TL;DR The imaginary part of the impedence tells you the reactive component of the impedance; this is responsible (among others) for the difference in phase between current and voltage and the reactive power used by the circuit.
The underlying principle is that any periodic signal can be treated as the sum of (sometimes) infinite sinewaves called harmonics, with equally spaced frequencies. Each of them can be treated separately, as a signal of its own.
For these signals you use a representation that is like:
$$ v(t) = V_{0} \cos (2 \pi f t + \phi) = \Re \{ V_{0}e^{j 2 \pi f t + \phi} \} $$
And you can see that we already jumped in the domain of complex numbers, because you can use a complex exponential to represent rotation.
So impedance can be active (resistance) or reactive (reactance); while the first one by definition doesn't affect the phase of signals (\$ \phi \$) the reactance does, so using complex numbers is possible to evaluate the variation in the phase that is introduced by the reactance.
So you obtain:
$$ V = I \cdot Z = I \cdot |Z| \cdot e^{j \theta} $$
where |Z| is the magnitude of the impedance, given by:
$$|Z|=\sqrt{R^2+X^2}$$
and theta is the phase introduced by the impedance, and is given by:
$$\theta = \arctan \left( \frac{X}{R} \right) $$
When applied to the previous function, it becomes:
$$ v(t) = \Re \{ I_{0}|Z|e^{j 2 \pi f t + \phi + \theta } \}
= I_{0} |Z| \cos (2 \pi f t + \phi + \theta ) $$
Let's consider the ideal capacitor: it's impedance will be \$ \frac{1}{j \omega C} = -\frac{j}{\omega C} \$ which is imaginary and negative; if you put it into the trigonometric circumference, you obtain a phase of -90°, which means that with a purely capacitive load the voltage will be 90° behind the current.
So why?
Let's say that you want to sum two impedances, 100 Ohm and 50+i50 Ohm (or, without complex numbers, \$ 70.7 \angle 45 ^\circ \$ ). Then with complex numbers you sum the real and imaginary part and obtain 150+i50 Ohm.
Without using complex numbers, the thing is quite more complicated, as you can either use cosines and sines (but it's the same of using complex numbers then) or get into a mess of magnitudes and phases. It's up to you :).
Theory
Some additional notions, trying to address your questions:
- The harmonics representation of signals is usually addressed by Fourier series decomposition:
$$ v(t) = \sum_{- \infty}^{+ \infty} c_{n}e^{jnt} , \text{ where }
c_{n} = \frac{1}{2 \pi } \int_{-\pi}^{\pi} v(t)e^{-jnt} \, dt $$
- The complex exponential is related to the cosine also by the Euler's formula:
$$ cos(x) = \frac{e^{ix}+e^{-ix}}{2} $$
First start with the approximations
L = L1 + L2 + L3
R = R1 + R3
C = C1
R2 is too large to do anything.
The influence of C2 could possibly be somehow added in to the simple RLC model, but I would ignore it unless there is some specific reason to include it. Once there is a reason to include it, then there should be a way to calculate its influence on the outcome.
Ignoring C2, it looks like your resonant frequency is about 377kHz, and the Q is about 6.4. This is not a great match for a 1MHz source. I assume that your program is going to tell you how to tune things up to get your big voltage to light up the gas tube.
Take care with the high voltage and the bright lights! I know a guy who remembered not to touch the high voltage but forget about the sunburn.
Best Answer
The graph you show (for short time durations) is governed by the specific heat capacity of the material in question. For longer/extended time periods it is governed by the thermal conductivity of the material. Specific heat capacity for most common materials (and possibly most uncommon materials) falls with temperature hence, at lower temperatures, a particular material will exhibit what your graph shows at short time durations.
If you want to find out more, examine the specific heat capacity for the material in the IGBT.
No, an RC model of the thermal properties of a material is not reflected in a change of capacitance - the model assumes that the thermal properties remain constant unless there is particular attention paid by the model designer to making capacitance change with local ambient temperature (I've never seen this done BTW).