Download Modeling of Capacitive and Electromagnetic Field Shielding Effects

MODELING OF CAPACITIVE AND ELECTROMAGNETIC FIELD SHIELDING EFFECTS
IN A CVT
William Sommerville and James Gover, Fellow IEEE, Kettering University
Robert Sanchez and Jimmy Bou, Sandia National Laboratories
Abstract: In the discharge of a capacitor the current
was measured with a current viewing transformer
(CVT). In addition to measuring the current flowing
through the CVT primary, a 51 MHz noise signal was
added to the primary current. When the CVT was
covered with a gold shield, the noise was eliminated.
Analysis of the measured results indicate that the gold
layer reflected the electromagnetic wave that was
generated by current flowing in the primary and that
the capacitance between the shield and the CVT
secondary had no measurable effect on the CVT
output.
Key Words: Shielding, Current Viewing Transformer
(CVT), CVT Capacitances, Electromagnetic Field
Coupling
L = lumped circuit inductance, including that of the CVT
C = capacitance of energy storage capacitor
R = lumped circuit resistance, including that reflected into
the primary of the CVT
V0 = initial voltage across the capacitor.
n  02   2
0 
1
LC

, f0 
(2)
0
2
1
2RC
(3)
(4)
Note that the values used in the actual circuit tested yield
the underdamped solution given in Equation (1).
I. INTRODUCTION
A common method for generating a pulse of electric
power is to charge a capacitor to a predetermined voltage
and discharge it though a load as shown in Figure 1. This
circuit is used in a wide range of applications including
camera flashes and detonator firing circuits such as is
found in the mining and demolition sectors.
Figure 1: Pulsed power circuit schematic. The CVT is a
current viewing transformer used to measure the
discharge current.
Writing the differential equations for this series RLC
circuit and solving it gives the following solution for the
under-damped current when the switch is closed at t = 0.
i t  
where
V0  t
e sin n t  ,
n L
(1)
When this circuit was discharged, a current waveform
typical of that shown in Figure 2 was measured. Note the
presence of the waveform predicted by Equation (1) in
which f0=1.1 MHz. Riding on the “primary” waveform is
a damped sinusoid of frequency 51±1 MHz.
Figure 2: Output from unshielded CVT.
When the CVT was covered or shielded with a grounded
gold conductor (except for the interior of the toroid) and
the energy storage capacitor was discharged, the CVT
measured the waveform shown in Figure 3. Note that the
51 MHz ringing signal was essentially eliminated by the
grounded shield.
In this model, the magnitude of the voltage of the noise
source is proportional to the mutual inductance between
the primary and secondary of the CVT and the time
derivative of the primary current. Ls is the total
inductance of the CVT secondary at 51 MHz. At this
frequency the permeability of the CVT core is the same as
the permeability of free space. Note that in a capacitive
coupling model, the noise source would be a current
source whose magnitude is proportional to the mutual
capacitance between the primary and secondary and the
time derivative of the voltage in the primary circuit.
II. EFFECTS ON THE PRIMARY
Figure 3: Output from shielded CVT.
The problems addressed in this paper are:
(1) What is the source of the high frequency noise?
(2) What is the physical role filled by the shield in
reducing the noise?
(a) Is the shield simply protecting the CVT from
picking up externally generated electromagnetic
signals?
(b) Does the change in inductance and distributed
capacitance of the CVT due to the shield act as a
high pass filter that removes the noise?
First, ringing on the flat cable transmission line
connecting the energy storage capacitor to the resistive
load was given consideration. The flat cable was
configured with a capacitor termination at the sending end
and an approximate short circuit at the load end. Due to
the length of the flat cable, the double transient time
would result in a noise frequency that was over 500 MHz
or about 10 times that observed. So reflections on the flat
cable as the source of the noise was eliminated.
In investigating the design of the CVT it was observed
that an area existed between the secondary turns and the
50 Ohm load in which one could induce an emf through
magnetic field coupling. An equivalent circuit for
magnetic field coupling is shown in Figure 4.
CVTs are commonly used to measure currents where high
isolation or minimal disruption of circuit performance is
desirable. The most obvious effect that a CVT has on a
circuit is that it reflects the secondary’s magnetizing
inductance and load resistance back into the primary
circuit with both diminished by the square of the turns’
ratio. Eddy current losses in the CVT core also contribute
to a resistance component that is included in the primary
circuit.
Empirical data shows that shielding the CVT reduces the
inductance and resistance of the primary. A regression
analysis of the secondary voltage of a CVT using
identical capacitive discharge primary currents allows one
to construct the equivalent series RLC circuit. Figures 2
and 3 show the voltages with respect to time that were
used for the regression analysis.
The shielding dramatically reduced the high frequency
ringing on the signal, but it also changed the overall shape
of the current pulse. Regressions of the signals found a 9
percent lower inductance value in the primary with the
shielded CVT as well as a 29 percent lower resistance.
The inductance was estimated to drop from 98 nH to 89
nH while the resistance dropped from 352mΩ to 250mΩ
when shielding was added.
III. EFFECTS ON THE SECONDARY
Several models have been created to approximate the
parasitic losses in transformers.[1] CVTs may be modeled
using similar, high-order models to achieve high accuracy
over a broad frequency range, but due to the separation
and relative isolation of the primary and the secondary, a
simple model was found to suffice for this analysis.
Figure 4: Equivalent circuit for noise generation.
The secondary was modeled as a parallel RLC network
with the component values estimated from the output
impedance measured as a function of frequency. As
expected from the model, the output impedance behaved
as a bandpass filter with the low frequencies dominated
by the inductive reactance, middle frequencies by the
resistance, and the higher frequencies by the capacitance.
Figure 5 shows an example of the frequency response of
the secondary impedance.
frequencies when the skin effect and proximity effect take
away the ideal conductor approximation for the entire
length of the wire.
The turn-to-turn capacitance in the secondary was found
assuming uniform spacing between 21 turns of wire
around the core geometry. The spacing on the interior of
the toroid was 725 μm and the exterior spacing was 1.46
mm. These values, along with the varying gap distance
along the sides of the core, produced an estimate using
Ansoft electrostatics software of 8.3 pF of turn-to-turn
capacitance.
Figure 5: Output impedance frequency response.
The shielded and unshielded secondary impedances have
similar capacitance and resistance values of 52 pF and 50
Ω, respectively.
IV. THEORETICAL ANALYSIS OF SECONDARY
The parasitic capacitance effects were modeled using
Ansoft’s Maxwell SV software and compared to each
other and the empirical data. This software package that
is available to students on-line solves Maxwell’s
Equations under static conditions. The capacitance was
determined by solving for the electric field and integrating
its square divided by the permittivity of the insulating
medium over the volume occupied by the electric field.
The core to shield capacitance calculated using the
parallel plate approximation over the appropriate surfaces
was found to be 12.9 pF. Fringing field effects were
approximated with Ansoft electrostatics software to be
about 5 pF to make a total core to shield capacitance of
about 18 pF.
Another capacitance considered was the secondary to
shield capacitance. From the specifications of the CVT, a
distance of 1.06 mm was found between the secondary
windings and shield. Using the wire geometry and
spacing along with the permittivity of the epoxy between
the windings and shielding, a capacitance of about 10 pF
was found between the secondary windings and the
shield. Like the core to shield capacitance, this could be
expected to increase by up to 40% due to fringe field
effects for a maximum turn to shield capacitance of 14
pF.
These capacitance calculations are shown in Figure 6.
The CVT was modeled using 21 turns of 31 AWG magnet
wire around the core. The inside diameter of the toroid
shaped core was 4.75 mm, the outside diameter was 9.53
mm, and the thickness was 3.18 mm. The edges were
assumed square for purposes of calculation. These
dimensions lead to a total wire length of 314 mm and a
DC resistance of 147 mΩ for a 21 turn secondary.
At higher frequencies, the skin effect begins to have an
effect, bringing the resistance up by a factor of six at 50
MHz. This is still much lower than the 50 Ω resistance in
parallel with the transformer and therefore had little effect
on the noise response of the CVT.
The secondary to core capacitance was modeled using the
wire geometry over a flat ferrite surface in Ansoft. The
relative permittivity was set to 3 to approximate the
polyester nylon insulation material. This resulted in a
differential capacitance of 138 pF/m with closely-packed
windings and 183 pF/m for loosely-packed windings.
The estimated secondary to core capacitance is therefore
47 to 62 pF at low frequencies, lowering at higher
Figure 6: Comparison of capacitance sources to the
measured total capacitance.
The turn to core capacitance is seen to dominate and is of
the same order of magnitude as the measured total
equivalent capacitance.
Although the turn to core
capacitance is larger, its contribution to the total
equivalent capacitance is not linear. Each capacitance
value listed above is distributed over 21 windings in a
complex series-parallel combination with resistances and
inductances.
Therefore, the calculated capacitances
appear to confirm the measured capacitance value.
The inductance of the CVT secondary is highly dependent
on the frequency of interest. This is due to the frequency
dependence of the permeability of the core. Data on the
core material shows that it has a relative permeability of
3000 at low frequencies, 2000 at the 1.1 MHz signal
frequency, and 1 at the 51 MHz ringing frequency. With
21 turns wound on a toroid geometry, these relative
permeabilities result in inductances of 563 μH, 376 μH,
and 188 nH, respectively.
V. EXPLANATION OF RINGING EFFECT
Since the ringing is evidently lightly damped, Equation
(3) can be used along with the inductance value calculated
for the core geometry at the ringing frequency to derive
an equivalent parallel capacitance. Using an inductance
value of 188 nH at 51 MHz, a capacitance of 51.8 pF was
calculated. Note that this is the same as the capacitance
determined from Figure 5 using the measured frequency
response of the secondary impedance.
Subtracting the ideal current waveform of the primary
circuit from the response measured from the unshielded
CVT, shown in Figure 2, allows isolation of the ringing
noise signal. This is shown in Figure 7 below.
Figure 7: Ringing signal extracted from unshielded CVT
response.
The under-damped, well-behaved 51 MHz sinusoid curve
shown in Figure 7 is a fit to the actual data which is also
shown in Figure 7. Note that the fit is excellent for late
times, but is not accurate at early times. The envelope of
the damped sinusoid is determined by the same damping
coefficient used in the series RLC circuit, in Equation (4).
If the resistance value of 50 Ohms is used, the capacitance
must be 715 pF to match the envelope shown in Figure 7.
This higher capacitance, however, would require the
equivalent inductance to be 13 nH, much lower than the
air core approximation. It would also disagree with the
theoretical and empirical data shown in Figure 6.
However, if the capacitance is set to 52 pF, the inductance
value can remain unchanged as well and the required
resistance is found to be 680 Ohms. Part of this could be
explained due to the skin effect acting in the resistor.
This increase in resistance at higher frequencies would
not necessarily conflict with the data shown in Figure 5
because the lower parallel reactance of the capacitor
determines the high frequency impedance measurements.
Data was taken similar to that in Figure 5 with no
difference in high frequency response and a derived
capacitance of 52 pF for both shielded and unshielded
models of the CVT. This and the relatively small
theoretical contributions of the shield to the total
capacitance shown in Figure 6 lead to the conclusion that
the shielding capacitance has little effect on the ringing
circuit and would not significantly attenuate the ringing
once it is triggered. It is therefore concluded that the
shielding prevents the initial high frequency energy from
reaching the secondary circuit and causing the ringing.
VI. SHIELDING OF THE SECONDARY
When current flows through the wire that threads the
center of the CVT’s toroid, an electromagnetic wave is
generated. When this wave reaches the inner diameter of
the core, it begins to diffuse into the high permeability
core as described in another paper at this conference. [2]
The wave also propagates into the region of the CVT’s
secondary where its magnetic field induces an emf in the
loop formed between the 50 Ohm load resistor and the
transformer secondary. There may well also be a mutual
capacitance contribution to the induced noise.
Experimental methods may be used to determine the
relative contributions from mutual capacitance and mutual
inductance coupling.
When the gold shield is placed around the CVT the ratio
of the magnitude of the transmitted TEM wave to the
magnitude of the incident wave is described by Equation
(5) below.
22
(5)
T
2  1
where,
T = fraction of the transmitted TEM wave,
η1 = the intrinsic impedance of free space, and
η2 = the intrinsic complex impedance of gold. Note that
the intrinsic impedance of gold, like other conductors, is a
complex number whose real and imaginary components
are equal.[3]
The intrinsic impedance of free space is 377 Ohms. The
magnitude of the intrinsic impedance of gold is found
using Equation (6),
2 
2
 2 2
, 2 
1
 f 2 2
(6)
where,
σ2 = conductivity of gold, 4.1x107 S/m,
δ2 = skin depth of gold,
f = frequency of the wave, and
μ2 = permeability of gold, equal to μ0. [4]
and that the shield design be reduced in area to the
minimum required to eliminate noise generation. It is
likely the noise could also be eliminated by precisely
locating the plane of the CVT secondary loop so that it is
normal to the direction of current flow in the primary
circuit. This could be accomplished by the design of a
simple mounting fixture.
VI. REFERENCES
At the signal frequency of 1.1 MHz, the skin depth of
gold is 74.9 μm, the intrinsic impedance of gold is 460
μΩ, and the transmitted wave magnitude is only 2.4x10-4
percent of the wave magnitude incident on the gold
shield. A wave component at 51 MHz would have
1.7x10-3 percent of the incident wave magnitude
transmitted into the gold shield.
Neglecting additional attenuation inside the gold shield,
and making use of Equation 5 allows one to conclude that
the magnitude of the wave leaving the gold shield in the
interior of the CVT is two times its magnitude inside the
shield. This is because the intrinsic impedance of the
epoxy will be several orders of magnitude larger than that
of the gold.
Thus, the shield is found to be effective at protecting the
interior of the CVT from electromagnetic radiation. It
does this by reflecting rather than attenuating the
electromagnetic wave. It is concluded that the gold shield
prevents an initial pulse of electromagnetic energy from
entering the loop of wire formed by the ends of the
secondary winding and the 50 Ohm resistor.
1.
2.
3.
4.
Lu, Y. L.; Zhu, J. G.; and Hui, S. Y.; Experimental
Determination of Stray Capacitances in High
Frequency Transformers, IEEE Transactions on
Power Electronics, Vol. 18, No. 5, September, 2003.
Sommerville, W. T.; Gover, J.; Sanchez, R.; and Bou,
J.; Modeling of Magnetic Field Diffusion Phenomena
in a CVT, Proceedings of 2005 EMCWA Conference,
Indianapolis, Indiana.
Paul, C. R. Introduction to Electromagnetic
Compatibility, 1992, John Wiley & Sons, Inc. New
York.
Paul, C. R., Electromagnetics for Engineers: With
Applications to Digital Systems and Electromagnetic
Interference, John Wiley & Sons, 2004.
VII. AUTHORS’ BIO
Mr. William Sommerville has completed coursework for
the B.S. in electrical engineering from Kettering
University. He will receive his degree in September of
2005 and graduate first in his class in December. He did
most of his co-op terms at Sandia National Laboratories
where he designed and built in MEMS technology the
“world’s smallest radio” for his senior thesis.
V. CONLUSIONS AND RECOMMENDATIONS
The ringing frequency of the noise induced in the CVT
agrees exactly with that calculated using measured values
of the capacitance of the CVT’s secondary and the
secondary’s inductance calculated using the standard
equation for the inductance of a coil wound on a toroid.
Thus, ringing in the unshielded CVT is the natural ringing
frequency of the CVT’s secondary. At this frequency of
51 MHz, the permeability of the core is that of free space.
The data and analysis indicate that the gold plating on the
CVT reflects the electromagnetic field generated by the
current pulse and keeps it from inducing noise in the
CVT. The capacitive effect of the shield was not
sufficient to eliminate the ringing.
The noise pulse could also be eliminated by use of a
band-stop filter centered at 51 MHz.
It is recommended that further experiments be conducted
to determine the mutual inductance and mutual
capacitance between the primary and secondary circuits
Dr. James Gover is Professor of Electrical Engineering at
Kettering University. In 1998 he retired from Sandia
National Laboratories. He received the IEEE Fellow
award for his work in radiation effects in nuclear systems
and his work on national policy research earned him the
IEEE-USA Citation of Honor. He has served IEEE as a
Congressional Fellow and Competitiveness Fellow.