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MEASUREMENT CIRCUITS
Keywords: supress magnetic interference audio pcb design time-varying magnetic fields circuit analysis
Mar 06, 2006
APPLICATION NOTE 3651
Circuit Analysis in the Presence of Time-Varying Magnetic Fields
Abstract: Power sources such as transformers, electric motors, and the inductor in a switching regulator
produce time-varying magnetic fields that can interfere with electronic circuitry by inducing noise.
Understanding this process allows the designer to better suppress magnetic interference.
Ohm's law and Kirchhoff's voltage law (KVL) are powerful tools for conventional circuit analysis (mesh analysis).
If, however, time-varying magnetic fields are present in a circuit, Faraday's law must be invoked as well. To
account for the additional current induced by a time-varying magnetic field, a term must be added to Ohm's law
and KVL. Introducing 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
the position of the voltmeter leads.
Circuit Analysis Without Time-Varying Magnetic Fields (a Review)
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 (Equation 1).
Consider the circuit of Figure 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
(Equation 2). (For the counter-clockwise direction chosen, note that voltages across the resistors are negative.)
Figure 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.
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You can solve for the component values in Figure 1 by 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.
Note that KVL can be written in integral form. Textbooks on electromagnetic theory define voltage as the vector
integral of the electric field (E) along a path (dl), like that from node A to node C in Figure 1 (Equation 6). With
the magnetic field off, the closed-loop integral from node A to C to E and back to A equals zero (Equation 7).
Thus, Kirchhoff's voltage law can be written in integral form: the closed-loop integral of the electric field is equal
to zero.
Circuit Analysis in the Presence of a Time-Varying Magnetic Field
Now, turn on the magnetic field in Figure 1. The field varies with time, which induces current in the loop, and
that condition requires use of Faraday's law. Faraday's law states: 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 (Equation 8):
where B is the magnetic field, A is the area of the surface in 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 Figure 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 both to the battery (IU1) and to magnetic induction (IMAG):
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Ohm's law must be modified (extended) to account for the additional current:
KVL must be extended as well. A comparison among Equations 1, 7, and 8 shows that KVL is extended by adding
the -d Φ/dt term to the right side of Equation 1:
Equations 2 through 5 can be rewritten to include the time-dependent magnetic field components:
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 KVL and Equation 15 is the extended Ohm's law, with the
sign of the d Φ/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 Figure 1 circuit with time-varying magnetic field. The
voltage across U1 (nodes A-F) is VAF = U1. But, VAF also equals current in the loop times the two resistances:
We now have two possible voltages for nodes A-F. In fact, there are two possible voltages for each pair of nodes
that include a component in Figure 1. See Equations 16 through 25. For a simple comparison, we arbitrarily set
U1 = 2V, d Φ/dt = 1V, R1 = 2kΩ, and R2 = 4kΩ. Then, the loop current according to Equation 15 is 0.5mA.
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Node B-C is especially interesting, because the current through R1 is non-zero, yet the voltage across it could be
zero. Similarly, node E-F is a short (zero ohm) wire, yet the voltage across it is non-zero. So which voltage is it?
The math is correct. 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 an electric
field along a given path. If a time-dependent 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 Figure 2, voltmeter #1 measures nodes A-F from the left-hand side, obtaining a measurement of
U1 = 2V. In contrast, voltmeter #2 measures nodes A-F (B-E is same as A-F) from the right-hand side, with the
result:
Figure 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.
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
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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. Note that the voltmeter probes are arranged to
minimize interference from the magnetic field.
The measured voltage appears to depend on the position of the probes. Voltmeter #1 acts like an electric field
integrator, integrating the electric field inside the battery U1. Voltmeter #2 integrates the electric field inside R1
and R2. Different integration paths yield different voltages.
We can demonstrate this position-dependent effect with another example. Consider Figure 3, in which an audio
signal (1kHz 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 1kΩ and 10kΩ
resistor connected in series; the loop that catches the magnetic flux was purposely enlarged to one inch square.
Two physical board layouts were tested (Figure 4).
Figure 3. This audio application circuit demonstrates how magnetic interference diminishes audio quality.
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Figure 4. Connecting the ground trace near the 10kΩ resistor (a) or on the top of the loop (b) shows that the
physical layout of the volume-control circuit affects magnetic interference.
Figures 5a and 5b show plots of the audio-amplifier output spectra. The 1kHz audio test tone is identical in
both cases, but the amplitude of 300Hz motor interference depends only on the ground connection. The worst
magnetic interference (-62dBc) is based on Figure 4a, in which the audio amplifier receives interference voltage
from the 10kΩ resistor (Figure 5a). In effect, the audio amplifier acts as an electric field integrator that
integrates the electric field inside the 10kΩ resistor. On the other hand, Figure 5b (the output spectra based on
Figure 4b) shows an interference voltage received from the 1kΩ resistor. The lesser interference shown in this
plot (-78.5dBc) represents an improvement of 16.5dB. (The expected interference ratio is 20dB, because the
resistor ratio is 10:1. However, the loading effect of the audio amplifier's input impedance lowers the
interference amplitude in Figure 5a.)
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Figure 5a. Magnetic interference from an electric motor in Figure 4a causes a peak at 300Hz, about 62dBc below
the audio test tone.
Figure 5b. Magnetic interference from an electric motor in Figure 4b causes a peak at 300Hz, about 78.5dBc
down from the audio test tone—an improvement of 16.5dB over the circuit of Figure 5a.
This phenomenon has been verified experimentally with a mathematical derivation of the two voltages¹
presented in Appendix A. 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
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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 path- or position-dependent, integrating along a different path yields a
different voltage. As Equations 9 through 15 in the previous section do not clearly predict the position-dependent
effect, they must be used very carefully.
PCB Layout in the Presence of Magnetic Interference
To repeat the conclusion above, magnetic interference from components such as electric motors and power
inductors in switching supplies can cause noise in the system. A good PC-board layout can minimize this
interference.
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Rule #1: Keep magnetic components far away from noise-sensitive circuits.
Rule #2: Keep the electric components (ICs, resistors, capacitors, etc.) that are part of a circuit loop
close together to minimize the area of the loop.
Rule #3: Use the analysis shown above to identify a ground connection that yields the least magnetic
interference.
Rule #4: If the ground connection in Rule #3 cannot be easily identified, use a solid ground plane
instead. Experiments show that a solid ground plane results in low magnetic interference.
Conclusion
One can safely apply Kirchhoff's voltage law and Ohm's law in circuit analysis if no time-varying magnetic fields
are present. But if such fields are present, you must extend KVL and Ohm's law with the help of Faraday's law.
As demonstrated above, the presence of a time-varying magnetic field produces two simultaneous voltages
across a pair of nodes. The dual-voltage effect appears to depend on the position of the voltmeter probes, which
makes the term "voltage" ambiguous. Voltage between two nodes is no longer expressed in a simple algebraic or
numerical form, but as a complex vector integral. It is also path dependent. Thus, understanding how magnetic
fields induce noise in a circuit can help a PC-board designer place the components to minimize magnetic
interference.
Appendix A
For simplicity, we set U1 of Figure 1 to zero. (U1 need not be set to zero, however, for the following discussion to
be valid.) Figure 1 is redrawn as Figure A1, with voltmeters attached and a time-varying magnetic field present.
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Figure A1. This circuit shows how a time-varying magnetic field induces two different voltages at nodes A-B, as
measured by the two voltmeters.
Voltage across the resistors in Figure A1 can be found using Faraday's and Ohm's law. We defined voltage as the
vector integral of the dot product of electric-field intensity (E) and path dl from node A to node B, along path C.
From Figure A1, Equations A1 and A2 depict V1 and V2.
Subtracting V2 from V1 gives Equation A3. By changing the integration from B to A in the right-hand term
(instead of A to B) and indicating the consequent change of sign, Equation A3 becomes Equation A4.
The right-hand side of Equation A4 is the line integral of electric field once along a closed loop enclosing the
magnetic field (indicated by the flux density B). From Faraday's law, Equation A5 is equivalent to Equation A4.
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It therefore follows that:
where A is the area bounded by the loop, and Φ is the total magnetic flux through that area. For simplicity, we
assume that the magnetic field increases linearly with time, so d Φ/dt = α.
The current through both resistors is the same, and Ohm's law relates the voltage drops and current to Equation
A7. Notice that the integration is in the same direction as the current. Equation A7 is rearranged to Equation A8
by changing the integration from node A to B on both sides, and adding a minus sign to the R1 term.
Because C1 and C1' form a closed loop that does not include the magnetic field, integration along path C1 is the
same as integration along C1'. Similarly, C2' can be replaced with C2. Then, substituting into Equation A8 the V1
and V2 expressions from Equations A1 and A2, produces Equation A9. Finally, solving the simultaneous equations
A6 and A9 yields the results we are looking for—voltages V1 and V2 (Equations A10 and A11).
Note that the polarity of V1 and V2 are opposite. Again, the voltage across a resistor is the integral of the electric
field along a path. If d Φ/dt ≠ 0, then the integration is path dependent. That effect is a consequence of what is
called a non-conservative electric field. Integrating the electric field from node A to B along path C1 (Figure A1)
gives a value different from that obtained by integrating along path C2. Thus, the measured voltage depends on
which path the voltmeter "sees."
Reference
¹Robert H. Romer, "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.
Application Note 3651: www.maxim-ic.com/an3651
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