ng=0 width=100%>Yahoo! is not affiliated with the authors of this page or responsible for its content.
Circuit Analysis Accounts For Magnetic Fields Power Electronics Technology November 2004
www.powerelectronics.com
30
Circuit Analysis Accounts
For Magnetic Fields
Power sources such as electric motors, transform-
ers and the inductor in a switching regulator pro-
duce time-varying magnetic fields that can induce
noise in electronic circuitry.
By Ken Yang, Senior Member of the Technical Staff
Maxim Integrated Products Inc., Sunnyvale, Calif.
O
hms law and Kirchhoff s voltage law
(KVL) provide powerful tools for con-
ventional circuit analysis (mesh analy-
sis), but if time-varying magnetic fields
are present, Faradays law must be in-
voked as well. The additional current induced by a time-
varying magnetic field must be accounted for by adding a
term to Ohms law and KVL. This introduction of Faradays
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.
Analysis Without Magnetic Fields
To understand the influence of time-varying magnetic
fields, we should first review the KVL and Ohms law equa-
tions for a circuit with no magnetic field present. Then, in
the next section we can compare them with the correspond-
ing equations in which a time-varying magnetic field is
present.
With no magnetic field present, one commonly uses
KVL and Ohms 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 volt-
ages 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.)
-V
R1
V
R2
+ U
1
= 0
(2)
You can solve for the component values in Fig. 1 by
applying Ohms 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:
V
R1
= IR
1
(3)
V
R2
= IR
2
(4)
(5)
Note that KVL can be written in integral form. Text-
books on electromagnetic theory define voltage as the vec-
tor integral of the
electric field (E)
along a path (dl),
such as that from
node A to node C
in Fig. 1 (Eq. 6).
With the mag-
netic field off, the
closed-loop inte-
gral from node A
to C to E and back
to A equals zero
(Eq. 7). Thus,
Kirchhoff s volt-
age law can be
written in inte-
gral form: the closed-loop integral of the electric field is
equal to zero.
(6)
(Bold font indicates vectors)
Fig. 1.
To illustrate the effect of a time-varying
magnetic field, consider the response of a
simple closed-loop circuit (formed by a battery
and two resistors) with and without the field.
+
A
B
C
D
F
R1
Magnetic
Field
U1
R2
E
I www.powerelectronics.com
Power Electronics Technology November 2004
31
CIRCUIT ANALYSIS
(7)
(KVL in integral form)
Analysis with Magnetic Fields
Now, turn on the magnetic field in Fig. 1. The field var-
ies with time, which induces current in the loop, and that
condition calls for the use of Faradays 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):
(8)
(Faradays Law)
where B is the magnetic field, A is the area of the surface in
question, and is the total magnetic flux through that area.
The direction of the induced current depends on the di-
rection 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 (I
U1
) and that due to magnetic induction (I
MAG
):
I = I
U1
+ I
MAG
(9)
Ohms law must be modified (extended) to account for
the additional current:
V = I
U1
R + I
MAG
R
(10)
(Extended Ohms Law)
Kirchhoff s voltage law must be extended as well. A com-
parison of equations 1, 7 and 8 shows that KVL is extended
by adding the -d /dt term to the right side of equation 1:
(11)
(Extended KVL)
Equations 2 through 5 can be rewritten to include the
time-dependent magnetic field components:
(12)
CIRCLE 223 on Reader Service Card or freeproductinfo.net/pet Power Electronics Technology November 2004
www.powerelectronics.com
32
CIRCUIT ANALYSIS
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.
Fig. 3.
This audio application circuit demonstrates how magnetic
interference diminishes audio quality.
V
R1
= (I
U1
+ I
MAG
)R
1
(13)
V
R2
= (I
U1
+ I
MAG
)R
2
(14)
(15)
Thus, equations 1 through 5 have been extended to ac-
count 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
Ohms law, with the sign of the d /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 V
AF
= U
1
. But, V
AF
also
equals current in the loop times the two resistances:
V
AF
= U
1
(16)
(17)
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 U
1
=
2 V,
d
dt = 1 V, R
1
= 2 k , and R
2
= 4 k . Then, the loop
current according to equation 15 is 0.5 mA.
V
AF
= 2V
(18)

(19)
V
BC
= I(R
1
) = 1V
(20)
V
BC
= U
1
I(R
2
) = 0V
(21)
V
DE
= I(R
2
) = 2V
(22)
V
DE
= U
1
I(R
1
) = 1V
(23)
V
EF
= I(0O) = 0V
(24)
V
EF
= U
1
I(R
1
) I(R
2
) = 1V
(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 non-
zero. 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-de-
+
A
B
C
F
R1
Magnetic
Field
U1
E
I
R1
Voltmeter #1
V= U1
Voltmeter #2
V= I(R1+R2)=U1+
d-F/dt. d-F/dt
Volume
Control
Spectrum
Analyzer
50
Load
Audio
Signal
Loop
+-
MAX4410
-
+
1uF
10k
R1
10k
100pF
CIRCLE 224 on Reader Service Card or freeproductinfo.net/pet www.powerelectronics.com
Power Electronics Technology November 2004
33
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 mea-
sures 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
Contrary to a common misconception, the induced
voltage is distributed not in the wires connecting the resis-
tors, 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 ex-
periments verify that the voltage drop across the connect-
ing 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-
Fig. 4.
Connecting the ground trace near the 10-k resistor (a) or on
the top of the loop (b) shows that the physical layout of the volume-
control circuit affects magnetic interference.
tion of the probes. Voltmeter #1 acts like an electric field
integrator that integrates the electric field inside the bat-
tery U1, and voltmeter #2 integrates the electric field in-
side R1 and R2. Different integration paths yield different
voltages. The following case further demonstrates this
position-dependent effect.
ONE INCH
COPPER
Audio Signal
Generator
Audio Signal
Generator
Magnetic flux
from nearby
motor
(a)
(b)
1k
10k
1k
10k
To MAX4410
Audio
Amplifier
To MAX4410
Audio
Amplifier
+-
+-
Magnetic flux
from nearby
motor
CIRCLE 225 on Reader Service Card or freeproductinfo.net/pet Power Electronics Technology November 2004
www.powerelectronics.com
34
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 inter