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			− V/R

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

Make both sides into a power of e.

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 = I₀

e	−Rt/L

The end.

The Physics Hypertextbook
© 1998–2013 Glenn Elert

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