# You are subscribed to this thread L di/dt vs. i dL/dt kmarinas86

1. Aug 25, 2009

### kmarinas86

"L di/dt" vs. "i dL/dt"

I have heard many times that the positive "inductive" part of a circuit's reactive power is equal to "L i di/dt".

I also know that "L di/dt" is just one of two terms that is derived from applying derivative of "L i" with respect to time. The other term is "i dL/dt".

My concern is the relationship between "L i" and the integration of "L di/dt" with respect to time t. I have heard that "L i" is equal to magnetic flux. However, I know that in cases when L changes over the course of time that it is possible that "L i" to be greater than integration of "L di/dt" with respect to time t.

What if the magnetic circuit is changing such that the inductance of the coil changes with respect to the surrounding and interior (magnetic) permeabilities? Taken to an extreme, if the inductance is increasing and the current is not changing, then it appears that the reactive power "L i di/dt" is still zero. But it seems apparent that the increases of inductance increases the potential energy in the field given the current already there.

Perhaps the problem I have originates from the convention where inductive reactive power is equal to "L i di/dt", ignoring cases where "i dL/dt" is non-zero. Perhaps inductive reactive power is really "L i di/dt + (1/2) i^2 dL/dt". That is the complete derivative of (1/2)Li^2.

Does anyone know of an independent peer-review source that gives credence to my concern?

Last edited: Aug 25, 2009
2. Aug 26, 2009

### bjacoby

Re: "L di/dt" vs. "i dL/dt"

I'd say your calculation is correct. The power consists of two terms both containing L(t). Thus at any time t the above two terms calculate the instantaneous power. Note that if the current is constant (di/dt =0) then only the second terms contributes. But this makes sense in that an increase of inductace means you are increasing the energy put into the system to increase the inductance which means more energy stored in the magnetic field. That increase shows up as the power represented by the second term. Hence one can store and remove energy from a magnetic field either by an increase or decrease in the current in the inductor and also by changing the physical configuration of the inductor to provide for more or less flux in space. And obviously one can also do both at the same time.

I see no problem worthy of a peer-reviewed source here.