The lumped element model

[Ahmad Kishki]

First assumption: the change of magnetic flux in time outside a conductor is zero.

Last semester in circuit analysis we treated circuits with sinusoidal voltage variation as steady state circuits while using the phasor notation. But now that i know electrodynamics i begin to doubt my concepts that were built in circuit analysis. For sinusoidal variation of voltage the electric field through the conductor will also vary sinusoidally which means that there will be a changing magnetic field enclosed by the circuit. What assumption here is needed to lead to the assumption that the change in magnetic flux outside a conductor is zero? Is it that the surface area of the circuit is too small? But that is not satisfactory...

Second assumption: the change of charge in time inside conducting elements is zero.

This means that displacement current is zero, right?

Source: http://en.m.wikipedia.org/wiki/Lumped_element_model

 

[Baluncore]

But that is not satisfactory...

If the assumptions made in your application are “not satisfied” sufficiently then you should not be using a lumped element model.

No simple model can ever be an exact representation of the reality it attempts to model. Assumptions are made to simplify the model to the point where it can be solved numerically in a reasonable time. The “lumped element model” has worked sufficiently well in most situations over the last 100 years. Without those assumptions, the physical dimensions of the components and the path taken by the connecting wires would need to be specified in an EM model. That would be most inconvenient when not essential.

 

[Ahmad Kishki]

If the assumptions made in your application are “not satisfied” sufficiently then you should not be using a lumped element model.

No simple model can ever be an exact representation of the reality it attempts to model. Assumptions are made to simplify the model to the point where it can be solved numerically in a reasonable time. The “lumped element model” has worked sufficiently well in most situations over the last 100 years. Without those assumptions, the physical dimensions of the components and the path taken by the connecting wires would need to be specified in an EM model. That would be most inconvenient when not essential.

You misunderstood me.. I am asking for the basis of these two assumptions in field theory...

 

[Babadag]

In my opinion, if you follow Kirchhoff laws for lumped circuit then Sum(Ii)=0 in a connection point that means [ in the connection point] i [result]= dq/dt =0.
The second law Sum(Vi)=0 in a loop that means no magnetic field change which could produce an EMF in the loop. dB/dt=0.

 

[Ahmad Kishki]

In my opinion, if you follow Kirchhoff laws for lumped circuit then Sum(Ii)=0 in a connection point that means [ in the connection point] i [result]= dq/dt =0.
The second law Sum(Vi)=0 in a loop that means no magnetic field change which could produce an EMF in the loop. dB/dt=0.

Yes i get this, but what assumption is required to say that dB/dt=0? I mean, a time changing electric field will lead to a time changing magnetic field... So how can this assumption hold when the voltage is sinusoidal wrt time?

 

[Baluncore]

what assumption is required to say that dB/dt=0

The average dB/dt over one sinusoidal cycle will be zero.

 

[Ahmad Kishki]

The average dB/dt over one sinusoidal cycle will be zero.

Hmmm...

 

[anorlunda]

AC circuit analysis applies to integral numbers of whole cycles, average quantities, and ignoring non-sinusoidal startup transients.

You are perfectly free to skip AC analysis, and to analyze the circuits transiently using differential equations. That is more general and more universal, but a whole lot more work. Where the simplifications of AC analysis are justified, they save a lot of time and effort.

 

[Babadag]

Yes i get this, but what assumption is required to say that dB/dt=0? I mean, a time changing electric field will lead to a time changing magnetic field... So how can this assumption hold when the voltage is sinusoidal wrt time?

You are right, Ahmad. But this is the simplification of the Kirchhoff laws. He neglected the magnetic flux outside the conductors. Actually, it is an EMF around the loop due to fact dB/dt>0.It is one of the self-imposed constraints by the Lumped Matter Discipline.