The process is similar to the iterative method used by computers to solve equations. A full description of the technique is provided by reference [4]. A copy of the Mathcad worksheet used to perform the computations can be downloaded from [5].

Representative circuit model.
The end result was a circuit model which characterises the cross-coupling mechanism of the assembly-under-review. The refined values for the capacitors were: These values are significantly higher than those of (2) because of the presence of insulating material. The average relative permittivities are much greater than unity. The values assigned to the inductors were: These are not much different from those of equation (1). Since this particular set of tests did not provide refined values for the resistors, the values assumed in (3) remain unchanged.
Assigning these new values to the components of Figure 1b results in the representative circuit model of the assembly; one which can be used to define the mechanisms involved in the propagation of electromagnetic interference. It can be modified to predict the EMI of a similar system, whatever the values of the interface components, and whatever the length of the assembly.
The first peak in the response illustrated by Figure 2a is due to resonance of the differentialmode current at its half-wave frequency. This was measured to be: 6 6.93 10 Hz fh1  (12) The first peak in the response of the common-mode loop is illustrated in Figure 2b. The halfwave frequency of the common-mode loop was measured to be: 6 10.26 10 Hz fh2  The fact that fh2 is higher than fh1 means that the common-mode currents and voltages travel faster than the differential-mode signals.

Charge propagation
One of the deductions of Electromagnetic Theory is that when a step voltage is applied to the near end of a lossless transmission line, a step current is induced [6]. Electromagnetic energy propagates along at a speed comparable to that of light in a vacuum and appears at the far end as a constant current which flows into the load. If the velocity of propagation is v, then the time taken to propagate along a line of length len is If the line is represented as a set of n segments of equal length, then the time for the step to flow along one segment is This definition means that the parameter dt is finite. During time dt, constant current In will flow into the near end and the charge delivered to the first segment at time 0 t dt  will be: This charge propagates to the far end and arrives there at time tT  . During the time dt, the current If delivered to the far end is: That is, the line can be represented as a set of segments which carry discrete packages of charge flowing at a near-light velocity. This reasoning can also be applied to the threeconductor line of Figure 1c.
The ratio of the amplitude of the voltage step to that of the current step is a constant, and can be defined as the characteristic resistance [7]. The characteristic resistances of the three conductors are: Page 10 of 25

Partial parameters
Transmission line theory introduces the concept of partial currents and partial voltages to simulate behaviour of a twin-conductor line. For a three-conductor line the number of interdependent variables increases significantly. Vna1 and Ina1 identify the voltage and current absorbed at the near end of the differential-mode loop, Vni1 and Ini1 identify the incident voltage and current at that end, while Vnr1 and Inr1 identify the reflected currents. At the far end, the letter n is replaced by the letter f. For the common-mode loop, the character 1 is replaced by the character 2.

Reflection equations
The set of equations defining their relationship is: In vector-matrix form, these equations are: Adding (29) and (31): This gives:

Propagation mechanism
Given knowledge of Ini, Vni can be calculated using (26). Then Ina can be calculated using (33). The partial currents Inr delivered to the cable can be derived from equation (25) and the charges delivered to the differential mode loop and the common mode loop during time dt will be After a short interval, these charges will arrive at the far end and the partial currents delivered at this end during time dt will be: If there are radiation losses, then the charges Qfi arriving at the far end will be less than Qnr.
The incident voltages will be: By analogy with equation (33), the currents Ifa absorbed at the far end will be The currents reflected at the far end will be: and the charges sent back down the line during time dt will be: The time taken for a charge package to propagate from one end of the differential-mode loop to the other is: And the tine taken for charges in the common-mode loop to travel the same distance is: Where fh1 and fh2 are defined by equations (12) and (13).

Computations
If it is assumed that the differential-mode loop is divided into 15 n1  segments, then the time increment dt will be 9 4.81 10 sec T1 dt n1 Charges in the common-mode loop travel much faster. So the number of segments which represents this loop will be: In Mathcad, the function ceil(x) returns the least integer which is greater or equal to x.
It has been observed that when a step pulse was delivered to the near end of a twin-conductor line which was open-circuit at the far end, the current arriving at that end increased exponentially. A simple R-C circuit was used to simulate the current lost. The current arriving at the far end was reasoned to be the current delivered to the near end, minus the current lost due to radiation [8]. The virtual component which carried the radiated current was assigned the name Crad. On that occasion, the value of this parameter was measured to be 250 pF.
Thermal losses due to currents in the series resistances of the conductors can be simulated by assuming they are the properties of the source. That is: The current transformer of Figure 1a makes its own contribution to the overall response of the system. So its effect needs to be included in any simulation. The model for frequency analysis is illustrated by Figure 7.2.7 of reference [2]. For transient analysis, it can be represented by a current source Isec in parallel with a resistance RT and an inductance LT.
The mesh equations for this model are: Where Iind in (46)

Ins
Inr Int  

Qns
Qns Ins dt     Page 17 of 25

Int Qns
The arrays Rn, Rf, Gn and Gf are created to avoid unnecessary repetition of the computations in the main program.
The half-wave frequencies fh1 and fh2 are defined by (12) and (13). The transit times T1 and T2 are as defined by equations (40) and (41). The equations for dt and n2 are defined by (42) and (43).
For transient testing, the signal generator is set to provide square waves at a defined frequency f. So the voltage source in the simulation needs to replicate this action. With a square wave, the mark-space ratio is unity and the time Tmark the voltage is positive is equal to half the period of the waveform. Hence: Since the simulation of time is restricted to a multiple of dt, the number of time steps M during the period Tmark is where the function floor(x) is the greatest integer less than or equal to x.
It is also necessary for the simulation to replicate the time Tscope it takes for the oscilloscope trace to scan from one side of the graticule to another. The number N of time steps involved is: The final line at the bottom of Figure  The function rad(Inr ,Qns ,Ro ,Crad) takes as input the value of the reflected current emanating from the near end, (Inr1 or Inr2) and calculates the amplitude of the current Int which actually arrives at the far end, as well as the total charge Qns which has departed into the environment. Reference [8] provides a full explanation of the underlying reasoning.
Page 18 of 25 The single-column vector F provided as input to the function forward(F ,Int, n) holds the value of the current Int at each section of the n segments of the line at a particular instant. The value of this current is placed in the first segment, the contents of every segment are moved forward, and the content of the final segment is provided as output. The action is similar to that of a shift register. The function back(B ,Ifr ,n) moves data from the far end back to the near end.
The function sqwv(n ,M) creates a square wave which switches from a value of zero to -Vg/2 at time 0 t dt  , to Vg/2 at t Tmark  and back to -Vg/2 at 2 t Tmark  . It continues to create a square wave at frequency f. The main program first defines the vectors which simulate the propagation of charges forward and backward along the line. The iterations are carried out for a period of a single scan of the oscilloscope Tscope, plus a time equal to Tmark. The time Tmark coincides with the hold-off period of the oscilloscope before the scan is triggered.
The first two program statements in the iterative section simulate the action of the signal generator. The statements which follow use the output of each function as the input of the next function. Each iteration calculates the values of all the parameters of Figure 1c at time t i . The parameter j is set to unity at the end of the simulated hold-off period and starts counting thereafter.
The output vector Vsig simulates the voltage which would be created by a current transformer clamped round the send conductor, Vret is proportional to the current in the return conductor, and Vgnd is proportional to the current in the structure. Vch1 is a record of the voltage which would be monitored by channel 1 of the oscilloscope. This worksheet can be downloaded [9].

Test results.
A series of tests was carried out on the rig of Figure 1a with the signal generator set to give a square wave output. As with the frequency response tests, Channel 1 of the oscilloscope was used to monitor the input voltage at the terminals at the near end of the line, and channel 2 was used to monitor the output of the current transformer. This was done at a number of frequencies f of the square wave, and the waveform observed on the screen was compared with that of the simulation.
Initially, there were significant deviations between theoretical and test results. However, when the model was modified to include the effect of the inductance of the current transformer and the effect of radiated emission, close correlation was achieved. voltage step varied to set the value of Vch1 to 0.8 V, and the time period on the oscilloscope selected to give the waveform of a single cycle. Figure 6a is a copy of the photograph taken of that display. Figure 6c is a photo of the waveform on channel 2.
The input parameters on the first page of the worksheet were adjusted to set the voltage Vg to 1.62 volts and the frequency of the simulation to 200 kHz, and a record taken of the graphs of the vectors Vch1 and Vch2. These are displayed by Figures 6b and 6d. Vch2 is a copy of the vector Vgnd in Figure 5.
The close correlation between the test and simulated waveforms provides a high degree of confidence in the modelling technique. Of particular interest is the sawtooth appearance of the waveforms of figures 6c and 6d. This is because the step change in the common-mode current arrives back at the near end before that of the differential-mode current.

Steady state conditions
The main program illustrated on Figure 5 was altered to compute the response of the model to a step pulse of 5 micro-seconds duration. Currents in all three conductors are simulated.
The red curve at the top of the graph of Figure 7a simulates the waveform of the current Isig in the send conductor. This is the differential-mode current. The dashed green curve at bottom of the graph simulates the current Ignd in conductor 3, the 15 mm copper tube. This curve correlates with Vch2 of Figure 6d. Ignd is the common-mode current. The blue curve simulates the current Iret in the return conductor. It can be seen that most of the return current flows in the return conductor. That is, during the first 5 micro-seconds, the differential-mode loop carries most of the electromagnetic energy.
When the time frame of the simulation is increased to 50 micro-seconds, as shown in Figure  7b, things begin to change. Current in the copper tube continues to rise, and after 250 microsecond, current in the common-mode loop exceeds that in the differential-mode loop.

Photons and charges
It has been shown that the transient behaviour of the electromagnetic coupling between the three conductors can be visualised and analysed as a system of moving charges which propagate forwards and backwards along the conductors. What is not immediately obvious is that charges propagate to and fro between the conductors. During the first 5 nanoseconds after the appearance of the step voltage, charges flow from the send conductor, across the gap into the other two conductors, back along those conductors, and then back across the gap into the send conductor. This deduction supports that made during the transient analysis of an open-circuit cable. [10].
In fact, this assumption is inherent in the basic transmission line equations The very existence of this pair of equations depends on the assumption that current flows from the return conductor back into the send conductor. Figure C.1 of reference [11] illustrates this. Current is due to the flow of charges. Voltage changes when charges move.
The problem with this visualisation is that charges do not flow through insulation material. The entity involved on the propagation of electromagnetic energy is the photon, and photons do not carry charge [12]. However, if it is assumed that when a photon departs from an atom, it leaves a charge on that atom and that when it arrives at another atom it creates a charge on that second atom, then a possible explanation emerges. It is photons which actually carry the electromagnetic energy; and the interaction of photons with atoms creates a system of moving charges which can be analysed using the concepts of circuit design.
In this particular example, it can be reasoned that most of the return current flows in the return conductor during the first 5 microseconds because its surface is closest to the send conductor. After 5 milliseconds, most of the return current flows via the structure, because its surface area is so much greater.

The equipotential ground
This concept is illustrated in books on electromagnetic theory as a line at the bottom of a circuit diagram which does not possess the properties of inductance, capacitance or resistance. ' Figure 12-3 Lumped constant representation of a transmission line' in reference [13] is one example. In reference [14], it is shown how this representation simplifies the analysis by subsuming the properties of the return conductor into those of the send conductor. The problem with this simplification is that it assumes that the interaction between the return conductor and the ground is not relevant. Invoking the concept renders impossible any analysis of the coupling between the three conductors of the signal link. It eliminates any chance of analysing EMI.
This analysis has shown that when the structure is used to carry any return current, it creates a situation where a vast number of charges are propagating backwards and forwards along that structure. This manifests itself as a high magnetic field which permeates the whole system. If the current in the send conductor switches off, the unwanted dynamic energy possessed by current in the structure will spread far and wide and reappear as interference in other parts of the system.
The fact that the resistance of the structure of vehicles is extremely low leads many engineers to utilise it as a general-purpose return conductor for power supplies, actuators, and sensors. It is reasoned that the inclusion of dedicated return conductors would result in needless increase in the weight and cost of the system under review. Such a design decision is the cause of innumerable EMI problems. It results in unexpected failures during formal EMC testing.
It is possible that the disastrous loss of Flight 800 [15] was due to transient current in the structure coupling energy into the wiring of the fuel level sensors in the centre tank. Any bad joint in the wiring could have caused a high voltage to develop across the gap and the resulting spark could have ignited the fuel. It may or may not be the case that such a mechanism was the actual cause of the disaster, but it is definitely a hazard condition which needs to be analysed during the design of aircraft.

The single-point ground.
A succinct definition [16] of this concept is: 'A single-point ground system is one in which subsystem ground returns are tied to a single point within that subsystem. The intent of using a single-point ground system is to prevent currents of two different subsystems from sharing the same return path and producing common impedance coupling.' The fallacy in this reasoning is the fact that it ignores the existence of the send conductors.
It is the send conductor which is the source of electromagnetic energy. First and foremost, this conductor acts as a transmitting antenna. The energy it transmits goes where it can. This energy can be visualised as the dynamic energy of moving charges. In conductors, this manifests itself as current flow. The return current follows whatever path it can find. It does not necessarily follow the path designated by the system designer.
If a designated conductor is provided to carry return current, and is located as close as possible to the send conductor, then the electromagnetic energy automatically follows the path defined by the routing of the cable. If the return conductor is deliberately separated from the send conductor, electromagnetic energy spreads far and wide.
The action of implementing a single-point ground in any system guarantees that such a system will experience EMI problems during its entire existence.