RL Circuits

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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.

Summary

Problems

practice

  1. Write something.
    • Answer it.
  2. Write something.
    • Answer it.
  3. Write something.
    • Answer it.
  4. Write something completely different.
    • Answer it.

numerical

  1. problems

Resources

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