How Does High-Frequency Affect the Line Equation in AC Circuits?

[CalvinB]

Homework Statement

Show that in the case of alternating currents of high frequencies the equation

can be approximated by the so-called high-frequency line equation

L = Inductance, C = Capacitance, R = Resistance, G = Conductance

Homework Equations

At very-high frequencies the inductor has a high reactance and acts almost like an open circuit. Thus, the current is low, so the resistance can be ignored (correct me if i am wrong :P). But how about G?

 

[berkeman]

Homework Statement

Show that in the case of alternating currents of high frequencies the equation

can be approximated by the so-called high-frequency line equation

L = Inductance, C = Capacitance, R = Resistance, G = Conductance

Homework Equations

At very-high frequencies the inductor has a high reactance and acts almost like an open circuit. Thus, the current is low, so the resistance can be ignored (correct me if i am wrong :P). But how about G?

First, I think that those 4 quantities (L, C, R and G) are per unit length, right?

Second, can you show how the first equation is derived?

Third, the R term is in series with the L term, and the G term is in parallel with the C term? Like in the Characteristic Impedance Zo model:

http://en.wikipedia.org/wiki/Characteristic_impedance

And if you want to say that the R term is negligible, you need to say that in contrast to the value of the impedance of the L term, not say that no current flows, since certainly current flows in a transmission line. Look at the impedance of the R resistance per unit length versus the impedance of the L inductance per unit length of a typical transmission line cable (like Cat-5). Do a similar comparison of impedances for C and G...

 

[CalvinB]

First, I think that those 4 quantities (L, C, R and G) are per unit length, right?

Second, can you show how the first equation is derived?

Third, the R term is in series with the L term, and the G term is in parallel with the C term? Like in the Characteristic Impedance Zo model:

http://en.wikipedia.org/wiki/Characteristic_impedance

And if you want to say that the R term is negligible, you need to say that in contrast to the value of the impedance of the L term, not say that no current flows, since certainly current flows in a transmission line. Look at the impedance of the R resistance per unit length versus the impedance of the L inductance per unit length of a typical transmission line cable (like Cat-5). Do a similar comparison of impedances for C and G...

I do have the first equation derived. I tried to type it in here, but somehow the Latex Reference keep messing up my equations >:(

And yes, since the problem did not clearly state how the RLCG are placed, i assumed they are like the Characteristic Impedance Zo model.

Since the impedance of L depends on the frequency, as the frequency increases, the impedance of L increases, so R can be neglected because the frequency does not change the resistance.(Again, correct me if I'm wrong :P) But how do i calculate the impedance for G?

 

[berkeman]

Since the impedance of L depends on the frequency, as the frequency increases, the impedance of L increases, so R can be neglected because the frequency does not change the resistance.(Again, correct me if I'm wrong :P) But how do i calculate the impedance for G?

I would start with the values of L', C', R', and G' (the primes often are used to indicate that the quantity is per unit length, like per meter or per foot) for a typical transmission line cable, like Cat-5 or RG-58. That will give you some numbers that you can use to justify your simplification of the equation, and will let you calculate a reasonable frequency for starting to use the 2nd equation in your post.

The G' value is basically the parasitic parallel leakage conductance, and is typically quite small.

 

[DeeNos]

Yes, you are correct in stating that at very high frequencies, the inductor has a high reactance and acts almost like an open circuit. This is because the reactance of an inductor is directly proportional to the frequency, and at high frequencies, the reactance becomes significantly larger than the resistance, making it negligible. However, the conductance (G) cannot be ignored in the high-frequency line equation as it represents the leakage or loss of energy in the circuit. This can be due to factors such as resistance in the wires, dielectric losses in the capacitor, or imperfect insulation. Therefore, the high-frequency line equation takes into account both the inductive and capacitive reactances, as well as the resistance and conductance in the circuit.