lumped circuits and trasmission lines

[Jtwain80]

Hi guys, I wanted to know the difference
between voltage and current in lumped circuits and
the current and voltage in transmission lines as in
coaxial cables

thanks

[Cribald]

Hi Jtwain80,

For transmission lines has no sense speak of voltages and currents but rather it need speak about waves of voltage and waves of currents.
When you solve a circuit like an audio amplifier or an operational amplifier circuit, applying the kirchoff laws and other electrical laws, implicitly you are assuming that the circuit is working under the
condition of lumped circuits. What does this mean? it means that considering the work frequency, the relative wavelength is very large compared with circuit dimensions. So you can assume that the change of electric signal (voltage and current) propagates instantaneously along the circuit.
Actually when the frequency rises and the wavelength become comparable with the circuit dimension, this approximation is no longer valid, now we are in the domain of distributed circuits and transmission lines. A little piece of coax cable (actually infinitesimal) or microstrip behave
like a little capacitor and a little inductor, that we could write as C*dx e L*dx , where x is the longitudinal lenght of cable. Recalling the constitutive equations of capacitor and inductor easily we get probably the most famous equations for e.m.fields engineers after the Maxwell equations,
known as the telegrapher’s equations:

Combining these equations we get two wave equations.
The solution of this equations is the sum of a forward wave and a backward wave.
If a voltage pulse start from source a voltage pulse and then a current pulse travel along the transmission line, they can be partially or completely absorbed by a load, or partially or completely reflected, etc.
If the source start from the middle of a trasmission line, a wave travel long x and a wave long -x.
The constanto Zo = SQRT(L/C) [Ohm] is called the characteristic impedance of the transmission line an is the ratio of voltage to current for the traveling waves.L [H/m]and C[F/m] depend on the physical characteristics
of line.

[Jtwain80]

thanks for your answer, cribald. transmission lines can be simulated with PSpice?
There are models of library or you have to create a macro?

[Cribald]

No macro. There are three types of models in PSpice:
T, that is the easiest way to model a loseless transmission line,
you have to insert characteristic impedance Zo value and
the electrical lenght TD (for example if the signal velocity in a cable is V= 0.75*c, where
c is the speed of light, TD=20ns it means that 0.75*c*20ns= 4.5mtr ).
Often Zo is known, or for a coax cable you can find out as SQRT(L/C) where L and C are provided
by vendor or as Zo= 60/SQRT(εr) * ln(b/a) where εr is the dielectric constant, a is the diameter of the inner conductor and b is the diameter of the inside surface.
Tlossy models a lossy transmission line and a loseless line
with R=G=0. For a coax cable R is modeled as Rs/6.28 *(1/b+ 1/a), where Rs is the
skin effect resistivity [Ohm], the G parameter is 6.28* σ/(ln(b/a)), where σ is the conductivity [Siemens/m].
At last we have the Tlumped model.This is made by a finite numbers of sections of cable and can be used when
TR/TD >=4 in the single section.(TR is the rise time of the pulse). More are the sections and more accurate is the result.

[Jtwain80]

Many thanks,
Your explainations were really useful !

[Author]

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