Can Maxwell's Method Be Used for High Frequency Circuit Calculations?

[mertcan]

Hi, initially I have seen that when circuits are exposed to high frequency kirchhoff laws become invalid. According to my search I can not find any derived circuit equations for that case, instead computational electromagnetism method is suggested using maxwell. Is it the only way to make calculation while high frequencies exist? Or any other methods exist as we have in low frequencies ?

 

[Cryo]

There are plenty of laws, all derived from Maxwell's laws under certain approximations :-), but the simple fact is that when the size of your device/apparatus/probe is comparable to wavelength, the physics becomes more complex (and more interesting)

What is the frequency range? What is application?
Have a look at theory of guided waves, transmission line theory, waveguides etc. IMHO. There are a lot of approximations and useful techniques developed for this field.

 

[mertcan]

There are plenty of laws, all derived from Maxwell's laws under certain approximations :-), but the simple fact is that when the size of your device/apparatus/probe is comparable to wavelength, the physics becomes more complex (and more interesting)

What is the frequency range? What is application?
Have a look at theory of guided waves, transmission line theory, waveguides etc. IMHO. There are a lot of approximations and useful techniques developed for this field.

Thanks for return, when I endeavor to dig information out of Internet I can not find valuable things could you share me nice sources book files or videos?

 

[mertcan]

There are plenty of laws, all derived from Maxwell's laws under certain approximations :-), but the simple fact is that when the size of your device/apparatus/probe is comparable to wavelength, the physics becomes more complex (and more interesting)

What is the frequency range? What is application?
Have a look at theory of guided waves, transmission line theory, waveguides etc. IMHO. There are a lot of approximations and useful techniques developed for this field.

Besides, I have seen that transmission line theory applies kirchhoff laws altough at high frequencies kirchhoff laws can not be applied. Could you explain this contradiction??

 

[Cryo]

Besides, I have seen that transmission line theory applies kirchhoff laws altough at high frequencies kirchhoff laws can not be applied. Could you explain this contradiction??

Not quite sure what you mean here. Do you mean that the common way to derive the telegraf equation is to introduce distributed inductance/capacitatne/resistance and then talk about a small section of the transmission line as if we were dealing with low-frequency circuits? Well this bit makes perfect sense. They start with a transmission line which is small in one direction (thickness), but large in the other (length-wise). The size here is relative to wavelength of the radiation. Then they isolate a small section of the transmission line, which is now small in both directions, so the low-freqency Kirchhoff's laws apply. Finally they link these small bits together and take a continuum limit. If you want to discuss this point (telegraf equation) you need to be more specific, i.e. give the derivation and ask about specific parts you don't like.

 

[Cryo]

Thanks for return, when I endeavor to dig information out of Internet I can not find valuable things could you share me nice sources book files or videos?

I don't work with guided waves myself. Most of my work is with propagating free-space waves, but I can suggest Pozar's "Microwave Engineering". Maybe experts around here will give better suggestions.

 

[mertcan]

@Cryo hi again I just would like to express I can not establish any connection between size of device being comparable to wavelength and kirchhoff law conservation. Could you help me understand better?

 

[tech99]

@Cryo hi again I just would like to express I can not establish any connection between size of device being comparable to wavelength and kirchhoff law conservation. Could you help me understand better?

If you imagine a network consisting of just one resistor at the end of long wires, at high frequencies the distributed L and C along the wires causes the wires to act as transmission lines. The resistance is then transformed to a totally different impedance at the opposite end of the wires. For this reason, complex networks which are physically large are a problem at high frequencies. I have not had to tackle a network of this sort before, but I think an RF bridge would be an example where Kirchoff might be necessary. For the most part, networks are ladder or tee types, where Kirchoff is not necessary.

 

[vanhees71]

The intuitive answer is that Kirchhoff's Laws of AC circuit theory are derived using the quasi-static approximations (approximations, because two different ones are involved), i.e., in conductors (Ohmic resistances and coils) the "displacement current" is neglected (magneto-quasi-static approximation), approximating the Ampere-Maxwell Law of the full theory by the magnetostatic Ampere Law. In the space between capacitor plates the displacement current cannot be neglected but $\dot{\vec{B}}$ in Faraday's Law of induction (electro-quasi-static approximation). The upshot is that the quasi-static approximations used are good as long as the spatial extension of the circuit is small compared with the wavelength of the electromagnetic waves with the frequency of the AC, i.e., if $\lambda \ll L$, where $$\lambda=2 \pi/k=2 \pi c/\omega=c/f.$$

 

[mertcan]

The intuitive answer is that Kirchhoff's Laws of AC circuit theory are derived using the quasi-static approximations (approximations, because two different ones are involved), i.e., in conductors (Ohmic resistances and coils) the "displacement current" is neglected (magneto-quasi-static approximation), approximating the Ampere-Maxwell Law of the full theory by the magnetostatic Ampere Law. In the space between capacitor plates the displacement current cannot be neglected but $\dot{\vec{B}}$ in Faraday's Law of induction (electro-quasi-static approximation). The upshot is that the quasi-static approximations used are good as long as the spatial extension of the circuit is small compared with the wavelength of the electromagnetic waves with the frequency of the AC, i.e., if $\lambda \ll L$, where $$\lambda=2 \pi/k=2 \pi c/\omega=c/f.$$

If you imagine a network consisting of just one resistor at the end of long wires, at high frequencies the distributed L and C along the wires causes the wires to act as transmission lines. The resistance is then transformed to a totally different impedance at the opposite end of the wires. For this reason, complex networks which are physically large are a problem at high frequencies. I have not had to tackle a network of this sort before, but I think an RF bridge would be an example where Kirchoff might be necessary. For the most part, networks are ladder or tee types, where Kirchoff is not necessary.

Thanks for your your return, My first question is : is there a proof when size of device on circuit gets smaller than wavelength, then kirchhoff law is much more true??
also I would like to say that I can not find nice examples and solution pertain to high frequency circuits on internet. Lots of stupid videos exist on youtube . Would you mind suggesting a book or pdf files or nice lecture notes or videos related to that topic in order to learn more effectively?

 

[vanhees71]

The standard textbook is of course Jackson, Classical electrodynamics.

 

[mertcan]

The standard textbook is of course Jackson, Classical electrodynamics.

Thanks @vanhees71 by the way is there mathematical proof when size of device on circuit gets smaller than wavelength, then kirchhoff law is much more true? ?

 

[vanhees71]

Sure, you can estimate the error made when doing the approximations leading from the full Maxwell equations to AC circuit theory. It's nicely discussed in Jackson, if I remember right.

 

[tech99]

Sure, you can estimate the error made when doing the approximations leading from the full Maxwell equations to AC circuit theory. It's nicely discussed in Jackson, if I remember right.

Does it mean that I am slightly in error if I calculate the input impedance of a line, which is in free space, using ordinary transmission line theory? For example, using Zo and Zload?

 

[mertcan]

Sure, you can estimate the error made when doing the approximations leading from the full Maxwell equations to AC circuit theory. It's nicely discussed in Jackson, if I remember right.

@vanhees71 I have already the book but I can not find the proof that when size of device on circuit gets smaller than wavelength, then kirchhoff law is much more true??

 

[Cryo]

Which Kirchoff law?

The current law simply states that current is incompressible (what goes into the node, must come out). In electrodynamics you would instead say that the integral of the current density over any closed surface must be zero (in simple terms current density*area = current )

Then, using Gauss' law on current density $\boldsymbol{J}$:

$\oint d^2r \boldsymbol{J}.\boldsymbol{\hat{n}}=\int d^3 r \boldsymbol{\nabla.J}$

So the Kirchhoff's Current Law requires the divergence of current density to vanish. Take the divergence of the fourth Maxwell's law:

$\boldsymbol{\nabla}.\boldsymbol{\nabla}\times\boldsymbol{B}=0=\mu_0\boldsymbol{\nabla}.\boldsymbol{J}+\frac{n^2}{c^2}\boldsymbol{\nabla}.\boldsymbol{\dot{E}}$

So to get the Kirchhoff' Current Law you need $\boldsymbol{\nabla}.\boldsymbol{\dot{E}} \to 0$

The Kirchoff's voltage law states, that voltages around a closed loop add up to zero. Voltage between points $a$ and $b$ is given by: $V=\int^{b}_{a} \boldsymbol{E}.d\boldsymbol{r}$. So KVL is basically, $\oint \vec{E}.d\boldsymbol{r}=0$. From Maxwell's third law $\boldsymbol{\nabla}\times\boldsymbol{E}=-\boldsymbol{\dot{B}}$, so to get KVL you need

$\boldsymbol{\dot{B}}\to 0$.

At this point you have to start hand-waveing. For example, you could say that both conditions are satisfied if the time-scale of your oscillations is long (time-derivatives vanish). But what does long mean? Wave-equations couple space and time, so long time-scales are equivalent to short distances. The conversion factor is the speed of light. Have a look at microwave engineering books for more details. Did you check Pozar?

 

[Charles Link]

Dr. Walter Lewin makes an addition to the Kirchhoff Voltage Law (KVL)=see e.g. https://www.physicsforums.com/insights/a-new-interpretation-of-dr-walter-lewins-paradox/ so that the requirement that $\frac{dB}{dt}=0$ is not necessary. $\\$ However, the assumption in KVL that the current is the same everywhere in the circuit requires that wavelengths involved are large compared to the size of the circuit. Kirchhoff's voltage law does not allow for a current of the form $i(x,t)=i_o \cos(kx-\omega t)$ that is different for different locations $x$ in the circuit. With Kirchhoff, we can have $i=i_o \cos(\omega t)$ , but that current is the same everywhere in the circuit, i.e. we can have a branch, etc, but what goes into the junction comes out of the junction, etc.$\\$ See also post 52 of https://www.physicsforums.com/threa...asure-a-voltage-across-inductor.880100/page-3 for Dr. Walter Lewin's video. IMO, Dr. Walter Lewin makes an important addition to the KVL theory with the paradox that he presents for the case where $\frac{dB}{dt} \neq 0$. (It would be helpful to watch this video before reading the more in-depth discussion of it in the "link" of the first paragraph). $\\$ Edit: Additional item: With KVL, there is no provision for a circuit to be able to radiate power. When power is radiated, KVL will not compute the power correctly. If there is significant power radiated, the results that KVL gets must thereby be an approximation. @vanhees71 Please confirm these last statements, but I believe they are correct.

 

[mertcan]

Which Kirchoff law?

The current law simply states that current is incompressible (what goes into the node, must come out). In electrodynamics you would instead say that the integral of the current density over any closed surface must be zero (in simple terms current density*area = current )

Then, using Gauss' law on current density $\boldsymbol{J}$:

$\oint d^2r \boldsymbol{J}.\boldsymbol{\hat{n}}=\int d^3 r \boldsymbol{\nabla.J}$

So the Kirchhoff's Current Law requires the divergence of current density to vanish. Take the divergence of the fourth Maxwell's law:

$\boldsymbol{\nabla}.\boldsymbol{\nabla}\times\boldsymbol{B}=0=\mu_0\boldsymbol{\nabla}.\boldsymbol{J}+\frac{n^2}{c^2}\boldsymbol{\nabla}.\boldsymbol{\dot{E}}$

So to get the Kirchhoff' Current Law you need $\boldsymbol{\nabla}.\boldsymbol{\dot{E}} \to 0$

The Kirchoff's voltage law states, that voltages around a closed loop add up to zero. Voltage between points $a$ and $b$ is given by: $V=\int^{b}_{a} \boldsymbol{E}.d\boldsymbol{r}$. So KVL is basically, $\oint \vec{E}.d\boldsymbol{r}=0$. From Maxwell's third law $\boldsymbol{\nabla}\times\boldsymbol{E}=-\boldsymbol{\dot{B}}$, so to get KVL you need

$\boldsymbol{\dot{B}}\to 0$.

At this point you have to start hand-waveing. For example, you could say that both conditions are satisfied if the time-scale of your oscillations is long (time-derivatives vanish). But what does long mean? Wave-equations couple space and time, so long time-scales are equivalent to short distances. The conversion factor is the speed of light. Have a look at microwave engineering books for more details. Did you check Pozar?

Yes I have checked Pozar but could not see any clue or derivation of the relation between wavelength size and kirchhoff laws? Could you help me a little bit more to understand it better?

 

[mertcan]

Dr. Walter Lewin makes an addition to the Kirchhoff Voltage Law (KVL)=see e.g. https://www.physicsforums.com/insights/a-new-interpretation-of-dr-walter-lewins-paradox/ so that the requirement that $\frac{dB}{dt}=0$ is not necessary. $\\$ However, the assumption in KVL that the current is the same everywhere in the circuit requires that wavelengths involved are large compared to the size of the circuit. Kirchhoff's voltage law does not allow for a current of the form $i(x,t)=i_o \cos(kx-\omega t)$ that is different for different locations $x$ in the circuit. With Kirchhoff, we can have $i=i_o \cos(\omega t)$ , but that current is the same everywhere in the circuit, i.e. we can have a branch, etc, but what goes into the junction comes out of the junction, etc.$\\$ See also post 52 of https://www.physicsforums.com/threa...asure-a-voltage-across-inductor.880100/page-3 for Dr. Walter Lewin's video. IMO, Dr. Walter Lewin makes an important addition to the KVL theory with the paradox that he presents for the case where $\frac{dB}{dt} \neq 0$. (It would be helpful to watch this video before reading the more in-depth discussion of it in the "link" of the first paragraph). $\\$ Edit: Additional item: With KVL, there is no provision for a circuit to be able to radiate power. When power is radiated, KVL will not compute the power correctly. If there is significant power radiated, the results that KVL gets must thereby be an approximation. @vanhees71 Please confirm these last statements, but I believe they are correct.

As far as I see Walter Lewin does not link wavelength up with kirchhoff conservation laws. It is always written that wavelength size affects kirchhoff laws conservation. When it comes to find some mathematical proof if it, I can not find valuable things. I do not want to memorize those things just verbally, also I would like to examine the mathematical derivation of the relation between wavelength size and kirchhoff conservation laws. Could you help me in that regard?

 

[vanhees71]

Unfortunately, I have written this up only in German in my lecture notes about E&M. Maybe you can nevertheless follow it since the equation density is quite high ;-)):

https://th.physik.uni-frankfurt.de/~hees/publ/theo2-l3.pdf

 

[mertcan]

Unfortunately, I have written this up only in German in my lecture notes about E&M. Maybe you can nevertheless follow it since the equation density is quite high ;-)):

https://th.physik.uni-frankfurt.de/~hees/publ/theo2-l3.pdf

Thanks for sharing actually I have a little bit german but English would be so nice so could you tell me on which page I should focus for my question?

 

[jasonRF]

My first question is : is there a proof when size of device on circuit gets smaller than wavelength, then kirchhoff law is much more true??

First, i hope you have looked at transmission line theory enough so that you have seen the equation after the title "input impedance of lossless transmission line" at
https://en.m.wikipedia.org/wiki/Transmission_line
This of course aasumes $\exp(j \omega t )$ time dependence. Consider the simple case where you have a transmission line of length $\ell$ and characteristic impedance $Z_0$ that is shorted on one end. Of course the low frequency approximation (Kirchoff law) tells us that the impedance looking into the other end will be zero. However, transmission line theory tells us the impedance looking into the other end will be
$$Z_{in}=j Z_0 \tan(2 \pi \ell / \lambda) $$
where $\lambda$ is the wavelength of a signal on the line. $Z_{in}$ is clearly not zero. However, when $2 \pi \ell << \lambda$ then we have
$$Z_{in}\approx j Z_0 2 \pi \ell / \lambda $$
This clearly approaches the Kirchoff law approximation of zero as $\ell/\lambda \rightarrow 0$.

While this is not a completely general proof it shows you the basic idea. Hope that helps.

Jason

 

[vanhees71]

Thanks for sharing actually I have a little bit german but English would be so nice so could you tell me on which page I should focus for my question?

It's the beginning of Chpt. 4 (p. 103).