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.