No initial current in the inductor, switch in open state find: (a) Immediately after Close, #I_1, I_2, I_3, & V_L#? (b) Close long #I_1, I_2, I_3, & V_L#? (c) Immediately after Open, #I_1, I_2, I_3, & V_L#? (d) Open Long, #I_1, I_2, I_3, & V_L#?

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Answer 1 · 2016-11-29T21:15:18

Considering two independent currents #I_1# and #I_2# with two independents loops we have

loop 1) #E=R_1I_1+R_1(I_1-I_2)#
loop 2) #R_2I_2+L dot I_2+R_1(I_2-I_1)=0# or

#{(2R_1 I_1-R_1I_2=E),(-R_1I_1+(R_1+R_2)I_2+L dot I_2=0):}#

Substituting #I_1=(E-R_1I_2)/(2R_1)# into the second equation we have

#E+(R_1+2R_2)I_2+2L dot I_2=0# Solving this linear differential equation we have

#I_2=C_0e^(-t/tau)+E/(R_1+2R_2)# with #tau=(2L)/(R_1+2R_2)#

The constant #C_0# is determined according to the initial conditions.

#I_2(0)=0# so

#0=C_0+E/(R_1+2R_2)#

Substituting #C_0# we have

#I_2=E/(R_1+2R_2)(1-e^(-t/tau))#

Now we can answer the items.

a) #I_2=0,I_1=10/8,V_L=10/8 4#
b) #I_2=10/(4+2 cdot8),I_1=?, V_L=0#
c) #I_2=?,I_1=0,V_L=?# we let those answers to the reader
d) #I_1=I_2=V_L=0#