16. Equations
NB Equations apply to all units regardless of quantity expressed, e.g. milli: m, micro: µ, etc.
Var. = variable; SI = International System of Units
SYMBOLS = Lucida Console font; supported symbols from MS Windows character map:
⌠ ⌡ ∫ │ ─ √ φ θ Θ ∂ δ ζ ξ ς λ ψ ω τ µ Ω ∆ Δ ∑ ∏ π ∞ ∝ Ξ ○ ≠ ³ ² ± ≤ ∴
ELECTRONICS
Q1 OHMS LAW
Q1a V = IR
Q1b I = V/R
Q1c R = V/I
Q1d P = IV
Q1e I = P/V
Q1f V = P/I
Q1g P = V²/R
Q1h V = √PR
Q1i R = V²/P
Q2 OSCILLOSCOPE BANDWIDTH & RISE TIME
Q2a BANDWIDTH wrt RISE TIME
A close approximation of oscilloscope bandwidth can be determined from:
Bandwidth = 0.35
Rise time
Q2b RISE TIME wrt BANDWIDTH
Rise time = 0.35
Bandwidth
Where
Variable Unit SI
Bandwidth = frequency Hz
Rise time = seconds s
Example:
The Tek 7104 has a nominal bandwidth of 1GHz, so its rise time = 0.35 / 1GHz = 350ps
https://w140.com/tekwiki/wiki/7104
The actual value will differ slightly from this due to manufacturing tolerances of the CRT. The original specification for the 7104 with a 7A29 states it is <350ps. Here are older 1GHz 519s rated at 300ps:
https://vintagetek.org/wp-content/uploads/2017/09/519-screen.jpg
https://vintagetek.org/519-1-ghz-oscilloscope/
https://w140.com/tekwiki/wiki/519
Q3 OSCILLOSCOPE MAINFRAME RISE TIME wrt SIGNAL RISE TIME
Q3a MAINFRAME RISE TIME
This webpage explains in detail how to calculate oscilloscope rise time from the following equation:
t²rise measurement = t²mainframe + t²signal
________________________________
Oscilloscope mainframe rise time = t mainframe = √ (t²rise measurement - t²signal)
This can be determined using a signal source with a known, calibrated rise time, such as Tek's own 7k series Calibration Fixture plugins, e.g. 067-0587-01, or the faster 067-0587-02 needed for the 7104.
Example:
Tek 1GHz 7104 + 067-0587-02 Calibration Fixture with nominal 150ps max rise time. The actual value will differ slightly from this due to manufacturing tolerances of the fixture. The above webpage assumes their fixture's rise time is 20% faster at 120ps, but I have used its specified nominal value of 150ps in the equation below.
On the above webpage, the rise time of this signal measured on the 7104 = 300ps.
The nominal rise time of the Calibration Fixture = 150ps
________________________________
t mainframe = √ (t²rise measurement - t²signal)
__________________
t 7104 = √ (300ps² - 150ps²) = 260ps.
So based on a 150ps Cal Fixture rise time, the 7104 mainframe rise time is 260ps.
Applying [Q2], Bandwidth = 1 x 0.35 = 1 x 0.35 = 1.346GHz
Rise time 260ps
Q3b TRUE SIGNAL RISE TIME
This same equation can be used to determine true signal rise time from the displayed value:
t²rise measurement = t²mainframe + t²signal
___________________________________
Actual signal rise time = t signal = √ (t²rise measurement - t²mainframe)
Example:
Tek 7104 1GHz + 067-0587-02 calibration fixture with nominal 150ps max rise time. The actual value will differ slightly from this due to manufacturing tolerances of the fixture. The above webpage assumes their fixture's rise time is 20% faster at 120ps but I have used its specified nominal value of 150ps in the equation below.
From equation [Q2] above, the calculated mainframe rise time = 260ps.
From the webpage above, the rise time of the Cal Fixture measured on the CRT = 300ps
__________________________________
t signal = √ (t²rise measurement - t²mainramw)
__________________
t 7104 = √ (300ps² - 260ps²) = 150ps.
So the actual rise time of the Cal Fixture signal = 150ps.
Q4 NYQUIST SAMPLING RATE
Harry Nyquist determined the optimum sampling rate that can be applied to DSOs.
If the signal consists of L discrete levels, Nyquist's theorem states:
Q4a BitRate = 2 x Bandwidth x log2(L) bits/sec.
Where
bandwidth is the bandwidth of the channel
L is the number of signal levels used to represent data,
BitRate is the bit rate in bits per second.
https://en.wikipedia.org/wiki/Nyquist_rate
This is referred to as the Nyquist sampling rate, and is usually simplified to x3 to x5:
[E58]: 'To capture the true shape of the signal, you need a scope with a bandwidth large enough to capture several of the signals harmonics, so ideally use a scope with 3x to 5x the bandwidth you calculated for your signal.'
Q4b MAXIMUM BANDWIDTH RATE FOR A PULSE (based on [4d] below)
Maximum signal frequency = 0.4 / rise time (20-80%)
Q4c PULSE RISE TIME FOR MAXIMUM BANDWIDTH (based on [4d] below)
rise time = 0.4 / Maximum signal frequency
Q4d MINIMUM BANDWIDTH SAMPLE RATE FOR A DSO TO CAPTURE A PULSE
Absolute minimum: maximum signal frequency x 3
Or better: maximum signal frequency x 3.5, for example:
From [I24], HP54854A 4GHz 20GS/s DSO datasheet, page 4:
Bandwidth required to measure rise time with 3% error. Example: 100ps rise time (20-80%)
Maximum signal frequency content = 0.4/rise time (20-80%) Maximum signal frequency = 4GHz
N.B. HP complicates this issue as their constant 0.4 differs from [Q2a,Q2b]: 0.35.
I suspect either is usable, and HP is increasing the signal quality by rising to 0.4.
I've also seen it relaxed to 0.3/Tr. [E57, ETS Current Probes Page 42].
Until I know better, I will leave both [Q2] and [Q4] here.
Scope bandwidth required = 1.4 x maximum frequency Required scope bandwidth = 5.6GHz
Minimum scope sample rate required = 2.5 x bandwidth Required scope sample rate = 14GS/s
IOW maximum signal frequency x3.5 (3.5 = 1.4 x 2.5)
Q4e OPTIMUM BANDWIDTH SAMPLE RATE FOR FOR A DSO
Maximum signal frequency x 5
PHYSICS
Q5: ENERGY IS POWER EXPENDED OVER TIME
E = P x T = energy in Joules, J
Where
SU Description Unit SI
E Energy Joules J
P Power Watts W
T Time Seconds s
Worked examples given E = 24mJ, P = 6MW, T = 4ns:
Q5a E = P x T = 6MW x 4ns = 24mJ
Q5b P = E = 24mJ = 6MW
T 4ns
Q5c T = E = 24mJ = 4ns
P 6MW
Q6: CAPACITIVE ENERGY
E = 0.5 x C x V² = energy in Joules, J
Where
SU Description Unit SI
C Capacitance Farads F
V Voltage Volts V
Worked example for MK367 Nd:YAG, given C = 20µF. V = 673V:
Q6a E = 0.5 x C x V² = 0.5 x 20µF x 673V² = 4.53J
________ ___________
Q6b V = / E = / 4.53J = 673V
√ 0.5 x C √ 0.5 x 20µF
Q6c C = E = 4.53J = 20µF
0.5 x V² 0.5 x 673²
Q7: FLASHLAMP PFN PULSE ENERGY
THIS NEEDS CHECKING - CHICKEN & EGG & NOT SURE IF IT'S CORRECT:
From [O19] CORD Module 3-2: Pulsed Laser Flashlamps and Power Supplies
Where
Var Description Unit SI
r = load resistance (lamp resistance) Ohms Ω
L = total network inductance Henries H
C = total network capacitance Farads F
T = pulse width at 70% of average peak amplitude seconds s
Worked examples given L = 22µH, C = 20µF:
Q7a SIMPLE #1 MESH LCR LOAD RESISTANCE
____
L = r²C therefore r = / 4L
4 √ C
___ _________
r = / 4L = / 4 x 22µH = 2.097Ω
√ C √ 20µF
Q7b CR CHARGING TIME
C = T therefore T = C x r
r
T = C x r = 20µF x 2.097Ω = 41.94µs
Q7c #5 OR MORE MESH PFN
___ _____
r = / L = / 22µH = 1.049Ω
√ C √ 20µF
___ ____________
T = 2√ LC = 2√ 22µH x 20µF = 41.952µs
Or,
L = T x r = 41.952µs x 1.049Ω = 22µH
2 2
Q10 GAS LAWS
https://en.wikipedia.org/wiki/Gas_laws
[Q10a] GAY-LASSAC'S LAW
'For a given mass and constant volume of an ideal gas, the pressure exerted on the sides of its container is directly proportional to its absolute temperature'.
P ∝ T or P/T = k or P1/T1 = P2/T2
where
P = pressure
T = absolute temperature
k = a proportionality constant.
Q10b BOYLE'S LAW
https://en.wikipedia.org/wiki/Boyle%27s_law
'The absolute pressure exerted by a given mass of an ideal gas is inversely proportional to the volume it occupies if the temperature and amount of gas remain unchanged within a closed system.
The volume of a given mass of a gas is inversely related to pressure when the temperature is constant.'
PV= k1
where
P = pressure of the system
V = volume of the gas
k1 = constant value representative of the temperature and volume of the system.
Q10c CHARLES'S LAW
https://en.wikipedia.org/wiki/Charles%27s_law
'for a given mass of an ideal gas at constant pressure, the volume is directly proportional to its absolute temperature, assuming in a closed system.
The volume (V) of a given mass of a gas, at constant pressure (P), is directly proportional to its temperature (T)'
V ∝ T or V/T = k2 or V1/T1 = V2/T2
here
P = pressure of the system
V = volume of the gas
k2 = proportionality constant
(k2 is not the same as the proportionality constants in the other equations in this section)
Q20 LASER EXTRA-CAVITY POWER
Power of beam area = watts / π x radius²
Example: 0.5J @ 10ns pulse = 50MW output pulse
For a 2mm dia 50MW beam = 50MW/(π x 0.10cm²) = 1.59GW/cm² = 1.60GW
For a 4mm dia 50MW beam = 50MW/(π x 0.20cm²) = 398MW/cm² = 0.40GW
For a 5mm dia 50MW beam = 50MW/(π x 0.25cm²) = 255MW/cm² = 0.26GW
For a 6mm dia 50MW beam = 50MW/(π x 0.30cm²) = 177MW/cm² = 0.18GW
Q21 OPO NLOs
This webpage is an excellent primer for NLOs in general:
http://www.repairfaq.org/sam/laserssl.htm#sslsrgs
'Optical Parametric Oscillation (OPO) is a nonlinear process in which a single input laser beam or "pump" beam is converted into two lower-energy beams known as the "signal" beam and the "idler" beam.
This nonlinear process enables a fixed wavelength laser beam to be converted into other wavelengths. The wavelengths (λ) / frequencies (f) of the three beams must satisfy:
1 1 1
λ(Pump) = λ(Signal) + λ(Idler)
or equivalently: f(Pump) = f(Signal) + f(Idler)
Q30 CONVERT ELECTRON-VOLT (eV) TO WAVELENGTH & VICE-VERSA
From Wikipedia (https://en.wikipedia.org/wiki/Electronvolt):
Were
Var Description Unit SI
E = energy electron-volt eV
F = frequency Hertz Hz
λ = wavelength of a photon Distance nm
h = Planck constant, 4.135667516 Speed m/s
c = speed of light constant, 299792458m/s Speed m/s
E = hF = hc (4.135667516 x 10^-15eV)(299792458 m/s)
λ λ (nm)
This reduces to:
E (eV) = 4.135667516 feVs x F (PHz) = 1239.84193 eV-nm
λ (nm)
Or,
λ (nm) = 1239.84193 eV-nm
E (eV)
MATHEMATICS
Q50 CIRCUMFERENCE OF A CIRCLE
distance = π x diameter
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