alma1969 4 years ago What is an expression for the transform current through resistor R ?(see attachment)

1. alma1969

2. abtrehearn

Is the source voltage an arbitrary function of time?

3. abtrehearn

The current flows from the source through the first inductor, and then it branches off; part of the current flows through a purely capacitive branch. and the rest of it flows into the other inductor, which is in series with a resistor. At any instant, the voltages across the branches are the same. We can use this information and Kirchoff's laws to set up a system of equations to determine the currents throughout the circuit, including the current through the resistor.

4. abtrehearn

I will be doing some pencil-and-paper calculations on this, so it may be some time before my next post here.

5. alma1969

Tank you for your time !!!!!!!

6. abtrehearn

I get\[L^{2}Ci'''_R + RLCi''_R + Li'_R = e(t)\]for the time-domain equation. Since we are taking Laplace transforms of derivatives up to third-order, \[i''_R(0), i'_R(0),\]and\[i_R(0)\]enter into my calculations. Since nothing is stated about initial values for the current across the resistor, I keep those three numbers arbitrary, leaving them in symbolic form. When I take the Laplace transform of both sides of the equation I came up with, I get \[I_R(s) = E(s)/(L^{2}Cs^{3}+ RLCs^{2} + Ls)\] \[+ L^{2}Ci_R(0)s^{2}/(L^{2}Cs^{3} + RLCs^{2}+ Ls)\] \[+ (L^{2}Ci'_R(0) + RLCi_R(0))s/(L^{2}Cs^{3} + RLCs^{2} + Ls)\] \[+ (L^{2}Ci''_R(0) + RLCi'_R(0) + LI_R(0))/(L^{2}Cs^{3} + RLCs^{2} + Ls).\]I get\[L^{2}Ci'''_R + RLCi''_R + Li'_R = e(t)\]for the time-domain equation. Since we are taking Laplace transforms of derivatives up to third-order, \[i''_R(0), i'_R(0),\]and\[i_R(0)\]enter into my calculations. Since nothing is stated about initial values for the current across the resistor, I keep those three numbers arbitrary, leaving them in symbolic form. When I take the Laplace transform of both sides of the equation I came up with, I get \[I_R(s) = E(s)/(L^{2}Cs^{3}+ RLCs^{2} + Ls)\] \[+ L^{2}Ci_R(0)s^{2}/(L^{2}Cs^{3} + RLCs^{2}+ Ls)\] \[+ (L^{2}Ci'_R(0) + RLCi_R(0))s/(L^{2}Cs^{3} + RLCs^{2} + Ls)\] \[+ (L^{2}Ci''_R(0) + RLCi'_R(0) + LI_R(0))/(L^{2}Cs^{3} + RLCs^{2} + Ls).\]

7. alma1969

Thank you !!!!!!!!

8. abtrehearn

You're welcome :^)