author Second order equation Leif Roschier

## Theory and background

Radio-frequency single-electron transistor (RF-SET) is a sensitive charge detector. It's charge sensitivity $\delta q [e/\sqrt{Hz}]\,$ in normal (not superconducting) operation is typically set by pre-amplifier noise temperature $T_0 [K]\,$, charging energy $E_C [J]\,$, transistor island electron temperature $T [K]\,$, SET high bias DC resistance $R_\Sigma [\Omega]\,$ and LC-transformer impedance $Z_{TR} [\Omega]\,$ according to relation [1]

$\delta q \approx \frac{2(3\frac{R_\Sigma}{Z_{TR}}+\frac{Z_{TR}}{Z_T}) \sqrt{k_B T_0 Z_T}}{2\times 0.41 t^{-1.74} 0.9 E_C/e^2}. \,$

#### References

1. L. Roschier, M. Sillanpää, W. Taihong, M. Ahlskog, S. Iijima and P. Hakonen, “Carbon nanotube radio-frequency single-electron transistor”, Journal of Low Temperature Physics, 136, 465 (2004).

## Construction of the nomograph

The equation is written as $-\log(\delta q)+x+\frac{1}{2}\log(k_B T_0 Z_T)-\log(0.9(k_B E_C)^{2.74}-\log(0.41(k_B T)^{-1.74}/e)\,$

split into three equations that each are blocks:

 $-\log(\delta q)+x+\frac{1}{2}\log(k_B T_0 Z_T)-\log(0.9(k_B E_C)^{2.74}-\log(0.41(k_B T)^{-1.74}/e)=0\,$ type 3 $\exp(x) =$ type 2 $\frac{R_1}{E}= \frac{F}{R_2} \,\,\,\,\,\,\,$ type 4

## Generated nomograph

Second order equation
Generated portable document file (pdf): File:Ex second order eq.pdf

## Source code

"""
ex_second_order_eq.py

Second order equation: z**2+p*z+q=0

This program is free software: you can redistribute it and/or modify
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <http://www.gnu.org/licenses/>.
"""
from nomographer import *

N_params_1={
'u_min':-10.0,
'u_max':10.0,
'function':lambda u:u,
'title':r'$p$',
'tick_levels':3,
'tick_text_levels':2,
'tick_side':'left'
}

N_params_2={
'u_min':-10.0,
'u_max':10.0,
'function':lambda u:u,
'title':r'$q$',
'tick_levels':3,
'tick_text_levels':2,
'tick_side':'right',
}

N_params_3={
'u_min':0.0,
'u_max':5.0,
'function_3':lambda u:u,
'function_4':lambda u:u**2,
'title':r'$z$',
'tick_levels':0,
'tick_text_levels':0,
'title_draw_center':True,
'title_opposite_tick':False,
'extra_params':[{'tick_side':'left',
'u_min':0.1,
'u_max':12.0,
'tick_text_levels':2,
'tick_levels':3
}]
}

block_1_params={
'block_type':'type_10',
'width':10.0,
'height':10.0,
'f1_params':N_params_1,
'f2_params':N_params_2,
'f3_params':N_params_3,
}

main_params={
'filename':'ex_second_order_eq.pdf',
'paper_height':10.0,
'paper_width':10.0,
'block_params':[block_1_params],
'transformations':[('rotate',0.01),('scale paper',)],
'title_str':r'$z^2+pz+q=0$'
}
Nomographer(main_params)