Circuit Analysis Accounts
For Magnetic Fields
By Ken Yang, Senior Member of the Technical Staff
Maxim Integrated Products Inc., Sunnyvale, Calif.
Power sources such as electric motors, transformers and the inductor in a switching regulator produce time-varying magnetic fields that can induce
noise in electronic circuitry.
hm’s law and Kirchhoff ’s voltage law
(KVL) provide powerful tools for conventional circuit analysis (mesh analysis), but if time-varying magnetic fields
are present, Faraday’s law must be invoked as well. The additional current induced by a timevarying magnetic field must be accounted for by adding a
term to Ohm’s law and KVL. This introduction of Faraday’s
law into the circuit-analysis equations produces unexpected
anomalies: Two voltages appear to coexist simultaneously
between two nodes in a circuit, and the voltages appear to
depend on position of the voltmeter leads.
applying Ohm’s law and KVL. First, obtain an equation for
loop current by substituting from equations 3 and 4 into
equation 2. Then, solving for the current yields equation 5:
(3)
VR1 = IR1
O
VR2 = IR2
(5)
Note that KVL can be written in integral form. Textbooks on electromagnetic theory define voltage as the vector integral of the
electric field (E)
B
C
along a path (dl),
A
R1
such as that from
node A to node C
D
+
I
in Fig. 1 (Eq. 6).
With the magU1
Magnetic
R2
Field
netic field off, the
closed-loop inteE
gral from node A
F
to C to E and back
to A equals zero Fig. 1. To illustrate the effect of a time-varying
(Eq. 7). Thus, magnetic field, consider the response of a
Kirchhoff ’s volt- simple closed-loop circuit (formed by a battery
age law can be and two resistors) with and without the field.
written in integral form: the closed-loop integral of the electric field is
equal to zero.
Analysis Without Magnetic Fields
To understand the influence of time-varying magnetic
fields, we should first review the KVL and Ohm’s law equations for a circuit with no magnetic field present. Then, in
the next section we can compare them with the corresponding equations in which a time-varying magnetic field is
present.
With no magnetic field present, one commonly uses
KVL and Ohm’s law to perform a circuit analysis based on
the mesh technique. As commonly stated in textbooks, KVL
says the algebraic sum of all voltages around a closed loop
equals zero (Eq. 1):
(1) (KVL)
Consider the circuit of Fig. 1 with the magnetic field
turned off. If you take a voltmeter and measure the voltages across each component in the loop, the sum of those
voltages equals zero as predicted by KVL (Eq. 2). (For the
counter-clockwise direction chosen, note that voltages
across the resistors are negative.)
-VR1 – VR2 + U1 = 0
(2)
You can solve for the component values in Fig. 1 by
Power Electronics Technology
November 2004
(4)
(6)
(Bold font indicates vectors)
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CIRCUIT ANALYSIS
question, and F is the total magnetic flux through that area.
The direction of the induced current depends on the direction of the magnetic field. In Fig. 1, the induced current
is clock-wise as shown if the magnetic field is pointing out
of the page. Total current is now the sum of current due to
the battery (IU1) and that due to magnetic induction (IMAG):
I = IU1 + IMAG
(7)
(KVL in integral form)
(9)
Ohm’s law must be modified (extended) to account for
the additional current:
V = IU1R + IMAGR
(10)
(Extended Ohm’s Law)
Kirchhoff ’s voltage law must be extended as well. A comparison of equations 1, 7 and 8 shows that KVL is extended
by adding the -dF/dt term to the right side of equation 1:
Analysis with Magnetic Fields
Now, turn on the magnetic field in Fig. 1. The field varies with time, which induces current in the loop, and that
condition calls for the use of Faraday’s law: The integral of
the tangential component of electric-field intensity around
a closed loop equals the time rate-of-change for magnetic
flux passing through a surface bounded by that loop
(Eq. 8):
(11)
(Extended KVL)
Equations 2 through 5 can be rewritten to include the
time-dependent magnetic field components:
(8)
(Faraday’s Law)
where B is the magnetic field, A is the area of the surface in
(12)
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CIRCUIT ANALYSIS
C
B
A
R1
+
U1
Voltmeter #1
V= U1
Magnetic
R1
Field
VR2 = (IU1 + IMAG)R2
(14)
Voltmeter #2
V= I(R1+R2)=U1+
d-F/dt. d-F/dt
(15)
E
Thus, equations 1 through 5 have been extended to account for current induced by the magnetic field, forming
Equations 11 through 15. Equation 11 is the extended
Kirchhoff voltage law and equation 15 is the extended
Ohm’s law, with the sign of the dF/dt term indicating the
direction of current. These equations look simple enough,
but they seem to describe a paradox.
Using equations 12 through 15, consider an analysis of
the Fig. 1 circuit with time-varying magnetic field. The
voltage across U1 (nodes A and F) is VAF = U1. But, VAF also
equals current in the loop times the two resistances:
Fig. 2. Both voltmeters are measuring the same nodes, but the
measured voltages are different. Voltmeter #1 integrates the
electric field inside U1, and voltmeter #2 integrates the electric
field inside R1 and R2.
100pF
Volume
Control
+
-
(13)
I
F
Audio
Signal
VR1 = (IU1 + IMAG)R1
10k
1uF
Loop
R1
10k
+
MAX4410
50
Load
VAF= U1
Spectrum
Analyzer
(16)
(17)
Fig. 3. This audio application circuit demonstrates how magnetic
interference diminishes audio quality.
We now have two possible voltages across nodes A and
F. In fact, there are two possible voltages for each pair of
nodes that include a component in Fig. 1. See squations 16
through 25. For a simple comparison, we arbitrarily set U1=
2 V,
dΦ
= 1 V, R1 = 2 kV, and R2 = 4 kV. Then, the loop
dt
current according to equation 15 is 0.5 mA.
VAF = 2V
(18)
(19)
VBC = I(R1) = 1V
VBC = U1 – I(R2) = 0V
VDE = I(R2) = 2V
VDE = U1 – I(R1) = 1V
VEF = I(0O) = 0V
VEF = U1 – I(R1) – I(R2) = –1V
(20)
(21)
(22)
(23)
(24)
(25)
Nodes B and C are especially interesting, because the
current through R1 is non-zero, yet the voltage across it
could be zero. Similarly, nodes E and F represent a short
(zero-ohm) wire, yet the voltage across these nodes is nonzero. So which voltage is it? Nothing is wrong with the math.
Two voltages do coexist simultaneously! Mathematically,
the voltage you get depends on the path of integration taken
in your measurement. Remember that voltage is a vector
integral of electric field along a given path. If a time-deCIRCLE 224 on Reader Service Card or freeproductinfo.net/pet
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CIRCUIT ANALYSIS
pendent magnetic field is present, the integration is path
dependent. In simpler terms, the voltage depends on how
the measuring circuit (the voltmeter) is connected to the
nodes.
As shown in Fig. 2, voltmeter #1 measures the voltage
across nodes A and F from the left-hand side, obtaining a
measurement of U1 = 2 V. In contrast, voltmeter #2 measures the voltage across nodes A and F (B and E are the
same as A and F) from the right-hand side, with the result
+
Audio Signal
Generator
Magnetic flux
from nearby
motor
1k
10k
To MAX4410
Audio
Amplifier
(a)
+
Audio Signal
Generator
Contrary to a common misconception, the induced
voltage is distributed not in the wires connecting the resistors, but within the resistors. The integral of electric field
inside a wire is zero, so the voltage across a wire is zero. By
sliding the probe contact from point A to point B, lab experiments verify that the voltage drop across the connecting wires is zero. Thus, the voltage on voltmeter #1 does
not change. Similarly, sliding the contact from F to E does
not change the voltage on voltmeter #1. The same applies
to voltmeter #2. Sliding the contact from B to A or from E
to F does not change the reading. The voltmeter probes
are arranged to minimize interference from the magnetic
field.
The measured voltage appears to depend on the posi-
Magnetic flux
from nearby
motor
1k
To MAX4410
Audio
Amplifier
10k
(b)
ONE INCH
COPPER
Fig. 4. Connecting the ground trace near the 10-kV resistor (a) or on
the top of the loop (b) shows that the physical layout of the volumecontrol circuit affects magnetic interference.
tion of the probes. Voltmeter #1 acts like an electric field
integrator that integrates the electric field inside the battery U1, and voltmeter #2 integrates the electric field inside R1 and R2. Different integration paths yield different
voltages. The following case further demonstrates this
position-dependent effect.
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CIRCUIT ANALYSIS
Fig. 5 In (5a), the magnetic interference from an electric motor in Fig. 4a causes a peak at 300 Hz, about 62 dBc below the audio test tone (a). In
(5b), magnetic interference from an electric motor in Fig. 4b causes a peak at 300 Hz, about 78.5 dBc down from the audio test tone—an
improvement of 16.5 dB over the circuit of Fig. 5a.
Consider Fig. 3, in which an audio signal (a 1-kHz
sinewave) is attenuated by a volume-control potentiometer
(R1) and fed to an audio amplifier, whose output is analyzed by a spectrum analyzer. A nearby electric motor creates magnetic interference in the loop formed by R1 and
the audio-signal source. For simplicity, R1 has been replaced
with a 1-kV and a 10-kV resistor connected in series, and
the loop that catches the magnetic flux was enlarged to
1 in2. Two physical board layouts were tested (Fig. 4).
Figs. 5a and 5b show plots of the audio-amplifier output spectra. The 1-kHz audio test tone is identical in both
cases, but the amplitude of 300-Hz motor interference depends only on the ground connection. The worst magnetic
interference (-62 dBc) is based on Fig. 4a, in which the
audio amplifier receives interference voltage from the 10kV resistor (Fig. 5a). In other words, the audio amplifier
acts as an electric field integrator that integrates the electric field inside the 10-kV resistor.
On the other hand, Fig. 5b shows the interference voltage received from the 1-kV resistor. The lesser interference shown in this plot (-78.5 dBc) represents an improvement of 16.5 dB. (The expected interference ratio is 20 dB,
because the resistor ratio is 10-to-1, but the loading effect
of the audio amplifier’s input impedance lowers the interference amplitude in Fig. 5a.)
This phenomenon has been experimentally verified.
Note that the voltage across two nodes is not clearly
defined; it can be either of the two voltages, depending on
how the wires are positioned. This experiment demonstrates that the voltage between two nodes is no longer a
simple algebraic expression, but a vector integral of the
electric field along a given path. Because the integral is pathor position-dependent, integrating along a different path
yields a different voltage. Equations 9 through 15 in the
previous section do not clearly predict the position-dependent effect, so they must be used very carefully.
PETech
References
1. Romer, Robert H. “What do ‘Voltmeters’ Measure?
Faraday’s Law in a Multiply-Connected Region,” American
Journal of Physics. Vol. 50, No. 12 (Dec. 1982), pp. 1089-1093.
For more information on this article,
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