# A transformer problem

1. Jun 21, 2009

### John 123

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

3. The attempt at a solution
I have solved for i_1[which agrees with the book answer].
$$i_1=C_1e^{at}+C_2e^{bt}$$
where
$$a,b=\frac{-(L_2R_1+L_1R_2)+/_\sqrt{(L_2R_1-L_1R_2)^2+4M^2R_1R_2}}{2(L_1L_2-M^2)}$$
However I cannot get the correct answer for i_2, which is as follows:
$$i_2=\frac{1}{MR_2}[(L_1L_2-M^2)(aC_1e^{at}+bC_2e^{bt})+L_2R_1(C_1e^{at}+C_2e^{bt}]$$
I used the method of solving the characteristic equations in m.
Thus for i_1 and i_2 both have the same characteristic equation:
$$(L_1L_2-M^2)m^2+(L_1R_2+L_2R_1)m+R_1R_2=0$$
Thus we may write
$$i_1=C_1e^{at}+C_2e^{bt}$$
and
$$i_2=k_1e^{at}+k_2e^{bt}$$
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