What is conservation of flux linkage ?? Please explain with suitable diagram. I understand charge conservation and textbooks usually say that flux linkage conservation is analogous to that, but they rarely explain this in detail.

• Please provide a link to somewhere that talks about it. – Andy aka Jul 24 '16 at 9:28
• This topic has been covered in all leading circuit theory texts, to name a few valkenburg, desoer-kuh etc – Debajyoti Datta Jul 24 '16 at 12:46

I wiil try to go through a simple application of flux conservation using an example.

I will start from something perhaps more familiar to most of us.

When you try to find final status of this circuit using charge conservation principle

simulate this circuit – Schematic created using CircuitLab

you will quickly come to the conclusion that charge at time 0 yields $$Q(0)=C_1v_1(0)+C_2v_2(0)$$ and will be conserved and shared between the two capacitor giving a final voltage $$v_1(\infty)=v_2(\infty)=\frac{Q(\infty)}{C_1+C_2}=\frac{Q(0)}{C_1+C_2}=\frac{C_1v_1(0)+C_2v_2(0)}{C_1+C_2}$$

So what have we done? We had a two node circuit made of two capacitors:

• this two capacitors were initially charged at some initial voltage
• initial charge was calculated considering positive the upper capacitor armatures
• this charge is conserved and shared between the two caps in final status. Upper node will now remain at constant voltage
• just note that at final state voltage across resistor is zero, otherwise dissipation would change system energy and so we were not at final status yet.

Now le's make it dual and use flux conservation principle.

Now we have a two mesh circuit made of two inductors:

• this two inductors were initially charged at some initial current

simulate this circuit

• initial flux is calculated considering positive clockwise mesh current

$$\Phi(0)=L_1i_1(0)+L_2i_2(0)$$

• this flux is conserved and shared between the two coils in final status Outer mesh will now remain at constant current

$$i_1(\infty)=i_2(\infty)=\frac{\Phi(\infty)}{L_1+L_2}=\frac{\Phi(0)}{L_1+L_2}=\frac{L_1i_1(0)+L_2i_2(0)}{L_1+L_2}$$

• just note that at final state current through resistor is zero, otherwise dissipation would change system energy and so we were not at final status yet.

I understand this is far from a thorough study but I believe could help to make some clear around that.

Notes:

• Both electric charge and magnetic flux conservation priciples could be named toghether flux conservation, electric flux the first one and magnetic flux the second one.

• Resistors in above schematics are there just to avoid singularities in equations. Those could be rigorously coped with Dirac distributions but so far I'd rather spare the subject.

• Could you please tell me whether i am wrong.........if i integrate kcl , then continuity of charge in a node can be found, and analogously integrating kvl would give flux linkage continuity in a loop – Debajyoti Datta Jul 24 '16 at 17:49

Yes Debajyoti Datta, you are right. Just make sure you do not run throught some circular reference.

I mean KCL proof may come from charge conservation, then you cannot use KCL to proof charge conservation.

The way I understand it is that the term flux linkage is used to describe how much energy is conserved going from magnetic field to current and other similar EM processes

Magnetic flux is the density of magnetic fields going through a plane * the surface area of that plane

A coil would be consider near the ideal case where the total magnetic flux would be the flux of one ring in the coil * # of rings in that coil. In this case the flux linkage would be the same as the total magnetic flux, because the current produce would almost have the same amount of energy, only losing some energy to heat dissipation.