User:Msiddalingaiah/high voltage generators

My specific goal is to design efficient machines. Generating high voltage through brute force techniques is not of interest to me. Efficiency can be measured in terms of minimizing energy loss or small physical size or both. The specific variable I want to maximize is voltage step up ratio or $$V_{out} / V_{in}$$, preferably at low average power. Specific output voltage is a design parameter that should be linearly scalable.

Pulsed machines
Pulsed machines generate high voltage by storing energy in either a capacitor or inductor at a low voltage and then transforming it to a high voltage. An automotive ignition system or Xenon strobe tube trigger coils are common examples. The maximum step up ratio is a consequence of conservation of energy. For example, if all of the charge on a large capacitor is transferred to a small capacitor, conservation of energy requires

$$\frac{1}{2} C_p v_p^2 = \frac{1}{2} C_s v_s^2$$

The voltage step up ratio is given by

$$\frac{v_s}{v_p} = \sqrt{\frac{C_p}{C_s}}$$

If energy were stored in an inductor rather than a capacitor:

$$\frac{1}{2} L_p i_p^2 = \frac{1}{2} C_s v_s^2$$

The voltage to current step up ratio is given by

$$\frac{v_s}{i_p} = \sqrt{\frac{L_p}{C_s}}$$

Some interesting results can be observed from these simple relationships:


 * 1) Step up ratio is ultimately limited by energy storage elements
 * 2) Output voltage does not depend on input power

In theory, arbitrarily high step ratios are possible. In practice, it is very difficult to reduce the secondary capacitor to less than a few picoFarads. If the primary capacitor is on the order of a few microFarads, it is possibly to achieve a voltage step up of 1000 or more.

Transferring energy efficiently from the primary capacitor to secondary is also a challenge. Pulse transformers are one solution, but this would require a high turns ratio which leads to increased secondary capacitance, which is not desirable. If a high permeability core is used, magnetic saturation and core losses limit efficiency. For these reasons, pulse transformers tend yield a step up ratio of not much more than 100 or so.

Double tuned transformers
If the primary of a double tuned transformer is driven by a steady state sinusoid, the following equation describes the voltages and currents of the circuit:

$$ \begin{bmatrix} R_1 & -j \omega M\\ -j \omega M & R_2 \end{bmatrix} \begin{bmatrix} i_1\\ i_2 \end{bmatrix} = \begin{bmatrix} v_1\\ 0 \end{bmatrix} $$

Where $$R_1$$ and $$R_2$$ are the resistances in the primary and secondary coils respectively. $$i_1$$ and $$i_2$$ are the currents in the primary and secondary coils and $$v_1$$ is the primary drive voltage. $$M$$ is the mutual inductance. This assumes that both the primary and secondary circuits are tuned to the same frequency $$\omega$$.

If the input is a steady state sinusoid and the output is measured across $$C_2$$, the voltage step up ratio is

$$\frac{v_2}{v_1} = \frac{\omega M}{(R_1 R_2 + \omega^2 M^2) C_2}$$

Mutual inductance is related to the coefficient of coupling:

$$M = \frac{k}{\sqrt{L_1 L_2}}$$

Substitution above exhibits a voltage step up dependent on $$k$$. Differentiating and solving for zeros yields a maximum at

$$k_c = \frac{1}{\sqrt{Q_1 Q_2}}$$

Where

$$Q_1 = \frac{\omega L_1}{R_1}$$

$$Q_2 = \frac{\omega L_2}{R_2}$$

Substituting above yields a maximum step up of

$$\frac{v_2}{v_1} = \sqrt{Q_1 Q_2}\sqrt{\frac{L_2}{L_1}}$$

Given that inductance is proportional to the square of the number of turns:

$$L_1 \sim N_1^2$$ and $$L_2 \sim N_2^2$$

$$\frac{v_2}{v_1} \approx \sqrt{Q_1 Q_2}\frac{N_2}{N_1}$$

It follows that double tuned transformers at critical coupling can multiply step by much more than the turns ratio. In practice a turns ratio of 100 and Q of 100 is achievable. This could yield a potential step up of 10,000 or more. This a significant improvement over pulsed machines described in the previous section.

Input drive power is given by:

$$P_{in} = \frac{v_1^2}{2 R_1}$$

Given that $$R_1$$ is small, input power can be very high, on the order of many kiloWatts. Average power can be reduced significantly by operating the source in bursts. Time domain analysis in closed form is very complex as it requires finding the roots of a fourth order polynomial. However, numerical simulation shows that roughly $$Q$$ cycles are sufficient to achieve near peak voltage. A low duty cycle can reduce average power to an arbitrary low level. This mode of operation has been used to excellent effect with solid state Tesla coil designs with an interrupter. It is often call a Dual Resonant Solid State Tesla coil or DRSSTC. Many resources for these designs are widely available.

Although double tuned transformers can be very efficient, their construction requires no less than four distinct values than must be tuned optimally to achieve maximum performance:


 * 1) Primary resonant frequency
 * 2) Secondary resonant frequency
 * 3) Coupling coefficient
 * 4) Input frequency

Practical machines have been built with very good performance, but I am of the opinion that multitude of tuning parameters increases complexity. A simpler design with fewer tuning parameters is highly desirable.

L section
Consider a high voltage machine as a matching network: low impedance to high impedance. A simple form of matching network is the L-section. If the L-section is driven by a sinusoid, current is given by:

$$i = \frac{v_1}{R + j \omega L + \frac{1}{j \omega C}}$$

At resonance, the current reduces to:

$$i = \frac{v_1}{R}$$

The output voltage across the capacitor is:

$$v_2 = \frac{v_1}{j \omega C R}$$

Given that

$$\omega = \frac{1}{\sqrt{L C}}$$ and $$Q = \frac{\omega L}{R}$$

The magnitude of the voltage step up ratio is:

$$\left | \frac{v_2}{v_1} \right | = -Q$$

This is similar to the results in the previous section, without the turns ratio multiplication. As we will see, this is not a drawback.

To determine how much time is required to reach the maximum output voltage, the Laplace transform of the step up ratio is

$$\frac{v_2}{v_1} = \frac{\omega}{s^2 + \omega^2}\frac{\frac{1}{s C}}{\frac{1}{s C} + s L + R}$$

The inverse Laplace transform is an exponential sinusoid, whose magnitude is approximately:

$$\left | \frac{v_2}{v_1} \right | \approx \left (Q + 0.5 \right ) e^{\frac{- \omega}{2 Q} t} - Q$$

This shows that the output voltage reaches it's maximum in roughly Q cycles, similar to the dual resonant transformer in the previous section. In practice a Q of 100 can easily be achieved. Given a resonant frequency of 100 kHz, approximately 1 to 2 milliseconds is sufficient to reach maximum voltage. This means that a burst mode drive can reduce average power to an arbitrarily low level.

Although the step up ratio appears to be much less than that of the dual resonant transformer above, high step up ratio can still be achieved by driving the L-section using a standard well coupled step up transformer. A high quality ferrite core toroidal transformer could easily step up drive voltage by a factor of 20, 40 or even 100. Given a Q of 100, it is possible to achieve a step up of from 2000 to as much as 10,000. Modern MOSFETs and IGBTs are capable of high pulsed power up to 1000 volts or more. Using such a drive, the potential output voltage could reach as much as 10 million volts with average power no more than 100 watts.

The advantage of the L-section over the dual resonant transformer is minimal tuning. In the dual resonant transformer, several parameters must be tuned to achieve optimum performance. The L-section need only be driven at its resonant frequency, requiring no tuning at all.

Realization
Although the L-section described above can be realized using discrete inductors and capacitors, a more practical and elegant solution is to model a distributed air core helical resonator as a lumped L-section. It is true that a real helical resonator cannot be fully described using lumped models over a wide band of frequencies, but over a very narrow band, specifically the resonant frequency, I believe lumped models are sufficient. Rather than estimating inductance and capacitance of a resonator using common formulas such as those of Wheeler and Medhurst, the resonant frequency is readily estimated with great accuracy as the quarter wave length of wire used:

$$f_o = \frac{3 \times 10^8}{4 l_w}$$

The Q of the resonator is difficult to calculate accurately, but in practice it takes values ranging 100 or more. Q is largely limited by skin depth in the coil. Q can be significantly increased by surrounding the coil with a shield, which turns the coil into cavity resonator. It is interesting to note that helical resonators are commonly used in RF circuits as narrowband filters. This article describes design equations for UHF resonators, but the ideas can be applied to lower frequencies as well. Q values in the thousands could be achieved, but will require significant insulation from the high voltage, business end of the resonator. Many insulators exist that could serve this purpose. Given a Q of 1000, peak voltages in the range of 100 MV could be achieved efficiently at low average power levels.

Helical resonator within a cylindrical shield
$$Q = 35.9 \cdot d \cdot \sqrt{f}$$

$$Q = 2129 \cdot \sqrt{g}$$

$$Q = \frac{4.889 \times 10^6}{Z_o \cdot \sqrt{f}}$$

$$Q = \frac{9.6 \times 10^4}{N \cdot \sqrt{f}}$$

$$Z_o = \frac{136190}{d \cdot f}$$

$$Z_o = 51 \cdot N$$

$$Q \sqrt{Z_o} = 13250 \sqrt{d}$$

$$h = 1.5 \cdot d$$


 * Q - quality factor (dimensionless)
 * $$Z_o$$ - resonator characteristic impedance (Ohms)
 * d - mean helix diameter (cm)
 * h - height of helix (cm)
 * f - frequency (MHz)
 * g - wire diameter (cm)
 * N - number of turns

It is generally assumed that the coil is space wound such that the gap between turns is equal to the diameter of the wire.

Helical resonator design

Helical resonator over a ground plane
$$Z_o = 60 \left [ ln(4 \cdot a_r) - 1 \right ] \sqrt{1 + \frac{10 \cdot N^2}{\sqrt{\pi \cdot a_r^5}}}$$

$$Q = \frac{4.889 \times 10^9}{Z_o \cdot \sqrt{f}}$$


 * $$Z_o$$ is the characteristic impedance in ohms
 * $$a_r$$ is the aspect ratio (height / width of winding)
 * N is the number of turns
 * f is frequency in Hz
 * Space winding is assumed, close winding reduces Q by a factor of 2.3 or more

Comparison with helical resonator within a shield (HRS)

 * An aspect ratio of 3 is roughly equivalent to HRS with aspect ratio 1.5, larger values reduce impedance further
 * An aspect ratio of 1.5 yields an impedance which is roughly 33% higher than that of HRS
 * Maximum impedance occurs at aspect ratio of 1.5
 * Impedance increases almost linearly with the number of turns
 * Lower impedance (fewer turns) yields higher Q

Summary
It is interesting to observe that the three types of high voltage machines described above compare in chronological order with Teslas own research and development. The first type are pulsed induction coils, the second are dual resonant transformers or traditional Tesla coils, the third are what Tesla called "extra coils".

Tesla concluded on July 11, 1899:


 * It is a notable observation that these "extra coils" with one of the terminals free enable the obtainment of practically any e.m.f. the limits being so far remote that I would not hesitate undertaking to produce sparks of thousands of feet in length in this manner. Owing to this feature, I expect that this method of raising the e.m.f. with an open coil will be recognized later as a material and beautiful advance in the art. No such pressures - even in the remotest degree, can be obtained with resonating circuits otherwise constituted with two terminals forming a closed path.

I am inclined to agree with Tesla. Of course, Tesla did not enjoy the luxury of modern high speed, high power semiconductors as are available today. Still the same principles apply, regardless of the switch.