Modeling a spark gap--How to solve a DE with a step function

[Motocross9]

Homework Statement

I wish to solve a system of differential equations; however, I am modeling a spark gap by using a step function. How could I solve? I'll provide the first equation below as an example:

Relevant Equations

$V_ocos(\omega*t)=\dot Q_1R_1+(Q_2/C_1)(1-U(Q_2-C_1V_o))$

Honestly not sure how to go about this. Again this is one equation of 4 that I have. I considered using Laplace transforms but taking the Laplace transform of a step function whose argument is one of the variables being solved for doesn't seem possible. Also, if there is an alternative way to model a spark gap in a circuit, I would love to be informed of it. Thanks in advance!

(also $Q_1$ and $Q_2$ are the functions of time I wish to solve for)

 

[Gordianus]

Could you post the circuit you're trying to model? The RHS suggests a resistor in series with a capacitor "switched" by the spark-gap. Is that O.K?

 

[Motocross9]

Could you post the circuit you're trying to model? The RHS suggests a resistor in series with a capacitor "switched" by the spark-gap. Is that O.K?

I am modeling the basic Tesla coil circuit. In particular, it is this one:

The resistor isn't shown in this diagram, however. I am treating the spark gap as a capacitor, whose voltage drops to zero once the voltage across it reaches $V_o$. I actually entered the first equation wrong--its fixed now.

 

[Gordianus]

Quite a tricky circuit. It has, at least, two widely different time constants. A slow one, related to the charge of the HV capacitor at mains frequency and a fast one, related to the discharge of the HV capacitor on the primary of the HV transformer. In a simple model I'd consider the spark gap as a non-linear resistance instead of a capacitor.