User:KX36

Child-Langmuir Law
$$I_a$$ is anode current.

$$S$$ is surface area of anode.

$$d$$ is distance between anode and cathode.

$$V_a$$ is the potential difference from anode to cathode.

$$P$$ is the perveance of the device.

$$I_a = \left ( \frac{4 \epsilon_0}{9} \sqrt{\frac{2 e}{m_e}} \right)\left( \frac{S}{d^2} \right ) V_a^\frac{3}{2}$$

$$I_a = k_1 \left ( \frac{S}{d^2} \right ) V_a^\frac{3}{2}$$

$$I_a = P V_a^\frac{3}{2}$$

$$I_a \propto V_a^\frac{3}{2}$$

$$ P \propto \frac{S}{d^2}$$

Triode equation (and rearrangements thereof)
$$\mu$$ is the amplification factor of the triode.

$$I_a = P \left( \frac{V_a}{\mu} + V_g \right) ^ \frac{3}{2}$$

$$V_a = \mu \left( \left( \frac{I_a}{P} \right) ^ \frac{2}{3} - V_g \right)$$

$$V_g = \left( \frac{I_a}{P} \right) ^ \frac{2}{3} - \frac{V_a}{\mu}$$

Derivatives of triode equation
$$r_a$$ is the anode resistance

$$g_m$$ is the transconductance of the device

$$r_a = \frac{\mu}{g_m}$$

$$\mu = {dV_a \over dV_g}$$

$$g_m = {dI_a \over dV_g} = \frac{3}{2}\ \sqrt[3]{P^2 I_a}$$

$$r_a = {dV_a \over dI_a} = \frac{2}{3}\ \frac{\mu}{\sqrt[3]{P^2 I_a}}$$

Type II Compensator transfer function
$$H(s) = \frac{V_e}{V}(s) = - \frac{R_3 C_1 s +1}{[R_1 (C_1 + C_2) s](R_3 \frac{C_1 C_2}{C_1 + C_2} s + 1)}$$

$$H(s) = \frac{V_e}{V}(s) \approx - \frac{R_3 C_1 s +1}{(R_1 C_1 s)(R_3 C_2 s + 1)}$$ when $$C_2 \ll C_1$$

Type III Compensator transfer function
$$H(s) = \frac{V_e}{V}(s) = - \frac{(R_3 C_1 s +1)[C_3 (R_1 + R_4) s + 1]}{[R_1 (C_1 + C_2) s](R_3 \frac{C_1 C_2}{C_1 + C_2} s + 1)(R_4 C_3 s + 1)}$$

$$H(s) = \frac{V_e}{V}(s) \approx - \frac{(R_3 C_1 s +1)[R_1 C_3 s + 1]}{(R_1 C_1 s)(R_3 C_2 s + 1)(R_4 C_3 s + 1)}$$ when $$C_2 \ll C_1$$ and $$R_4 \ll R_1$$

Formula for R3
For type III compensation,

$$R_3=\frac{F_c}{F_{lc}}\frac{V_{osc}}{V_{in}}R_1$$

Voltage Feedforward
The following formulae are for one particular voltage feedforward circuit which uses a bias winding to charge a capacitor through a resistor to create the ramp proportional to Vbias, which is proportional to Vsec

$$V_{osc} = \frac{N_{b2}}{N_s} V_{sec} \left( 1 - e^{\frac{-1}{F_s R C}} \right)$$

Comparator gain [N.B. gain is independant of Vsec (proportional to input voltage). Classical voltage mode control has a comparator gain (Vsec/Vosc) which proportiobal to Vsec, which causes problems. The idea of votlage feedforward is to make Vosc proportional to Vsec so that comparator gain is constant]:

$$\frac{V_{sec}}{V_{osc}} = \frac{N_s}{ N_{b2} \left( 1 - e^{\frac{-1}{F_s R C}} \right)}$$

Find RC for minimum acceptable Vosc at minimum Vpri (input voltage).

$$R C = \frac{-1}{F_s ln \left( 1-\frac{N_p}{N_{b2}} \frac{V_{osc}}{V_{pri}} \right) }$$

Unfortunately any Offset voltage introduces an error:

Vosc=Nb2/Ns * Vsec * [1-e^(-1/FsRC)] + Voffset

Vsec/Vosc = Vsec/ [Nb2/Ns * Vsec * [1-e^(-1/FsRC)] + Voffset]

Vsec/Vosc = Vsec/ Vsec[Nb2/Ns * [1-e^(-1/FsRC)] + Voffset/Vsec]

Vsec/Vosc = 1/ [Nb2/Ns * [1-e^(-1/FsRC)] + Voffset/Vsec]

Offset error keeps Vsec in the equation, Vsec proportional to Vin, so PWM comparator gain is a function of input voltage.

EA gain
From Summing Amplifier equation, with V2=0V and IN+=Vref.

$$V_{out} = -R_3 \left( \frac{V_{in}-V_{ref}}{R_1} - \frac{V_{ref}}{R_2} \right) + V_{ref}$$

$$V_{out} = V_{ref} \left( \frac{R_3}{R_2} + \frac{R_3}{R_1} + 1 \right) - \frac{R_3}{R_1} V_{in}$$

For the target condition, Vout = Vref (i.e. opamp output equal to non-inverting input). When this is true, the above can be simplified to:

$$V_{in} = V_{ref} \left( \frac{R_1}{R_2} + 1 \right)$$

or to find R2 for a target Vin:

$$R_2 = \frac{R_1}{\frac{V_{in}}{V_{ref}} - 1}$$

A slightly different configuration can be found by swapping Vref and ground to make the circuit a typical summing amplifier. This may be useful when Vin is negative when referenced to ground.

$$V_{out} = - R_3 \left( \frac{V_{in}}{R_1} + \frac{V_{ref}}{R_2} \right)$$

Note, the target condition is now Vout=0V as the non-inverting input is grounded. When this is the case, above can be simplified to:

$$V_{in} = - V_{ref} \frac{R_1}{R_2}$$

K factor - phase boost
Based on H. Dean Venable's well known paper which defines a pole and a zero around a centre frequency for a given phase boost at that frequency. Phase boost will be maximum at this frequency.

Below these well documented formulae are 2 more sets of formulae which I have derived for finding the phase boost (&theta;) at any frequency (f) between a zero-pole pair rather than just the centre frequency, and for finding the frequencies between these zero-pole pairs which give a known phase boost.

RLC series circuits
$$X_L = 2 \pi f L$$    $$X_C = \frac{1}{2 \pi f C}$$

Total reactance $$X = X_L - X_C$$ because XL and XC are 180&deg; out of phase.

$$Z = \sqrt{R^2 + X^2} = \sqrt{R^2 + (X_L - X_C)^2}$$

At resonance XC=XL, therefore X=0, Z=R.

Gain of a low pass RLC filter
$$Gain = \frac{X_C}{Z} = \frac{X_C}{\sqrt{R^2 + (X_L - X_C)^2}}$$

$$Gain = \frac{X_C}{Z} = \frac{1}{2 \pi f C \sqrt{R^2 + \left(2 \pi f L - \frac{1}{2 \pi f C} \right) ^2}}$$

At resonance, Gain = XC/R.

Gain of a high pass RLC filter
$$Gain = \frac{X_L}{Z} = \frac{X_L}{\sqrt{R^2 + (X_L - X_C)^2}}$$

$$Gain = \frac{X_L}{Z} = \frac{2 \pi f L}{\sqrt{R^2 + \left(2 \pi f L - \frac{1}{2 \pi f C} \right) ^2}}$$

At resonance, Gain = XL/R.

Phase of a low pass RLC filter
$$\theta = - \arctan{ \left[ Q \left( \frac{f}{f_p} - \frac{f_p}{f} \right) \right]} - 90^\circ$$

where $$Q = \frac{1}{R} \sqrt{\frac{L}{C}}$$ and $$f_p = \frac{1}{2 \pi \sqrt{LC}}$$.

Therefore,

$$\theta = - \arctan{\left[ \frac{1}{R} \sqrt{\frac{L}{C}} \left( 2 \pi f \sqrt{LC} - \frac{1}{2 \pi f \sqrt{LC}} \right) \right]} - 90^\circ$$

Phase of a high pass RLC filter
$$\theta = - \arctan{ \left[ Q \left( \frac{f}{f_p} - \frac{f_p}{f} \right) \right]} + 90^\circ$$

$$\theta = - \arctan{\left[ \frac{1}{R} \sqrt{\frac{L}{C}} \left( 2 \pi f \sqrt{LC} - \frac{1}{2 \pi f \sqrt{LC}} \right) \right]} + 90^\circ$$

Inductor gap
$$l_g = \frac{\mu_0 L I^2_{max}}{B^2_{max} A_c}$$

$$B_{max} = \sqrt{\frac{\mu_0 L I^2_{max}}{l_g A_c}}$$

$$I_{max} = \sqrt{\frac{l_g B^2_{max} A_c}{\mu_0 L}}$$

$$\mu_0 = 4 \pi \cdot 10^{-7}$$ permeability of free space (H/m)

$$l_g$$ gap length (m)

$$L$$ Inductance (H)

$$I_{max}$$ max peak current (A)

$$B_{max}$$ max flux density (T)

$$A_c$$ core cross-sectional area (m2)

Snubber design
For an unknown inductance L resonating with an unknown capacitance C1 at a measurable frequency F, add capacitance C2 in parallel with C1 until the resonant frequency halves. From the LC frequency equation we know that for frequency to half, capacitance must be quadrupled, therefore C2=3*C1 and we can calculate C1 from C2. From F and C1, we can calculate the resonant impedance (reactance of C1 at the resonant frequency) and this becomes the snubber resistance Rsnb. Power lost in the snubber is proportional to Csnb so its value should be minimised while being significantly greater than C1. C2 may be used in the final snubber as its capacitance is arguably "significantly" greater than C1 (Usually in electronics significatly greater means 5-10x greater).

Modelling thermal system
'''THIS IS JUST ME THINKING OUT LOUD. DON'T TAKE THIS SECTION AS ANY RELIABLE SOURCE'''

Discharging capacitor through a resistor
A charged capacitor discharging through a resistor from voltage V0 to 0

$$V(t)=V_0 e^{-\frac{t}{\tau}}$$

$$\tau$$ is the time taken for voltage to fall from $$V_0$$ to $$V_0/e$$ when the end point is V=0.

$$\tau = RC$$

Transfer function:

$$H_C(s) = { V_C(s) \over V_{in}(s) } = { 1 \over 1 + RCs } = { 1 \over 1 + \tau s  } $$

Thermal equivalent
A hot object falling from starting temperature $$T_0$$ to room temperature $$T_{rt}$$:

$$T(t)=T_0 e^{-\frac{t}{\tau}}$$

$$\tau$$ is the time taken for temperature to fall to $$\frac{1}{e}(T_0 - T_{rt})$$

Transfer function:

$$H(s) = { T(s) \over T_{in}(s) }  = { 1 \over 1 + \tau s  } $$

Really I need a transfer function of T(s)/Pin(s).

This is a good start because it means I only have to measure temperature and time to get Tau, and not need to know the thermal equivalents of R and C. The end result should be a way to correctly tune a PID controller for an critically damped step response in a thermal system.