Why at microwave frequencies the simple voltage and current equations not applicable? Why we have to consider the waves associated with the currents?
Well, the concepts embodied by Ohm's laws don't completely lose validity. ... Ohm's laws can be drived directly from Maxwell's equations (of course), but only with the principal assumption that the electronic component has 'zero' physical extent. When the wavelength of the EM fields approaches the scale of the individual components under analysis, those assumptions no longer apply. For example, a simple carbon resistor has dimensions on the scale of millimeters, or even centimeters (eg: power resistors). If the resistor has a significant "electrical length" as compared to the wavelength (typical at microwave frequencies), the EM fields in and around the resistor are no longer homogenous across the component. That is, one part of the component may be at a 'peak' in field strength, when another part might be at a minimum, or even the opposite polarity. Clearly, Ohm's laws do not consider that situation. Now, what about wave guides and free space transmission? There's no 'component' there at all, just free space. The concept of "Ohm's law" might still be applied (though with careful constraints). Consider Power = V^2/R. In the case of waveguide transmission, one can still calculate the RMS field strength via E = sqrt(P/R) where P is the power through a certain area and R is the free space impedance. Note, however that this is not calculating the instantaneous field strength. Nor is this calculating the spatial distributions of the fields (both E and H). For these, one must resort to Maxwell's equations.
Did you mean Ohm "law" in the form
Voltage = Resistance*Current ?
Well, the concepts embodied by Ohm's laws don't completely lose validity. ... Ohm's laws can be drived directly from Maxwell's equations (of course), but only with the principal assumption that the electronic component has 'zero' physical extent. When the wavelength of the EM fields approaches the scale of the individual components under analysis, those assumptions no longer apply. For example, a simple carbon resistor has dimensions on the scale of millimeters, or even centimeters (eg: power resistors). If the resistor has a significant "electrical length" as compared to the wavelength (typical at microwave frequencies), the EM fields in and around the resistor are no longer homogenous across the component. That is, one part of the component may be at a 'peak' in field strength, when another part might be at a minimum, or even the opposite polarity. Clearly, Ohm's laws do not consider that situation. Now, what about wave guides and free space transmission? There's no 'component' there at all, just free space. The concept of "Ohm's law" might still be applied (though with careful constraints). Consider Power = V^2/R. In the case of waveguide transmission, one can still calculate the RMS field strength via E = sqrt(P/R) where P is the power through a certain area and R is the free space impedance. Note, however that this is not calculating the instantaneous field strength. Nor is this calculating the spatial distributions of the fields (both E and H). For these, one must resort to Maxwell's equations.
I agree on the state by Prof. Patrick Bisson , but i like add the following note
at microwave frequencies where it becomes difficult to measure voltages and currents directly. On the other hand, incident and reflected power are easy to measure using directional couplers.
Patrick explanation is pretty fine. I want to add that you can "always" (when working with linear systems, so the superposition principle holds) have the relation
Voltage(omega) = Impedance(omega)*Current(omega)
for a single frequency. Where the voltage, the impedance and the current is a global property of your device but not an intrinsic one of the material. So, if your not interested in the distribution of the electric field and current density in your problem this is fine for you. But you can always compute the global variables computing the integrals of the electric field and current densities and relate those two variables to compute the impedance.
In some sense, it is like the difference between the mass density of a body subjected to gravity (and force densities) and the mass lumped in the center of mass and the total force m*g.
Regards,
Nicolas
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