Compte-rendu d'APP
Thème:Résoudre un problème à l'aide des diagrammes de Bode
Collaborateurs: Agnus Jeremy, Fessard Valentin, Dorsemaine Thomas, Maignan Jeremy
I.Problématiques
Un oscilloscope permet des mesures sur un dispositif électronique, et les moyens assurant ces mesures peuvent pertuber plus ou moins le fonctionnement de ces dispositifs. Nous possédons un câble coaxial (assimilable à un condensateur de capacité de 100pF),et d'une sonde de tension passive.
Nous avons alors les problèmes suivants : La sonde permet elle une atténuation indépendante de la fréquence de travail? Quelle est la bande passante sous une impédance de source 1k ohm? Quelles sont les avantages d'une sonde par rapport à l'utilisation d'un cordon coaxial?
I.1 Objectifs
1) Tracer un diagramme de Bode avec une impédance source moyenne de 1k ohm pour le cas de l'utilisation d'une sonde. Déterminer au préalable la fonction de tranfert de ce diagramme.
2) Refaire ce diagramme pour le cas de l'utilisation d'une sonde avec compensation optimale. Puis évaluer les avantages et inconvénients que peut apporter une sonde par rapport au cable
3) A partir des résultats précédents,tracer les diagrammes asymptotiques de Bode dans le cas d'une surcompensation, d'une sous-compensation, et une compensation optimale.
I.2 Définitions importantes
Il convient au préalable de définir les termes suivants:
Fonction de transfert : relation entre la tension de sortie par rapport à la tension d'entrée.
Bande passante : Plage de fréquence correspondant à la différence entre la plus haute et la plus basse fréquence.
Augmenter une bande passante: Augmenter la différence entre la plus haute et la plus basse fréquence.
Tout d'abord nous utilisons un cable coaxial, ce qui donne le schéma suivant:
Z représentant l'équivalence de R1 et C1 en parallèle
A partir de ce schéma nous en tirons les égalités suivantes à l'aide de la loi d'ohm:
==>
R1 = ohm
Rg = ohm
C1 = F
(omega, f)
Z =
Grâce à nos relations on a :
i = = =
Soit
Ce qui donne la fonction de tranfert: =
(Vs, Ve, F)
F(omega) =
Phi
Phi =
P
(G, fo)
G =
-0.01429513651
Error in lines 1-1
Traceback (most recent call last):
File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute
exec compile(block+'\n', '', 'single') in namespace, locals
File "", line 1, in <module>
NameError: name 'G' is not defined
II.2 Cas de la sonde
Maintenant nous rajoutons une sonde, équivalent à une résistance R2=9 Mégaohm et à un condensateur C2=13.3pF en parallèles. Ce qui nous donne le schéma suivant:
Z2 représentant l'équivalence de R2 et C2 en parallèle.
R2 = ohm
C2 = F
Z2 =
Grâce à nos relations on a :
i = = =
Ce qui donne la fonction de tranfert: =
F2
F2(omega) =
P2
G2
G2 =
-20.04407480
II.3 Cas du cable et de la sonde réunis
(y1, y2, y3, y4, y5, omega, vs, ve, T, c1, c2, c3, R, i, jw)
(G, i)
Error in lines 3-3
Traceback (most recent call last):
File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1044, in execute
exec compile(block+'\n', '', 'single', flags=compile_flags) in namespace, locals
File "", line 1, in <module>
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 564, in wrapper
return func(*args, **options)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/plot/plot.py", line 1931, in plot
G = funcs.plot(*args, **original_opts)
File "sage/symbolic/expression.pyx", line 11796, in sage.symbolic.expression.Expression.plot (build/cythonized/sage/symbolic/expression.cpp:67212)
f = self._plot_fast_callable(param)
File "sage/symbolic/expression.pyx", line 11842, in sage.symbolic.expression.Expression._plot_fast_callable (build/cythonized/sage/symbolic/expression.cpp:67707)
return fast_callable(self, vars=vars, expect_one_var=True)
File "sage/ext/fast_callable.pyx", line 465, in sage.ext.fast_callable.fast_callable (build/cythonized/sage/ext/fast_callable.c:4633)
et = x._fast_callable_(etb)
File "sage/symbolic/expression.pyx", line 11679, in sage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:66335)
return fast_callable(self, etb)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1757, in fast_callable
return FastCallableConverter(ex, etb)()
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 217, in __call__
return self.arithmetic(div, div.operator())
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1685, in arithmetic
return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1685, in <lambda>
return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
File "sage/ext/fast_callable.pyx", line 751, in sage.ext.fast_callable.ExpressionTreeBuilder.call (build/cythonized/sage/ext/fast_callable.c:7236)
return ExpressionCall(self, fn, map(self, args))
File "sage/ext/fast_callable.pyx", line 626, in sage.ext.fast_callable.ExpressionTreeBuilder.__call__ (build/cythonized/sage/ext/fast_callable.c:6241)
return fc(self)
File "sage/symbolic/expression.pyx", line 11679, in sage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:66335)
return fast_callable(self, etb)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1757, in fast_callable
return FastCallableConverter(ex, etb)()
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 226, in __call__
return self.composition(ex, operator)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1723, in composition
return self.etb.call(function, *ex.operands())
File "sage/ext/fast_callable.pyx", line 751, in sage.ext.fast_callable.ExpressionTreeBuilder.call (build/cythonized/sage/ext/fast_callable.c:7236)
return ExpressionCall(self, fn, map(self, args))
File "sage/ext/fast_callable.pyx", line 626, in sage.ext.fast_callable.ExpressionTreeBuilder.__call__ (build/cythonized/sage/ext/fast_callable.c:6241)
return fc(self)
File "sage/symbolic/expression.pyx", line 11679, in sage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:66335)
return fast_callable(self, etb)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1757, in fast_callable
return FastCallableConverter(ex, etb)()
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 217, in __call__
return self.arithmetic(div, div.operator())
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1685, in arithmetic
return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1685, in <lambda>
return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
File "sage/ext/fast_callable.pyx", line 751, in sage.ext.fast_callable.ExpressionTreeBuilder.call (build/cythonized/sage/ext/fast_callable.c:7236)
return ExpressionCall(self, fn, map(self, args))
File "sage/ext/fast_callable.pyx", line 626, in sage.ext.fast_callable.ExpressionTreeBuilder.__call__ (build/cythonized/sage/ext/fast_callable.c:6241)
return fc(self)
File "sage/symbolic/expression.pyx", line 11679, in sage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:66335)
return fast_callable(self, etb)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1757, in fast_callable
return FastCallableConverter(ex, etb)()
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 218, in __call__
return self.arithmetic(ex, operator)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1685, in arithmetic
return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1685, in <lambda>
return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
File "sage/ext/fast_callable.pyx", line 751, in sage.ext.fast_callable.ExpressionTreeBuilder.call (build/cythonized/sage/ext/fast_callable.c:7236)
return ExpressionCall(self, fn, map(self, args))
File "sage/ext/fast_callable.pyx", line 626, in sage.ext.fast_callable.ExpressionTreeBuilder.__call__ (build/cythonized/sage/ext/fast_callable.c:6241)
return fc(self)
File "sage/symbolic/expression.pyx", line 11679, in sage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:66335)
return fast_callable(self, etb)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1757, in fast_callable
return FastCallableConverter(ex, etb)()
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 217, in __call__
return self.arithmetic(div, div.operator())
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1685, in arithmetic
return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1685, in <lambda>
return reduce(lambda x,y: self.etb.call(operator, x,y), operands)
File "sage/ext/fast_callable.pyx", line 751, in sage.ext.fast_callable.ExpressionTreeBuilder.call (build/cythonized/sage/ext/fast_callable.c:7236)
return ExpressionCall(self, fn, map(self, args))
File "sage/ext/fast_callable.pyx", line 626, in sage.ext.fast_callable.ExpressionTreeBuilder.__call__ (build/cythonized/sage/ext/fast_callable.c:6241)
return fc(self)
File "sage/symbolic/expression.pyx", line 11679, in sage.symbolic.expression.Expression._fast_callable_ (build/cythonized/sage/symbolic/expression.cpp:66335)
return fast_callable(self, etb)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1757, in fast_callable
return FastCallableConverter(ex, etb)()
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 212, in __call__
return self.symbol(ex)
File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1706, in symbol
return self.etb.var(SR(ex))
File "sage/ext/fast_callable.pyx", line 698, in sage.ext.fast_callable.ExpressionTreeBuilder.var (build/cythonized/sage/ext/fast_callable.c:6817)
raise ValueError("Variable '%s' not found" % var_name)
ValueError: Variable 'c1' not found