# Transient analysis with Laplace and nodal analysis error

Have the schematic:  There is an error in my equations with nodal analysis but I can't find where? May anyone help me?

I also add the solution using approach of series RLC circuit: But it seems to be A1 and A2 constant sign is incorrect comparing my result with that in textbook.

I also tried to solve with Laplace but using Mesh Current Method, my equations are (I1 - left mesh current, I2 - right): But same thing - I can't get properly result, may be something wrong but I can't understand what!

Thevenin Equiv circuit for v1 goes from 60V to 40% or 24V with Rs=30//20=12 ohms. The steady state v(t) across cap is then +24-30= -6Vdc

Then we know resonance ω= √(L/C)= v(0.5) = 0.707 and Q=XL/R = 2 π 0.5/12 which is <1 and thus overdamped and the initial voltage across cap will be 0V.

Does this method make it easier than Laplace?

• Yes, it seems to be easy but applies to current circuit. I need more general solution that applies to any 2-order circuit but your tips are good! Jan 12 '18 at 4:34
• I add my solution using your approach and I get an answer. Only difference with textbook answer that the signs of A1 and A2 are inversed in textbook answer. Why it so? Jan 12 '18 at 5:03

The node equations are correct, and the expression for $v_2$ is also correct. Check your analysis for $v_1$.

• I add the inverse Laplace Transform from MathCAD to post text and the result voltage I think must be the difference of equations for v1 - v2. But there is a strange result. It must be in an exponentional form but not what I get. There is a trick somewhere? Jan 12 '18 at 4:33
• By inspection of the circuit, the final value of v(t) is: 24 - 30 = -6 V, which is the same as your final equation gives.
– Chu
Jan 12 '18 at 9:03
• But I must find the equation of v(t) not simply final value. The problem is I use same steps in other tasks and the inverse Laplace of node voltage give me the right result but not here, I can't understand why? May the MathCAD not work properly with my equation or not I don't know. Jan 12 '18 at 9:28
• I know you want v(t), my point is that your equations are giving -6V at infinity, which you say is the correct answer.
– Chu
Jan 12 '18 at 14:53
• Is there any suggestion why I get an incorrect inverse Laplace transform? Jan 12 '18 at 16:50

There is an error in my equations with nodal analysis but I can't find where?

Your nodal equations are correct (assuming zero initial conditions). The expressions for $$\V_1(s)\$$ and $$\V_2(s)\$$ (in the frequency domain) are also correct. Regarding the expressions for $$\v_1(t)\$$ and $$\v_2(t)\$$ (in the time domain), I'm not sure if yours (using hyperbolic functions) are correct, but using exponential functions below I show them along with a plot. The following simulation checks the previous result: 