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Opus in profectus

RL Circuits

Discussion

energizing

deenergizing

Start with Kirchhoff's circuit rule.

V = IR + L	dI
	dt

0 = IR + L	dI
	dt

Separate the variables.

V − IR		=	dI
L			dt

V/R − I		=	dI
L/R			dt

−	dt	=	dI
	L/R		I − V/R

−	IR	=	dI
	L		dt

−	I	=	dI
	L/R		dt

−	dt	=	dI
	L/R		I

Integrate both sides over the appropriate limits.

	t			I
−	⌠ ⎮ ⌡	dt	=	⌠ ⎮ ⌡	dI
−	⌠ ⎮ ⌡	L/R	=	⌠ ⎮ ⌡	I − V/R
	0			0
−		t	= ln	⎛ ⎜ ⎝	I − V/R	⎞ ⎟ ⎠
−		L/R	= ln	⎛ ⎜ ⎝	− V/R	⎞ ⎟ ⎠

	t			I
−	⌠ ⎮ ⌡	dt	=	⌠ ⎮ ⌡	dI
−	⌠ ⎮ ⌡	L/R	=	⌠ ⎮ ⌡	I
	0			I₀
−		t	= ln	⎛ ⎜ ⎝	I	⎞ ⎟ ⎠
−		L/R	= ln	⎛ ⎜ ⎝	I₀	⎞ ⎟ ⎠

Make both sides into a power ofe.

e^−Rt/L = 1 −	IR
	V

e^−Rt/L =	I
	I₀

Solve for current as a function of time.

I =	V		⎛ ⎜ ⎝	1 − e^−Rt/L	⎞ ⎟ ⎠
	R

I = I₀e^−Rt/L

The end.