[SI-LIST] Re: Concept of Voltage

Shawn,
There are two possible definitions of voltage. 

The first one is classical with the integral of electric field between two
points - that definition assumes that the field is conservative (no
difference in voltage value if the integration path is changes). This
definition can be applied to lumped elements or TEM transmission lines only
in case if distance between the integration (measurement) points is much
smaller than the wavelength (locally the field is always conservative). It
pairs with the current definition through a surface integral over a
conductor cross-section (or port cross-section).

The second definition is specific to waveguiding structures (or transmission
lines) in microwave theory. Electric field in a cross-section of a waveguide
(or MTL) can be expressed as a sum of eigen-waves with some coefficients.
Those coefficients are treated as the voltages in the theory of multiports.
Technically voltages is this case are Fourier coefficients with the base
functions defined by a set of eigen-modes (each wave has its own voltage).
Currents in the multiport theory are introduced in a similar way through the
projections of magnetic field on the magnetic fields of the eigen-waves.
There are no restrictions on the size of cross-section with such definition
of voltage and current. Both wave-guide ports and local or lumped ports can
be used to define a multiport. This allows to build multi-port models to
combine distributed and lumped structures. Circuit theory can be used to
analyze connections of such multiports without any restrictions as soon as
the voltages and currents are defined identically for the connected ports.

Considering multi-conductor transmission lines, if it is analyzed with a
static field solver, the electric field is conservative by definition and
voltage can be uniquely defined in the model. Though, the model breaks if
the distance between strips or reference conductors becomes comparable with
the wavelength. In case of electromagnetic analysis of multi-conductor line,
the voltage can be defined following the first definition only in case if
cross-section size is much smaller than the wavelength. Such solution will
be identical to the obtained with the static field solver. See more on
estimations of frequency boundaries in I.V. Lindell: On the quasi-TEM modes
in inhomogeneous multiconductor transmission lines, IEEE Transactions on
MTT, vol.29, no.8, pp.812-817, 1981 or Electromagnetic waveguides and
transmission lines By Frank Olyslager. In case if cross-section becomes
comparable with the wave-length, the second or projection definition of
voltage can be used to turn MTL into a multiport. Note that the current
definition through the a conductor cross-section integral stays valid up to
higher frequencies than the conservative voltage definition. This fact can
be used to define voltage through the current and power transmitted by wave.

Finally, at high frequencies only power of propagating waves is measurable.
That is why waves and scattering parameters are used for interconnect
analysis at microwave frequencies. See more on definitions of multiports and
S-parameters in presentation #2010_01 at
http://www.simberian.com/TechnicalPresentations.php (tutorial materials from
DesignCon2010).

Best regards,
Yuriy

Yuriy Shlepnev
www.simberian.com   



-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx] On
Behalf Of Shawn Hermite
Sent: Sunday, July 17, 2011 11:06 PM
To: si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Concept of Voltage

To SI experts, especially those used Maxwell's equations in their books:
The concept of voltage has been deeply engrained in our minds. It's the
foundation of the circuit theory and electrical engineering at large.
Recently, when collecting materials for a presentation advocating the
importance of signal integrity, I realized that the concept of voltage is
based on the assumption of the electric field being conservative. The
general differential form of Maxwell's equations (the Faraday's Law part in
particular) contains the dB/dt term that ruins the validity of curl {E} = 0.

There are two categories of arguments:
(1) With the integral form of the Faraday's Law, we can treat the
time-varying magnetic flux term as an electromotive-force (like battery),
hence fix the KVL, the general idea of voltage is still being the line
integral of the electric field between two points.
(2) Switching to the frequency-domain, the surface integral of the B-field
is related to the characteristic dimension of the system (D) and the
operating frequency. It goes with some hand-waving arguments, and the claim
is that the unfriendly term is nearly zero when D << wavelength. This
argument also goes hand-in-hand with validity of lumped vs. distributed
element modeling.

I have also observed that in a multi-conductor transmission-line (MTL)
system where the TEM mode is propagating, the E-filed is conservative on
each cross-section, thus voltage is well defined between a signal conductor
and the common-reference conductor.  Here, the general electrodynamic
Maxwell's equations are all satisfied. The MTL theory explains why
RF/microwave testing has to be done using co-ax cables. It also makes sense
why 'port' (instead of circuit node) voltage is used in constructing
S-parameters.

If the second explanation holds true, I am really worried about what the
frequency limit is before we get into trouble with freely applying knowledge
developed with DC or low-frequency circuits.

Any way, looking for a better answer or proof on the validity of the concept
of voltage.

Thanks


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