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A transformer problem

  1. Jun 21, 2009 #1
    1. The problem statement, all variables and given/known data
    Primary coil of a transformer with an emf E(t),resistance R_1 and inductance L_1.Secondary coil resistance R_2 and inductance L_2.
    Let i_1 and i_2 be the respective currents in each coil.



    2. Relevant equations
    [tex]
    L_1\frac{di_1}{dt}+R_1i_1+M\frac{di_2}{dt}=E(t)
    [/tex]
    and
    [tex]
    L_2\frac{di_2}{dt}+R_2i_2+M\frac{di_1}{dt}=0
    [/tex]
    where M =mutual inductance.
    Solve the above system if E(t)=0
    [tex]
    M^2<L_1L_2
    [/tex]


    3. The attempt at a solution
    I have solved for i_1[which agrees with the book answer].
    [tex]
    i_1=C_1e^{at}+C_2e^{bt}
    [/tex]
    where
    [tex]
    a,b=\frac{-(L_2R_1+L_1R_2)+/_\sqrt{(L_2R_1-L_1R_2)^2+4M^2R_1R_2}}{2(L_1L_2-M^2)}
    [/tex]
    However I cannot get the correct answer for i_2, which is as follows:
    [tex]
    i_2=\frac{1}{MR_2}[(L_1L_2-M^2)(aC_1e^{at}+bC_2e^{bt})+L_2R_1(C_1e^{at}+C_2e^{bt}]
    [/tex]
    I used the method of solving the characteristic equations in m.
    Thus for i_1 and i_2 both have the same characteristic equation:
    [tex]
    (L_1L_2-M^2)m^2+(L_1R_2+L_2R_1)m+R_1R_2=0
    [/tex]
    Thus we may write
    [tex]
    i_1=C_1e^{at}+C_2e^{bt}
    [/tex]
    and
    [tex]
    i_2=k_1e^{at}+k_2e^{bt}
    [/tex]
    where a,b are as defined above.
    The problem arises when I write k_1 and k_2 in terms of C_1 and C_2 which can be achieved by substituting i_1 and i_2 solutions back into the first equation.
    And although I can manipulate the result to be similar to the book answer for i_2 I cannot get complete agreement.
    Help!!
    John

    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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