The issue that needs to be understood is that all circuits are all distributed at all frequencies except DC. Ohms law is a DC law, But at low frequencies, resistors behave much as they do at DC and Ohms law can be used. At higher frequencies, the distributed behavior becomes dominant. The same thing happens with very long power lines. Even at as low a frequency as 60 Hz, a very long power line (eg 500 miles) shows it distributed nature as must be treated as such. The underlying reason is that the actual electrical signals propagating in the circuits are electromagnetic waves, not volts and amps, which are induced by the electromagnetic wave. The conductors just act as guides for the waves. So a coaxial cable, for instance, will show Z=75 ohms (or some other Z, depending on the cable), even though its dc resistance is negligible.

At higher frequencies you have to use Maxwell equations, which in addition to boundary conditions require constitutive relations which describe media properties. In general, constitutive relations are in form D=D(E), B=B(H), J=J(E). For linear media, this comes in form D=eps*E, B=mu*H, and Ohms law IN LOCAL FORM J=sigma*E. However, bare in mind that sigma=sigma(position, frequency), i.e it is basically a scalar field which depends on frequency. At lower freqs you can neglect propagation effect, i.e the circuit, media is small compared to propagating wave, and you have sigma = sigma(frequency). In this case it is also simpler to define voltage and current so we arrive at Ohm law. At higher freqs basically you don't have the ability to define voltage in a simple way.

In continuation of what is being said : EM waves travel or energy transfer at such high frequency is not largely in the bulk but on the surface (skin effect) due to which the resistance is reduced (R=rho*l/A). The transfer of energy is based on charging and discharging of electro-magnetic field and the storage elements are capacitance and inductive respectively (the material property). For the signal that propagates it seems like a series of {CLRG} that are distributed.

Ohm's law for lumped elements derives from the more general Ohm's law definition of J=sigma*E.

Thus, no law is broken at high frequencies. At these frequencies the wavelength becomes small (lamda=c/f), and the fields, voltages, and currents along transmission lines vary (having minima and maxima at each multiple of lambda/2). The impedance U/I you see at one point will therefore be different from the impedance you see at another point. Ohm's law is still valid, the unconventional issue is that we are used to ONE value for currents, voltage, impedance on a line, and not a spatially dependent distribution.

Simone Winkler:

In Ohm's circuit law V=R.I, voltage is assumed to be constant, but at very high frequency voltage is vary due to propagation effect so we can' apply V=R.I and ohm's field theory is still valid because in this law voltage source isn't assumed to be constant...( This i conclude from above discussion)

My another doubt is " microprocessor having speed greater than 4 GHZ is not possible to design and the reason behind this is breaching of V= R.I relation at this frequency ? "

Nandan,

1) No, Z=U/I can still be applied.

All it is is that it isn't a constant quantity anymore. At position z=3 cm into the wire, the impedance Z will generally be different compared to position z=5 cm. In fact, Z will also vary in the same way as the voltage and currents according to the wavelength, with maxima and minima along the line.

2) Microprocessors at 4 GHz:

Here is what we do at low frequency when we connect two devices:

Design the output of the first device at the lowest possible output impedance, and design the input of the second device to have a very high input impedance. That way, there is almost no voltage drop across the output impedance of the first device, and almost all the voltage will be available to the second device without drawing much current from the output of the first device.

At high frequency, this concept does not work. As soon as you have an impedance discontinuity between two devices, a (usually very large) part of the signal (remember, the signal is now a wave) will be reflected, just like a water wave would be reflected if it hit a concrete wall. This way you can never get continuous power flow across a low-high impedance boundary. For high frequency systems, what you do instead, is design all ports to have the same impedance (usually that is 50 Ohms). This way, there are no reflections, and the wave can pass without disruption through a chain of devices. This HAS to be done, but also complicates certain things, and I won't go into detail in order to avoid confusion.

Microcontrollers have high-Z inputs, therefore, if you hit them with a high-frequency signals, you would get the aforesaid reflections. You would then only have part of the signal get through to the microcontroller. You will not be able to tell how much unless you know your input device very well, as well as the length of the cable that connects this device to the microcontroller, and so on. In general, because the ratio between output and input impedances for low frequency devices like microcontrollers is so high, almost nothing will get through.

Another complication is that at high frequencies, because we now have waves, these waves can couple to the air above the conductor and can propagate as a wireless signal. With low frequencies this doesn't happen because you would need a conductor that is a kilometer long. Now if these waves are radiated into the air, they can also be picked up by other conductors in the circuit and then interfere with the signal on that second conductor. That is called coupling, cross-talk, or similar and is usually undesired.

Hope this helps.

Nice explanation Simone. Also at such high frequency PC BOARD design is a real art as wave propagation is also affected by surrounding material properties.

As usual, the answers focus on EM theory, but the question concerned the validity of Ohm's law in a conductor, which is very much a solid-state system. J = sigma * E does have to be modified at very high frequencies, due to the inertial mass of the charge carriers (which are generally electrons). In a very simple, but generally accurate, model the inertial lag in electron motion can be interpreted as an inductance in series with the Ohm's law resistor. Note that this inductance has nothing to do with magnetic fields, only with carrier mass. If you evaluate the resistance R and inertial inductance L for a small volume of conductor, the L/R ratio is equal to the mean time between scattering events, which for semiconductors is usually within the order of magnitude centered on 100 fs. (The analysis of metals is complicated by Fermi degeneracy, which prevents all the electrons from participating in the dissipative processes) Thus, these inertial effects will show up around 1 THz; the normal EM distributed -circuit analyses are quite adequate at lower frequencies.

For a discussion of carrier inertia, see Champlin, Armstrong and Gunderson, Proc. IEEE 52, pp. 677-85 (1964).

Wiiliam you are correct but it is a case of "horses for courses" and what the original question meant by "high frequency" and which was apparently classified as dc-GHz.

But then I suppose 1000Ghz is a 1 THz :)

All math used in engineering, including V=IR, are approximate models of a real phenomena. This particular equation gives good answers when the frequency is low enough. The approximation, however, loses its accuracy as the frequency goes up, and more complex math models need to be used. Ohm's law is always there and is never breached. You just have to use the appropriate model for every situation.

Situating the problem can be done as follows in my opinion:

Suppose you have a regular 50Hz AC network. Such a network has a wavelength of lambda = c / f or about 300000km/s / 50 = 6000km. Now, what does this mean? Suppose you draw a infinitely long line and connect it to 50Hz AC. Then at 6000km down the line you would measure identical voltage at any given time instant as at the beginning of the line.

The question to be asked now is: What happens in between?

Well, suppose you measure all voltages at a single time instant, eg. when the beginning is at 0V. Then the voltage accross the line equals a sine wave with a period of 6000km. Hence at 3000km you get zero. At 1500 and 4500km you get the maximum respectively minimum.

The behavior sketched above is what actually happens in a transmission line. Not all parts of the line are at equal voltage at any given time instant. In order to avoid having these issues you should have short lines in comparison to the wavelength of the signal. Going back to our 50Hz AC line: If you measure the voltage at the beginning and at 1m, then they will be as good as identical.

It is typically assumed that a wire length smaller than 1/10th of the wavelength results in lumped element behavior (ie. the beginning and the end of the line are assumed to be on identical voltage).

So in essence your 4GHz limit allows for wire lengths of up to 0,75cm to be considerable as lumped elements, above them you need to consider it a transmission line.

I should mention that the speed of electricity in copper or semiconductor is lower than the speed of light, more towards 200000km/s instead of the used 300000km/s. So, adjust the values accordingly.

Further study on the subject can be found on courses on transmission lines and potentially wave propagation (aka understanding Maxwell's equations)

Hope this helps you forward

Lieven thanks it all makes good sense, similar to Simone re lumped / distributed, reflections etc.

As Mehrdard correctly notes resistance is used to model heat generated in a material due to the presence of EM fields penetrating the material (skin effect)

When a material becomes superconducting internal EM fields are expelled and no resistance or heat loss results. Maybe William can comment.

Ohms law (or if you want, the concept of linearity between voltage and current) is valid over a very huge dynamic range (more than 30 orders of magnitude) for all metals from DC to beyond optical frequencies.(>500 Thz). However the term "voltage " is often not clearly defined for anything higher than DC and could/should be replaced by Intergal Eds where the the value of this Integral is (in contrast to DC, as scalar electric potentials are not valid above DC) not independent of the path of integration. These caveats are covered by the wave approach and by most numerical codes dealy with high frequency simulations. It should be recalled that the electrical conductivity of metals is practically constant to somewhere around 300 THz (depending on the metal).Above all metals become a more or less lossy dielectric. In practice the surface roughness starts playing an increasinly important role for frequencies above 100 Mhz for the metallic losses, but again this is all still linear (thus Ohms law applies)

Frtiz I think the discussion concurs but the method of analysis differs or must be modified at higher frequency is what appears to differ.

Lumped element models can only be used if the wavelengths

of the signals are much greater than the size of the circuits

The issue that needs to be understood is that all circuits are all distributed at all frequencies except DC. Ohms law is a DC law, But at low frequencies, resistors behave much as they do at DC and Ohms law can be used. At higher frequencies, the distributed behavior becomes dominant. The same thing happens with very long power lines. Even at as low a frequency as 60 Hz, a very long power line (eg 500 miles) shows it distributed nature as must be treated as such. The underlying reason is that the actual electrical signals propagating in the circuits are electromagnetic waves, not volts and amps, which are induced by the electromagnetic wave. The conductors just act as guides for the waves. So a coaxial cable, for instance, will show Z=75 ohms (or some other Z, depending on the cable), even though its dc resistance is negligible.

Hi Dave nice to hear from you hope you are keeping well. This discussion appears truly NORTH, SOUTH, EAST and WEST represented :)

Sorry but for others interest Dave taught me in my undergraduate years in the early eighties one aspect of which was circuit theory.

Maybe answer the question like so:The ohm‘s law is now demonstrated to be correct in many frequencies，but there are different shows with different frequency ranges for a identical conductor. u=zi there u,i,and z all vary with frequncies,there will be a skin effect on the surface of the conduct when it work at high frequencies. "the surface roughness starts playing an increasinly important role for frequencies above 100 Mhz for the metallic losses"(see above Fritz's answer).when the frequency is more than 4Ghz there need a special measuring way to verify the ohm's law. it is impossible to prove it using traditional method.

Ohm's law is applied to lumped problems where R is a constant and current is also not dependent from the position in the cable where you are measuring. Electromagnetic theory is applied normally to problems when wavelength(m)= 300 / f(Mhz) is in the range of the problem size. Typical PCB size is 10cm, the frequency range and beyond where you problem can not be considered as lumped model is: f(Mhz) =300/0.1m=3GHz. In this case J= σ.E is applied in conjunction with the other Maxwell's equations, but not alone.

Ohm's law is J=sigma X E, which is a basic law of electromagnetics, and holds true for all frequencies and conditions. J is the current density in Amps/square meter, Sigma is the conductivity of the medium in Siemens, and E is the electric field imposed in Volts/meter. In most general case, J and E are vectors and sigma is a 3X3 matrix. The other equation referred to here is V=I X R is a limited derivation of Ohms law that applies to DC and low frequency circuits with a number of conditions. So, the assertion that Ohms law breaks at high frequencies is incorrect.

100% agree with Mehrdad. Law that I referred that breaks at high frequency is V=I X R not Ohm's Law. But I insist that Ohm's law can not be consider as ISOLATED LAW to solve high frequency problems as it is considered at DC or low frequency problems. At high frequency you have to consider Ohm's law with the other Maxwell's equations to solve the problems.

The answer that we have to consider Ohm's law together with Maxwell's equations

is correct. We recently wrote a paper on this issue: Ruehli,Antonini,Jiang,"Skin-effect loss models for time- and frequency-domain PEEC solver", IEEE Proceedings, No. 2, vol. 101, pp. 451-471, Feb. 2013 Essentially, Maxwell's equations are transformed into the circuit domain and Ohm's law is part of the result.

Actually ohm's law is applied to lumped problems where R is a constant but at high frequencies impedance varies due to variation in inductor's reactance as XL = 2πf,

Reading previous comments much of what has been expressed since SURENDRA is a repeat or confirmation of what was said earlier. Skin and surface effects dominate at very high frequencies as noted by Albert Ruehli.

David Levy of University of Sydney has the best explanation to the original question. Good work!

These are very important issues and they confuse a lot of electrical engineers. Simple reason for this is that EE majors now days hardly learn any physics or electricity and magnetism. First of all one should note that when conductivity is analyzed at DC or low frequencies, the inertia term in electron momentum equation is ignored; the electrical force on the electrons is balanced by the friction resulting from colliding current-carrying (free) electrons with the lattice structure in the metal. This picture is not valid at high frequencies; time derivative term in the electrons' momentum equation becomes important. This term is proportional to frequency, thus conductivity can become frequency dependent.

Another related issue here is that the electrical and magnetic phenomena get increasingly coupled as frequency increases. This follows from Faraday's and modified Ampere's laws; time varying magnetic field produces electric field ( EMF) and time varying electric field produces magnetic field. A simple thumb rule is that when size of a electric or magnetic system (circuit) becomes comparable to or smaller than the wave length ( lamda = c/f), coupling becomes significant. In such cases, a resistor can develop large inductive impedance at high frequencies. An inductor develops capacitance due to the electrical coupling between turns of the coils in the inductor. The displacement current in a capacitor becoes large enough to generate magnetic field, contributing a inductance to the structure meant to be capacitance at low frequencies.

I must mention that at high frequencies the well known skin effect becomes important making the inner part of the conductors useless for carrying currents; this effect also results from the coupling of the electrical and magnetic phenomena.

As electrical system are getting smaller and smaller, and faster and faster ( higher and higher frequencies), it is important that all EE and also computer engineers must begin to pay attention to fundamentals of electricity and magnetism, and not simply rely on 'soft ware packages' coming to them as black boxes.

Thank you Nagendra Singh for a very basic and lucid explanation......and i felt throughout my engineering course that we are partial to science.

I had a little error in my narration while describing the coupling between electric and magnetic phenomena. It should have read as follows:

A simple thumb rule is that when size, L,of a electric or magnetic system (circuit) becomes comparable in the order of magnitude or larger than the wave length ( Lamda = c/f), coupling becomes significant:

The low frequency circuit concepts are valid if Lamda >> L. ( Circuit theory, KCL, KVL, usual Ohm's law ). For example at 60 Hz Lambda is as large as the distance between LA and New York ( 5,000 km) and circuit size L ~ 1 m)

High frequency effects kick in when L ~ or > Lambda (EM theory, Lambda is in tens of centimeters or smaller)

Pure conductor, like a pure dielectric, are idealizations. At very high frecuncy not only real Ohmic current, but also Maxwell displacement current run trough the metall. So, conductor serves as a cappacitor as well at very high f.

Just to clarify, it is not the amount of power flow from one device to another which is important in a high-frequency microprocessor system, it is the time taken for the voltage to stabilise to its final value, and this is why impedance mismatched causing reflections are undesirable because they slow the system down and introduce noise which can cause the system to crash. This is why point-to-point communication such as SATA are favoured in modern PCs rather than the old IDE bus (which typically required an external block of terminators, only partially effective). In an analogue system such as a radio transmitter, components are frequently covered in a layer of silver to reduce the skin depth (because of its low resistivity). Also high frequency is a relative term, in a power station at 50Hz the conductors are so thick they have to be made in an H cross-section to minimise losses due to skin depth. Another application is very low frequency (VLF) communication with nuclear submarines, the skin depth in sea-water for radio signals is much less, the higher in frequency you go.

One should to use the telegrapher equations for distributed line with distributed capacitances, inductances etc.

As Physicists know, Ohm's and Kirchoff's expressions are just simplified (thus more practical) cases of Maxwell's equations for very small orders of magnitude of the involved physical "concepts". All the rest "phenomena" and techniques researched and applied by electronic engineers to solve "problems" of energy control/transmission of a "signal" are just specialized simplification "tricks" to make calculations more accurate and easier for specific practical problems that arise along the way (innacurate estimations) when you avoid to apply Maxwell in the first place. But, that's the way technology progresses faster... by applying simplifications of limited validity... considering that applyers know the limits of validity.

On High frequency an electromagnetic field will be buildup across the conductor, and strength of the field is directly proposal to frequency so... this thing will be happens..

There are a combination of things going on. The first is the so-called "skin effect". At high freqs, due to magnetic induction, current concentrates in the outer skin of the conductor. Thus the area over which the current can flow in the conductor decreases, which increases the resistance.

The second phenomenon is series inductance. Every conductor has a minute amount of series inductance associated with it. The impedance (resistance) of an inductor increases linearly with frequency. At low freqs, this inductance has a negligible effect, but becomes significant at high (GigaHz) frequencies.

I have just landed here and quickly I have had a llok to the answers to the original question. Three basic explanations have been provided:

1) Skin effect.

2) Distributed circuit effect.

3) Dynamic of electrons at very high frequencies.

All those things can happen (even together) but they start to be obvious over different range of frequencies. Thus, skin effect (the current distribution inside the conductor is no more uniform, as in DC, due to Faraday's law) appears even at relatively low frequencies (say, MHz and above). Current density tends to concentrate close to the surface of the conductor thus increasing the resistance per unit length and making relevant the inductive part of the impedance. In this situation V = Z * I , in the complex domain, of course, with Z being a complex impedance accounting for a frequency-dependent resistance and a frequency-dependent inductance. The important parameter here is the skin depth (for metals, delta = \sqrt{ 2 / omega mu sigma}. If delta (skin depth) is large in comparison with the cross sectional dimensions of the wire, the conventional Ohm's law can be applied. If delta starts to be comparable to those dimensions, skin effect becomes relevant. In the full-developed skin effect regime, the current flows througha thin layer (microns) in teh surface and resistance and reactance per unit length are identical.

A different thing is the distributed nature of the problem involving wires at high frequency. Distributed models based on full Maxwell equations, taking into account wave propagation along the wires, are necessary provided the LENGTH of the cable is not a tiny portion of the wavelength at the operation frequency. Usually the skin effect becomes obvious at frequencies well below the distributed nature of the problem has to be taken into account. However, for the long power lines (500 miles long) mentioned above, you might work in a regime where skin effect is negligible (current flows uniformly within the conductors) BUT propagation effects along the wires have to be taken into account.

Finally, electron dynamics inside a conductor becomes "special" (in such a way that the conductivity relating electric field and current density can not be considered a real quantity) only in the proximity of optical frequencies (infrared and above). The important parameter here is the average time between collisions of electrons. In DC and low frequencies (in this context 1 THz is low frequency) this time is much smaller than the period of the sinusoidal signal and the electron dynamics is dominated by collisions. In such situation the average speed of electrons is proportional to the electric field (i.e., to the actuating force). We are ignoring inertial effects!. However, when the time interval during which the electrons "freely fly" is not negligible in comparison with the signal period, macroscopic current is no more "in phase" with the applied electric field. However, there is still a linear relation between excitation (electric field) and response (current density) that can be account for through a complex frequency-dependent permittivity (Drude model for metals at optical frequencies).

I hope this can serve as a summary of the interesting comments above. Thanks.

The best answer was Erik Lindberg's::

"Lumped element models can only be used if the wavelengths

of the signals are much greater than the size of the circuits."

His answer is short and sweet, but says it all.

Therefore, every time you halve the size of your circuit, you get another octave of frequency range for ohms law (and vice versa). Rule of thumb is to keep your circuit size < 0.1 wavelength. This is why 3 GHz CMOS uP work in which one transistor drain load can drive the next stage transistor's gate. It's all about size...size of circuit and size of waves or speed of risetime. Make the circuit small enough, and within that you can still use many LF techniques....and not even skin effect is a worry. Of course you are not going to see microwave power amps done this way because the smaller you make the circuit the less power it can handle.

In the expression J= σ*E, E is axial intensity. It is not usually underlined because that in the most practical cases we have relatively little (or zero for DC) frequency when the intensity of electrical field is assumed to be an axial one. Increasing of frequency leads to increasing of field's radial component and your conductor became operate as antenna beginning from some frequency.