Difference between revisions of "Example:Radio-frequency single electron transistor"

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(Theory and background)
(Construction of the nomograph)
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== Construction of the nomograph ==
 
== Construction of the nomograph ==
A [[type 10]] is the block for the construction.
+
The equation is written as
 +
<math>-\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)\,</math>
 +
 
 +
split into three equations that each are blocks:
 +
 
 +
{| style="background:wihte; color:black" border="1" cellpadding="5" cellspacing="0"
 +
|-
 +
| <math>-\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\,</math> || [[type 3]]
 +
|-
 +
| <math>\exp(x) = </math> || [[type 2]]
 +
|-
 +
| <math>\frac{R_1}{E}= \frac{F}{R_2} \,\,\,\,\,\,\,</math> || [[type 4]]
 +
|}
  
 
== Generated nomograph ==
 
== Generated nomograph ==

Revision as of 21:04, 13 September 2008

Second order equation
Ex second order eq.png
author Leif Roschier

Theory and background

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

[math]\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}. \,[/math]

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 [math]-\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)\,[/math]

split into three equations that each are blocks:

[math]-\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\,[/math] type 3
[math]\exp(x) = [/math] type 2
[math]\frac{R_1}{E}= \frac{F}{R_2} \,\,\,\,\,\,\,[/math] type 4

Generated nomograph

Second order equation
Ex second order eq.png
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
 
    Copyright (C) 2007-2008  Leif Roschier
 
    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    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)