Circuitos Rc
INTRODUCCIN
El documento a continuacin presentado, muestra la teora general
utilizada para el anlisis de circuitos RC, RL y RLC. Se demostrarn
sus ecuaciones normales y algunas de sus propiedades fsicas.
CIRCUITOS RC
Los circuitos RC son circuitos que estn compuestos por una
resistencia y un condensador.
Se caracteriza por que la corriente puede variar con el tiempo.
Cuando el tiempo es igual a cero, el condensador est descargado, en
el momento que empieza a correr el tiempo, el condensador comienza
a cargarse ya que hay una corriente en
el circuito. Debido al espacio entre las placas del condensador,
en el circuito no circula corriente, es por eso que se utiliza una
resistencia.
Cuando el condensador se carga completamente, la corriente en el
circuito es igual a cero.
La segunda regla de Kirchoff dice: V = (IR) (q/C)
Donde q/C es la diferencia de potencial en el condensador.
En un tiempo igual a cero, la corriente ser: I = V/R cuando el
condensador no se ha cargado.
Cuando el condensador se ha cargado completamente, la corriente
es cero y la carga ser igual a: Q = CV
CARGA DE UN CONDENSADOR
Ya se conoce que las variables dependiendo del tiempo sern I y
q. Y la corriente I se sustituye por dq/dt (variacin de la carga
dependiendo de la variacin del tiempo):
(dq/dt)R = V (q/C)
dq/dt = V/R (q/(RC))
Esta es una ecuacin
Diferencial. Se pueden dq/dt = (VC q)/(RC)
Separar variable dq/(q VC) = - dt/(RC)
Al integrar se tiene ln [ - (q VC)/VC)] = -t/(RC)
Despejando q q dt = C V [(1 e-t/RC )] = q (1- e-t/RC ) El
voltaje ser ) = V DESCARGA DE UN CONDENSADOR
Debido a que la diferencia de potencial en el condensador es IR
= q/C, la razn de cambio de carga en el condensador determinar la
corriente en el circuito, por lo tanto, la ecuacin que resulte de
la relacin entre el cambio de la cantidad de carga dependiendo del
cambio en el tiempo y la corriente en el circuito, estar dada
remplazando I = dq/dt en la ecuacin de diferencia de potencial en
el condensador:
q = Q e-t/RC
Donde Q es la carga mxima
La corriente en funcin del tiempo entonces, resultar al derivar
esta ecuacin respecto al tiempo:
I = Q/(RC) e-t/RCSe puede concluir entonces, que la corriente y
la carga decaen de forma exponencial.CIRCUITOS RL
Los circuitos RL son aquellos que contienen una bobina
(inductor) que tiene autoinductancia, esto quiere decir que evita
cambios instantneos en la corriente. Siempre se desprecia la
autoinductancia en el resto del circuito puesto que se considera
mucho menor a la del inductor.
Para un tiempo igual a cero, la corriente comenzar a crecer y el
inductor producir igualmente una fuerza electromotriz en sentido
contrario, lo cual har que la corriente no aumente. A esto se le
conoce como fuerza contraelectromotriz.
Esta fem est dada por: V = -L (inductancia) dI/dt
Debido a que la corriente aumentar con el tiempo, el cambio ser
positivo (dI/dt) y la tensin ser negativa al haber una cada de la
misma en el inductor.
Segn kirchhoff: V = (IR) + [L (dI / dt)]
IR = Cada de voltaje a travs de la resistencia.
Esta es una ecuacin diferencial y se puede hacer la
sustitucin:
x = (V/R) I
es decir;dx = -dI
Sustituyendo en la ecuacin: x + [(L/R)(dx/dt)] = 0
dx/x = - (R/L) dt
Integrando: ln (x/xo) = -(R/L) t
Despejando x: x = xo e Rt / LDebido a que xo = V/R
El tiempo es cero
Y corriente cero V/R I = V/R e Rt / L
I = (V/R) (1 - e Rt / L)El tiempo del circuito est representado
por ( = L/R
I = (V/R) (1 e 1/()Donde para un tiempo infinito, la corriente
de la malla ser I = V/R. Y se puede considerar entonces el cambio
de la corriente en el tiempo como cero.
Para verificar la ecuacin que implica a ( y a I, se deriva una
vez y se reemplaza en la inicial: dI/dt = V/L e 1/(Se sustituye: V
= (IR) + [L (dI / dt)]
V = [ (V/R) (1 e 1/()R + (L V/ L e 1/()]
V V e 1/( = V V e 1/(OSCILACIONES EN UN CIRCUITO LC
Cuando un condensador se conecta a un inductor, tanto la
corriente como la carga den el condensador oscila. Cuando existe
una resistencia, hay una disipacin de energa en el sistema porque
una cuanta se convierte en calor en la resistencia, por lo tanto
las oscilaciones son amortiguadas. Por el momento, se ignorar la
resistencia.
En un tiempo igual a cero, la carga en el condensador es mxima y
la energa almacenada en el campo elctrico entre las placas es U =
Q2mx/(2C). Despus de un tiempo igual a cero, la corriente en el
circuito comienza a aumentar y parte de la energa en el condensador
se transfiere al inductor. Cuando la carga almacenada en el
condensador es cero, la corriente es mxima y toda la energa est
almacenada en el campo elctrico del inductor. Este proceso se
repite de forma inversa y as comienza a oscilar.
En un tiempo determinado, la energa total del sistema es igual a
la suma de las dos energas (inductor y condensador): U = Uc + ULU =
[ Q2/(2C) ] + ( LI2/2 )
CIRCUITO RLC
Un circuito RLC es aquel que tiene como componentes una
resistencia, un condensador y un inductor conectados en serie
En un tiempo igual a cero, el condensador tiene una carga mxima
(Qmx). Despus de un tiempo igual a cero, la energa total del
sistema est dada por la ecuacin presentada en la seccin de
oscilaciones en circuitos LC
U = [ Q2/(2C) ] + ( LI2/2 )
En las oscilaciones en circuitos LC se haba mencionado que las
oscilaciones no eran amortiguadas puesto que la energa total se
mantena constante. En circuitos RLC, ya que hay una resistencia,
hay oscilaciones amortiguadas porque hay una parte de la energa que
se transforma en calor en la resistencia.El cambio de la energa
total del sistema dependiendo del tiempo est dado por la disipacin
de energa en una resistencia:
dU/dt = - I2R
Luego se deriva la ecuacin de la energa total respecto al tiempo
y se remplaza la dada: LQ + RQ + (Q/C) = 0
Se puede observar que el circuito RCL tiene un comportamiento
oscilatorio amortiguado:
m(d2x/dt2) + b(dx/dt) + kx = 0
Si se tomara una resistencia pequea, la ecuacin cambiara a :
Q = Qmx e -(Rt/2L)Cos wt
w = [ (1/LC) (R/2L)2 ] 1/2Entre ms alto el valor de la
resistencia, la oscilacin tendr amortiguamiento ms veloz puesto que
absorbera ms energa del sistema. Si R es igual a (4L/C) el sistema
se encuentra sobreamortiguado.
carga
tiempo
CONCLUSIONES
Se visualiz la configuracin general para los circuitos RC, RL y
RLC.
Se present las propiedades fsicas generales de los circuitos RC,
RL y RLC.
Se establecieron las ecuaciones para carga y descarga de un
condensador en los circuitos RC.
Se mostr la ecuacin general para la corriente en un circuito RL,
as como el tiempo dado por la relacin entre resistencia e
inductancia.
Se entendieron las propiedades de los circuitos RLC.
Se expuso las ecuaciones generales para el anlisis de circuitos
RLC.
BIBLIOGRAFA
SERWAY. Fsica Tomo II Cuarta edicin. Ed Mc Graw Hill.
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61.2-0
61.3-0
61.4-0
61.5-0
61.6-0
61.7-0
61.8-0
61.9-0
62-0
62.1-0
62.2-0
62.30
62.40
62.50
62.60
62.70
62.80
62.90
630
63.10
63.20
63.30
63.40
63.50
63.6-0
63.7-0
63.8-0
63.9-0
64-0
64.1-0
64.2-0
64.3-0
64.4-0
64.5-0
64.6-0
64.7-0
64.8-0
64.90
650
65.10
65.20
65.30
65.40
65.50
65.60
65.70
65.80
65.90
660
66.10
66.2-0
66.3-0
66.4-0
66.5-0
66.6-0
66.7-0
66.8-0
66.9-0
67-0
67.1-0
67.2-0
67.3-0
67.4-0
67.50
67.60
67.70
67.80
67.90
680
68.10
68.20
68.30
68.40
68.50
68.60
68.70
68.8-0
68.9-0
69-0
69.1-0
69.2-0
69.3-0
69.4-0
69.5-0
69.6-0
69.7-0
69.8-0
69.9-0
70-0
70.10
70.20
70.30
70.40
70.50
70.60
70.70
70.80
70.90
710
71.10
71.20
71.30
71.40
71.5-0
71.6-0
71.7-0
71.8-0
71.9-0
72-0
72.1-0
72.2-0
72.3-0
72.4-0
72.5-0
72.6-0
72.7-0
72.80
72.90
730
73.10
73.20
73.30
73.40
73.50
73.60
73.70
73.80
73.90
740
74.1-0
74.2-0
74.3-0
74.4-0
74.5-0
74.6-0
74.7-0
74.8-0
74.9-0
75-0
75.1-0
75.2-0
75.3-0
75.40
75.50
75.60
75.70
75.80
75.90
760
76.10
76.20
76.30
76.40
76.50
76.60
76.7-0
76.8-0
76.9-0
77-0
77.1-0
77.2-0
77.3-0
77.4-0
77.5-0
77.6-0
77.7-0
77.8-0
77.9-0
780
78.10
78.20
78.30
78.40
78.50
78.60
78.70
78.80
78.90
790
79.10
79.20
79.3-0
79.4-0
79.5-0
79.6-0
79.7-0
79.8-0
79.9-0
80-0
80.1-0
80.2-0
80.3-0
80.4-0
80.5-0
80.60
80.70
80.80
80.90
810
81.10
81.20
81.30
81.40
81.50
81.60
81.70
81.80
81.9-0
82-0
82.1-0
82.2-0
82.3-0
82.4-0
82.5-0
82.6-0
82.7-0
82.8-0
82.9-0
83-0
83.1-0
83.20
83.30
83.40
83.50
83.60
83.70
83.80
83.90
840
84.10
84.20
84.30
84.40
84.50
84.6-0
84.7-0
84.8-0
84.9-0
85-0
85.1-0
85.2-0
85.3-0
85.4-0
85.5-0
85.6-0
85.7-0
85.8-0
85.90
860
86.10
86.20
86.30
86.40
86.50
86.60
86.70
86.80
86.90
870
87.10
87.2-0
87.3-0
87.4-0
87.5-0
87.6-0
87.7-0
87.8-0
87.9-0
88-0
88.1-0
88.2-0
88.3-0
88.4-0
88.50
88.60
88.70
88.80
88.90
890
89.10
89.20
89.30
89.40
89.50
89.60
89.70
89.8-0
89.9-0
90-0
90.1-0
90.2-0
90.3-0
90.4-0
90.5-0
90.6-0
90.7-0
90.8-0
90.9-0
91-0
91.10
91.20
91.30
91.40
91.50
91.60
91.70
91.80
91.90
920
92.10
92.20
92.30
92.4-0
92.5-0
92.6-0
92.7-0
92.8-0
92.9-0
93-0
93.1-0
93.2-0
93.3-0
93.4-0
93.5-0
93.6-0
93.70
93.80
93.90
940
94.10
94.20
94.30
94.40
94.50
94.60
94.70
94.80
94.90
95-0
95.1-0
95.2-0
95.3-0
95.4-0
95.5-0
95.6-0
95.7-0
95.8-0
95.9-0
96-0
96.1-0
96.2-0
96.30
96.40
96.50
96.60
96.70
96.80
96.90
970
97.10
97.20
97.30
97.40
97.50
97.60
97.7-0
97.8-0
97.9-0
98-0
98.1-0
98.2-0
98.3-0
98.4-0
98.5-0
98.6-0
98.7-0
98.8-0
98.9-0
990
99.10
99.20
99.30
99.40
99.50
99.60
99.70
99.80
99.90
1000
100.10
100.20
100.3-0
100.4-0
100.5-0
100.6-0
100.7-0
100.8-0
100.9-0
101-0
101.1-0
101.2-0
101.3-0
101.4-0
101.5-0
101.60
101.70
101.80
101.90
1020
102.10
102.20
102.30
102.40
102.50
102.60
102.70
102.80
102.9-0
103-0
103.1-0
103.2-0
103.3-0
103.4-0
103.5-0
103.6-0
103.7-0
103.8-0
103.9-0
104-0
104.1-0
104.20
104.30
104.40
104.50
104.60
104.70
104.80
104.90
1050
105.10
105.20
105.30
105.40
105.5-0
105.6-0
105.7-0
105.8-0
105.9-0
106-0
106.1-0
106.2-0
106.3-0
106.4-0
106.5-0
106.6-0
106.7-0
106.80
106.90
1070
107.10
107.20
107.30
107.40
107.50
107.60
107.70
107.80
107.90
1080
108.1-0
108.2-0
108.3-0
108.4-0
108.5-0
108.6-0
108.7-0
108.8-0
108.9-0
109-0
109.1-0
109.2-0
109.3-0
109.40
109.50
109.60
109.70
109.80
109.90
1100
110.10
110.20
110.30
110.40
110.50
110.60
110.70
110.8-0
110.9-0
111-0
111.1-0
111.2-0
111.3-0
111.4-0
111.5-0
111.6-0
111.7-0
111.8-0
111.9-0
112-0
112.10
112.20
112.30
112.40
112.50
112.60
112.70
112.80
112.90
1130
113.10
113.20
113.30
113.4-0
113.5-0
113.6-0
113.7-0
113.8-0
113.9-0
114-0
114.1-0
114.2-0
114.3-0
114.4-0
114.5-0
114.6-0
114.70
114.80
114.90
1150
115.10
115.20
115.30
115.40
115.50
115.60
115.70
115.80
115.90
116-0
116.1-0
116.2-0
116.3-0
116.4-0
116.5-0
116.6-0
116.7-0
116.8-0
116.9-0
117-0
117.1-0
117.2-0
117.30
117.40
117.50
117.60
117.70
117.80
117.90
1180
118.10
118.20
118.30
118.40
118.50
118.6-0
118.7-0
118.8-0
118.9-0
119-0
119.1-0
119.2-0
119.3-0
119.4-0
119.5-0
119.6-0
119.7-0
119.8-0
119.90
1200
120.10
120.20
120.30
120.40
120.50
120.60
120.70
120.80
120.90
1210
121.10
121.2-0
121.3-0
121.4-0
121.5-0
121.6-0
121.7-0
121.8-0
121.9-0
122-0
122.1-0
122.2-0
122.3-0
122.4-0
122.50
122.60
122.70
122.80
122.90
1230
123.10
123.20
123.30
123.40
123.50
123.60
123.70
123.80
123.9-0
124-0
124.1-0
124.2-0
124.3-0
124.4-0
124.5-0
124.6-0
124.7-0
124.8-0
124.9-0
125-0
125.1-0
125.20
125.30
125.40
125.50
125.60
125.70
125.80
125.90
1260
126.10
126.20
126.30
126.40
126.5-0
126.6-0
126.7-0
126.8-0
126.9-0
127-0
127.1-0
127.2-0
127.3-0
127.4-0
127.5-0
127.6-0
127.7-0
127.80
127.90
1280
128.10
128.20
128.30
128.40
128.50
128.60
128.70
128.80
128.90
1290
129.1-0
129.2-0
129.3-0
129.4-0
129.5-0
129.6-0
129.7-0
129.8-0
129.9-0
130-0
130.1-0
130.2-0
130.3-0
130.40
130.50
130.60
130.70
130.80
130.90
1310
131.10
131.20
131.30
131.40
131.50
131.60
131.7-0
131.8-0
131.9-0
132-0
132.1-0
132.2-0
132.3-0
132.4-0
132.5-0
132.6-0
132.7-0
132.8-0
132.9-0
1330
133.10
133.20
133.30
133.40
133.50
133.60
133.70
133.80
133.90
1340
134.10
134.20
134.3-0
134.4-0
134.5-0
134.6-0
134.7-0
134.8-0
134.9-0
135-0
135.1-0
135.2-0
135.3-0
135.4-0
135.5-0
135.60
135.70
135.80
135.90
1360
136.10
136.20
136.30
136.40
136.50
136.60
136.70
136.80
136.90
137-0
137.1-0
137.2-0
137.3-0
137.4-0
137.5-0
137.6-0
137.7-0
137.8-0
137.9-0
138-0
138.1-0
138.2-0
138.30
138.40
138.50
138.60
138.70
138.80
138.90
1390
139.10
139.20
139.30
139.40
139.50
139.6-0
139.7-0
139.8-0
139.9-0
140-0
140.1-0
140.2-0
140.3-0
140.4-0
140.5-0
140.6-0
140.7-0
140.8-0
140.90
1410
141.10
141.20
141.30
141.40
141.50
141.60
141.70
141.80
141.90
1420
142.10
142.2-0
142.3-0
142.4-0
142.5-0
142.6-0
142.7-0
142.8-0
142.9-0
143-0
143.1-0
143.2-0
143.3-0
143.4-0
143.50
143.60
143.70
143.80
143.90
1440
144.10
144.20
144.30
144.40
144.50
144.60
144.70
144.8-0
144.9-0
145-0
145.1-0
145.2-0
145.3-0
145.4-0
145.5-0
145.6-0
145.7-0
145.8-0
145.9-0
146-0
146.10
146.20
146.30
146.40
146.50
146.60
146.70
146.80
146.90
1470
147.10
147.20
147.30
147.4-0
147.5-0
147.6-0
147.7-0
147.8-0
147.9-0
148-0
148.1-0
148.2-0
148.3-0
148.4-0
148.5-0
148.6-0
148.7-0
148.80
148.90
1490
149.10
149.20
149.30
149.40
149.50
149.60
149.70
149.80
149.90
1500
150.1-0
150.2-0
150.3-0
150.4-0
150.5-0
150.6-0
150.7-0
150.8-0
150.9-0
151-0
151.1-0
151.2-0
151.3-0
151.40
151.50
151.60
151.70
151.80
151.90
1520
152.10
152.20
152.30
152.40
152.50
152.60
152.7-0
152.8-0
152.9-0
153-0
153.1-0
153.2-0
153.3-0
153.4-0
153.5-0
153.6-0
153.7-0
153.8-0
153.9-0
1540
154.10
154.20
154.30
154.40
154.50
154.60
154.70
154.80
154.90
1550
155.10
155.20
155.3-0
155.4-0
155.5-0
155.6-0
155.7-0
155.8-0
155.9-0
156-0
156.1-0
156.2-0
156.3-0
156.4-0
156.5-0
156.60
156.70
156.80
156.90
1570
157.10
157.20
157.30
157.40
157.50
157.60
157.70
157.80
157.9-0
158-0
158.1-0
158.2-0
158.3-0
158.4-0
158.5-0
158.6-0
158.7-0
158.8-0
158.9-0
159-0
159.1-0
159.20
159.30
159.40
159.50
159.60
159.70
159.80
159.90
1600
160.10
160.20
160.30
160.40
160.5-0
160.6-0
160.7-0
160.8-0
160.9-0
161-0
161.1-0
161.2-0
161.3-0
161.4-0
161.5-0
161.6-0
161.7-0
161.8-0
161.90
1620
162.10
162.20
162.30
162.40
162.50
162.60
162.70
162.80
162.90
1630
163.10
163.2-0
163.3-0
163.4-0
163.5-0
163.6-0
163.7-0
163.8-0
163.9-0
164-0
164.1-0
164.2-0
164.3-0
164.4-0
164.50
164.60
164.70
164.80
164.90
1650
165.10
165.20
165.30
165.40
165.50
165.60
165.70
165.8-0
165.9-0
166-0
166.1-0
166.2-0
166.3-0
166.4-0
166.5-0
166.6-0
166.7-0
166.8-0
166.9-0
167-0
167.10
167.20
167.30
167.40
167.50
167.60
167.70
167.80
167.90
1680
168.10
168.20
168.30
168.4-0
168.5-0
168.6-0
168.7-0
168.8-0
168.9-0
169-0
169.1-0
169.2-0
169.3-0
169.4-0
169.5-0
169.6-0
169.70
169.80
169.90
1700
170.10
170.20
170.30
170.40
170.50
170.60
170.70
170.80
170.90
171-0
171.1-0
171.2-0
171.3-0
171.4-0
171.5-0
171.6-0
171.7-0
171.8-0
171.9-0
172-0
172.1-0
172.2-0
172.30
172.40
172.50
172.60
172.70
172.80
172.90
1730
173.10
173.20
173.30
173.40
173.50
173.6-0
173.7-0
173.8-0
173.9-0
174-0
174.1-0
174.2-0
174.3-0
174.4-0
174.5-0
174.6-0
174.7-0
174.8-0
174.9-0
1750
175.10
175.20
175.30
175.40
175.50
175.60
175.70
175.80
175.90
1760
176.10
176.20
176.3-0
176.4-0
176.5-0
176.6-0
176.7-0
176.8-0
176.9-0
177-0
177.1-0
177.2-0
177.3-0
177.4-0
177.5-0
177.60
177.70
177.80
177.90
1780
178.10
178.20
178.30
178.40
178.50
178.60
178.70
178.80
178.9-0
179-0
179.1-0
179.2-0
179.3-0
179.4-0
179.5-0
179.6-0
179.7-0
179.8-0
179.9-0
180-0
180.1-0
180.20
180.30
180.40
180.50
180.60
180.70
180.80
180.90
1810
181.10
181.20
181.30
181.40
181.5-0
181.6-0
181.7-0
181.8-0
181.9-0
182-0
182.1-0
182.2-0
182.3-0
182.4-0
182.5-0
182.6-0
182.7-0
182.80
182.90
1830
183.10
183.20
183.30
183.40
183.50
183.60
183.70
183.80
183.90
1840
184.1-0
184.2-0
184.3-0
184.4-0
184.5-0
184.6-0
184.7-0
184.8-0
184.9-0
185-0
185.1-0
185.2-0
185.3-0
185.40
185.50
185.60
185.70
185.80
185.90
1860
186.10
186.20
186.30
186.40
186.50
186.60
186.7-0
186.8-0
186.9-0
187-0
187.1-0
187.2-0
187.3-0
187.4-0
187.5-0
187.6-0
187.7-0
187.8-0
187.9-0
188-0
188.10
188.20
188.30
188.40
188.50
188.60
188.70
188.80
188.90
1890
189.10
189.20
189.30
189.4-0
189.5-0
189.6-0
189.7-0
189.8-0
189.9-0
190-0
190.1-0
190.2-0
190.3-0
190.4-0
190.5-0
190.6-0
190.70
190.80
190.90
1910
191.10
191.20
191.30
191.40
191.50
191.60
191.70
191.80
191.90
192-0
192.1-0
192.2-0
192.3-0
192.4-0
192.5-0
192.6-0
192.7-0
192.8-0
192.9-0
193-0
193.1-0
193.2-0
193.30
193.40
193.50
193.60
193.70
193.80
193.90
1940
194.10
194.20
194.30
194.40
194.50
194.6-0
194.7-0
194.8-0
194.9-0
195-0
195.1-0
195.2-0
195.3-0
195.4-0
195.5-0
195.6-0
195.7-0
195.8-0
195.90
1960
196.10
196.20
196.30
196.40
196.50
196.60
196.70
196.80
196.90
1970
197.10
197.2-0
197.3-0
197.4-0
197.5-0
197.6-0
197.7-0
197.8-0
197.9-0
198-0
198.1-0
198.2-0
198.3-0
198.4-0
198.50
198.60
198.70
198.80
198.90
1990
199.10
199.20
199.30
199.40
199.50
199.60
199.70
199.8-0
199.9-0
200-0
200.1-0
200.2-0
200.3-0
200.4-0
200.5-0
200.6-0
200.7-0
200.8-0
200.9-0
201-0
201.1-0
201.20
201.30
201.40
201.50
201.60
201.70
201.80
201.90
2020
202.10
202.20
202.30
202.40
202.5-0
202.6-0
202.7-0
202.8-0
202.9-0
203-0
203.1-0
203.2-0
203.3-0
203.4-0
203.5-0
203.6-0
203.7-0
203.80
203.90
2040
204.10
204.20
204.30
204.40
204.50
204.60
204.70
204.80
204.90
2050
205.1-0
205.2-0
205.3-0
205.4-0
205.5-0
205.6-0
205.7-0
205.8-0
205.9-0
206-0
206.1-0
206.2-0
206.3-0
206.40
206.50
206.60
206.70
206.80
206.90
2070
207.10
207.20
207.30
207.40
207.50
207.60
207.7-0
207.8-0
207.9-0
208-0
208.1-0
208.2-0
208.3-0
208.4-0
208.5-0
208.6-0
208.7-0
208.8-0
208.9-0
2090
209.10
209.20
209.30
209.40
210.4-0
211.4-0
212.40
213.4-0
214.40
215.40
216.4-0
217.40
218.4-0
219.4-0
220.40
221.4-0
222.40
223.4-0
224.4-0
225.40
226.4-0
227.40
228.40
229.4-0
230.40
231.4-0
232.4-0
233.40
234.4-0
235.40
236.40
237.4-0
238.40
239.4-0
240.4-0
241.40
242.4-0
243.40
244.4-0
245.4-0
246.40
247.4-0
248.40
249.40
250.4-0
251.40
252.4-0
253.4-0
254.40
255.4-0
256.40
257.40
258.4-0
259.40
260.4-0
261.40
262.40
263.4-0
264.40
265.4-0
266.4-0
267.40
268.4-0
269.40
270.40
271.4-0
272.40
273.4-0
274.4-0
275.40
276.4-0
277.40
278.40
279.4-0
280.40
281.4-0
282.40
283.40
284.4-0
285.40
286.4-0
287.4-0
288.40
289.4-0
290.40
291.40
292.4-0
293.40
294.4-0
295.4-0
296.40
297.4-0
298.40
299.4-0
300.4-0
301.40
302.4-0
303.40
304.40
305.4-0
306.40
307.4-0
308.4-0
309.40
310.4-0
311.40
312.40
313.4-0
314.40
315.4-0
316.40
317.40
318.4-0
319.40
320.4-0
321.4-0
322.40
323.4-0
324.40
325.40
326.4-0
327.40
328.4-0
329.4-0
330.40
331.4-0
332.40
333.40
334.4-0
335.40
336.4-0
337.40
338.40
339.4-0
340.40
341.4-0
342.4-0
343.40
344.4-0
345.40
346.40
347.4-0
348.40
349.4-0
350.4-0
351.40
352.4-0
353.40
354.4-0
355.4-0
356.40
357.4-0
358.40
359.40
360.4-0
361.40
362.4-0
363.4-0
364.40
365.4-0
366.40
367.40
368.4-0
369.40
370.4-0
371.4-0
372.40
373.4-0
374.40
375.4-0
376.4-0
377.40
378.4-0
379.40
380.40
381.4-0
382.40
383.4-0
384.4-0
385.40
386.4-0
387.40
388.40
389.4-0
390.40
391.4-0
392.40
393.40
394.4-0
395.40
396.4-0
397.4-0
398.40
399.4-0
400.40
401.40
402.4-0
403.40
404.4-0
405.4-0
406.40
407.4-0
408.40
409.40
410.4-0
411.40
412.4-0
413.40
414.40
415.4-0
416.40
417.4-0
418.4-0
419.40
420.4-0
421.40
422.40
423.4-0
424.40
425.4-0
426.4-0
427.40
428.4-0
429.40
430.4-0
431.4-0
432.40
433.4-0
434.40
435.40
436.4-0
437.40
438.4-0
439.4-0
440.40
441.4-0
442.40
443.40
444.4-0
445.40
446.4-0
447.4-0
448.40
449.4-0
450.40
451.4-0
452.4-0
453.40
454.4-0
455.40
456.40
457.4-0
458.40
459.4-0
460.4-0
461.40
462.4-0
463.40
464.40
465.4-0
466.40
467.4-0
468.40
469.40
470.4-0
471.40
472.4-0
473.4-0
474.40
475.4-0
476.40
477.40
478.4-0
479.40
480.4-0
481.4-0
482.40
483.4-0
484.40
485.40
486.4-0
487.40
488.4-0
489.40
490.40
491.4-0
492.40
493.4-0
494.4-0
495.40
496.4-0
497.40
498.40
499.4-0
500.40
501.4-0
502.4-0
503.40
504.4-0
505.40
506.4-0
507.4-0
508.40
509.4-0
510.40
511.40
512.4-0
513.40
514.4-0
515.4-0
516.40
517.4-0
518.40
519.40
520.4-0
521.40
522.4-0
523.40
524.40
525.4-0
526.40
527.4-0
528.4-0
529.40
530.4-0
531.40
532.40
533.4-0
534.40
535.4-0
536.4-0
537.40
538.4-0
539.40
540.40
541.4-0
542.40
543.4-0
544.40
545.40
546.4-0
547.40
548.4-0
549.4-0
550.40
551.4-0
552.40
553.40
554.4-0
555.40
556.4-0
557.4-0
558.40
559.4-0
560.40
561.4-0
562.4-0
563.40
564.4-0
565.40
566.40
567.4-0
568.40
569.4-0
570.4-0
571.40
572.4-0
573.40
574.40
575.4-0
576.40
577.4-0
578.4-0
579.40
580.4-0
581.40
582.4-0
583.4-0
584.40
585.4-0
586.40
587.40
588.4-0
589.40
590.4-0
591.4-0
592.40
593.4-0
594.40
595.40
596.4-0
597.40
598.4-0
599.40
600.40
601.4-0
602.40
603.4-0
604.4-0
605.40
606.4-0
607.40
608.40
609.4-0
610.40
611.4-0
612.4-0
613.40
614.4-0
615.40
616.40
617.4-0
618.40
619.4-0
620.40
621.40
622.4-0
623.40
624.4-0
625.4-0
626.40
627.4-0
628.40
629.40
630.4-0
631.40
632.4-0
633.4-0
634.40
635.4-0
636.40
637.4-0
638.4-0
639.40
640.4-0
641.40
642.40
643.4-0
644.40
645.4-0
646.4-0
647.40
648.4-0
649.40
650.40
651.4-0
652.40
653.4-0
654.4-0
655.40
656.4-0
657.40
658.4-0
659.4-0
660.40
661.4-0
662.40
663.40
664.4-0
665.40
666.4-0
667.4-0
668.40
669.4-0
670.40
671.40
672.4-0
673.40
674.4-0
675.40
676.40
677.4-0
678.40
679.4-0
680.4-0
681.40
682.4-0
683.40
684.40
685.4-0
686.40
687.4-0
688.4-0
689.40
690.4-0
691.40
692.40
693.4-0
694.40
695.4-0
696.40
697.40
698.4-0
699.40
700.4-0
701.4-0
702.40
703.4-0
704.40
705.40
706.4-0
707.40
708.4-0
709.4-0
710.40
711.4-0
712.40
713.4-0
714.4-0
715.40
716.4-0
717.40
718.40
719.4-0
720.40
721.4-0
722.4-0
723.40
724.4-0
725.40
726.40
727.4-0
728.40
729.4-0
730.4-0
731.40
732.4-0
733.40
734.4-0
735.4-0
736.40
737.4-0
738.40
739.40
740.4-0
741.40
742.4-0
743.4-0
744.40
745.4-0
746.40
747.40
748.4-0
749.40
750.4-0
751.40
752.40
753.4-0
754.40
755.4-0
756.4-0
757.40
758.4-0
759.40
760.40
761.4-0
762.40
763.4-0
764.4-0
765.40
766.4-0
767.40
768.4-0
769.4-0
770.40
771.4-0
772.40
773.40
774.4-0
775.40
776.4-0
777.4-0
778.40
779.4-0
780.40
781.40
782.4-0
783.40
784.4-0
785.4-0
786.40
787.4-0
788.40
789.4-0
790.4-0
791.40
792.4-0
793.40
794.40
795.4-0
796.40
797.4-0
798.4-0
799.40
800.4-0
801.40
802.40
803.4-0
804.40
805.4-0
806.40
807.40
808.4-0
809.40
810.4-0
811.4-0
812.40
813.4-0
814.40
815.40
816.4-0
817.40
818.4-0
819.4-0
820.40
821.4-0
822.40
823.40
824.4-0
825.40
826.4-0
827.40
828.40
829.4-0
830.40
831.4-0
832.4-0
833.40
834.4-0
835.40
836.40
837.4-0
838.40
839.4-0
840.4-0
841.40
842.4-0
843.40
844.4-0
845.4-0
846.40
847.4-0
848.40
849.40
850.4-0
851.40
852.4-0
853.4-0
854.40
855.4-0
856.40
857.40
858.4-0
859.40
860.4-0
861.4-0
862.40
863.4-0
864.40
865.4-0
866.4-0
867.40
868.4-0
869.40
870.40
871.4-0
872.40
873.4-0
874.4-0
875.40
876.4-0
877.40
878.40
879.4-0
880.40
881.4-0
882.40
883.40
884.4-0
885.40
886.4-0
887.4-0
888.40
889.4-0
890.40
891.40
892.4-0
893.40
894.4-0
895.4-0
896.40
897.4-0
898.40
899.40
900.4-0
901.40
902.4-0
903.40
904.40
905.4-0
906.40
907.4-0
908.4-0
909.40
910.4-0
911.40
912.40
913.4-0
914.40
915.4-0
916.4-0
917.40
918.4-0
919.40
920.4-0
921.4-0
922.40
923.4-0
924.40
925.40
926.4-0
927.40
928.4-0
929.4-0
930.40
931.4-0
932.40
933.40
934.4-0
935.40
936.4-0
937.4-0
938.40
939.4-0
940.40
941.4-0
942.4-0
943.40
944.4-0
945.40
946.40
947.4-0
948.40
949.4-0
950.4-0
951.40
952.4-0
953.40
954.40
955.4-0
956.40
957.4-0
958.40
959.40
960.4-0
961.40
962.4-0
963.4-0
964.40
965.4-0
966.40
967.40
968.4-0
969.40
970.4-0
971.4-0
972.40
973.4-0
974.40
975.4-0
976.4-0
977.40
978.4-0
979.40
980.40
981.4-0
982.40
983.4-0
984.4-0
985.40
986.4-0
987.40
988.40
989.4-0
990.40
991.4-0
992.4-0
993.40
994.4-0
995.40
996.4-0
997.4-0
998.40
999.4-0
1000.40
1001.40
1002.4-0
1003.40
1004.4-0
1005.4-0
1006.40
1007.4-0
1008.40
1009.40
1010.4-0
1011.40
1012.4-0
1013.40
1014.40
1015.4-0
1016.40
1017.4-0
1018.4-0
1019.40
1020.4-0
1021.40
1022.40
1023.4-0
1024.40
1025.4-0
1026.4-0
1027.40
1028.4-0
1029.40
1030.40
1031.4-0
1032.40
1033.4-0
1034.40
1035.40
1036.4-0
1037.40
1038.4-0
1039.4-0
1040.40
1041.4-0
1042.40
1043.40
1044.4-0
1045.40
1046.4-0
1047.4-0
1048.40
1049.4-0
1050.40
1051.4-0
1052.4-0
1053.40
1054.4-0
1055.40
1056.40
1057.4-0
1058.40
1059.4-0
1060.4-0
1061.40
1062.4-0
1063.40
1064.40
1065.4-0
1066.40
1067.4-0
1068.4-0
1069.40
1070.4-0
1071.40
1072.4-0
1073.4-0
1074.40
1075.4-0
1076.40
1077.40
1078.4-0
1079.40
1080.4-0
1081.4-0
1082.40
1083.4-0
1084.40
1085.40
1086.4-0
1087.40
1088.4-0
1089.40
1090.40
1091.4-0
1092.40
1093.4-0
1094.4-0
1095.40
1096.4-0
1097.40
1098.40
1099.4-0
1100.40
1101.4-0
1102.4-0
1103.40
1104.4-0
1105.40
1106.40
1107.4-0
1108.40
1109.4-0
1110.40
1111.40
1112.4-0
1113.40
1114.4-0
1115.4-0
1116.40
1117.4-0
1118.40
1119.40
1120.4-0
1121.40
1122.4-0
1123.4-0
1124.40
1125.4-0
1126.40
1127.4-0
1128.4-0
1129.40
1130.4-0
1131.40
1132.40
1133.4-0
1134.40
1135.4-0
1136.4-0
1137.40
1138.4-0
1139.40
1140.40
1141.4-0
1142.40
1143.4-0
1144.4-0
1145.40
1146.4-0
1147.40
1148.4-0
1149.4-0
1150.40
1151.4-0
1152.40
1153.40
1154.4-0
1155.40
1156.4-0
1157.4-0
1158.40
1159.4-0
1160.40
1161.40
1162.4-0
1163.40
1164.4-0
1165.40
1166.40
1167.4-0
1168.40
1169.4-0
1170.4-0
1171.40
1172.4-0
1173.40
1174.40
1175.4-0
1176.40
1177.4-0
1178.4-0
1179.40
1180.4-0
1181.40
1182.4-0
1183.4-0
1184.40
1185.4-0
1186.40
1187.40
1188.4-0
1189.40
1190.4-0
1191.4-0
1192.40
1193.4-0
1194.40
1195.40
1196.4-0
1197.40
1198.4-0
1199.4-0
1200.40
1201.4-0
1202.40
1203.4-0
1204.4-0
1205.40
1206.4-0
1207.40
1208.40
1209.4-0
1210.40
1211.4-0
1212.4-0
1213.40
1214.4-0
1215.40
1216.40
1217.4-0
1218.40
1219.4-0
1220.40
1221.40
1222.4-0
1223.40
1224.4-0
1225.4-0
1226.40
1227.4-0
1228.40
1229.40
1230.4-0
1231.40
1232.4-0
1233.4-0
1234.40
1235.4-0
1236.40
1237.40
1238.4-0
1239.40
1240.4-0
1241.40
1242.40
1243.4-0
1244.40
1245.4-0
1246.4-0
1247.40
1248.4-0
1249.40
1250.40
1251.4-0
1252.40
1253.4-0
1254.4-0
1255.40
1256.4-0
1257.40
1258.4-0
1259.4-0
1260.40
1261.4-0
1262.40
1263.40
1264.4-0
1265.40
1266.4-0
1267.4-0
1268.40
1269.4-0
1270.40
1271.40
1272.4-0
1273.40
1274.4-0
1275.4-0
1276.40
1277.4-0
1278.40
1279.4-0
1280.4-0
1281.40
1282.4-0
1283.40
1284.40
1285.4-0
1286.40
1287.4-0
1288.4-0
1289.40
1290.4-0
1291.40
1292.40
1293.4-0
1294.40
1295.4-0
1296.40
1297.40
1298.4-0
1299.40
1300.4-0
1301.4-0
1302.40
1303.4-0
1304.40
1305.40
1306.4-0
1307.40
1308.4-0
1309.4-0
1310.40
1311.4-0
1312.40
1313.40
1314.4-0
1315.40
1316.4-0
1317.40
1318.40
1319.4-0
1320.40
1321.4-0
1322.4-0
1323.40
1324.4-0
1325.40
1326.40
1327.4-0
1328.40
1329.4-0
1330.4-0
1331.40
1332.4-0
1333.40
1334.4-0
1335.4-0
1336.40
1337.4-0
1338.40
1339.40
1340.4-0
1341.40
1342.4-0
1343.4-0
1344.40
1345.4-0
1346.40
1347.40
1348.4-0
1349.40
1350.4-0
1351.4-0
1352.40
1353.4-0
1354.40
1355.4-0
1356.4-0
1357.40
1358.4-0
1359.40
1360.40
1361.4-0
1362.40
1363.4-0
1364.4-0
1365.40
1366.4-0
1367.40
1368.40
1369.4-0
1370.40
1371.4-0
1372.40
1373.40
1374.4-0
1375.40
1376.4-0
1377.4-0
1378.40
1379.4-0
1380.40
1381.40
1382.4-0
1383.40
1384.4-0
1385.4-0
1386.40
1387.4-0
1388.40
1389.4-0
1390.4-0
1391.40
1392.4-0
1393.40
1394.40
1395.4-0
1396.40
1397.4-0
1398.4-0
1399.40
1400.4-0
1401.40
1402.40
1403.4-0
1404.40
1405.4-0
1406.4-0
1407.40
1408.4-0
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2787.40
2788.40
2789.40
2790.40
2791.40
2792.40
2793.40
2794.40
2795.40
2796.40
2797.40
2798.40
2799.40
2800.40
2801.40
2802.40
2803.40
2804.40
2805.40
2806.40
2807.40
2808.40
2809.40
2810.40
2811.40
2812.40
2813.40
2814.40
2815.40
2816.40
2817.40
2818.40
2819.40
2820.40
2821.40
2822.40
2823.40
2824.40
2825.40
2826.40
2827.40
2828.40
2829.40
2830.40
2831.40
2832.40
2833.40
2834.40
2835.40
2836.40
2837.40
2838.40
2839.40
2840.40
2841.40
2842.40
2843.40
2844.40
2845.40
2846.40
2847.40
2848.40
2849.40
2850.40
2851.40
2852.40
2853.40
2854.40
2855.40
2856.40
2857.40
2858.40
2859.40
2860.40
2861.40
2862.40
2863.40
2864.40
2865.40
2866.40
2867.40
2868.40
2869.40
2870.40
2871.40
2872.40
2873.40
2874.40
2875.40
2876.40
2877.40
2878.40
2879.40
2880.40
2881.40
2882.40
2883.40
2884.40
2885.40
2886.40
2887.40
2888.40
2889.40
2890.40
2891.40
2892.40
2893.40
2894.40
2895.40
2896.40
2897.40
2898.40
2899.40
2900.40
2901.40
2902.40
2903.40
2904.40
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2927.40
2928.40
2929.40
2930.40
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2932.40
2933.40
2934.40
2935.40
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2940.40
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2942.40
2943.40
2944.40
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2946.40
2947.40
2948.40
2949.40
2950.40
2951.40
2952.40
2953.40
2954.40
2955.40
2956.40
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2958.40
2959.40
2960.40
2961.40
2962.40
2963.40
2964.40
2965.40
2966.40
2967.40
2968.40
2969.40
2970.40
2971.40
2972.40
2973.40
2974.40
2975.40
2976.40
2977.40
2978.40
2979.40
2980.40
2981.40
2982.40
2983.40
2984.40
2985.40
2986.40
2987.40
2988.40
2989.40
2990.40
2991.40
2992.40
2993.40
2994.40
2995.40
2996.40
2997.40
2998.40
2999.40
3000.40
3001.40
3002.40
3003.40
3004.40
3005.40
3006.40
3007.40
3008.40
3009.40
3010.40
3011.40
3012.40
3013.40
3014.40
3015.40
3016.40
3017.40
3018.40
3019.40
3020.40
3021.40
3022.40
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3086.40
3087.40
3088.40
3089.40
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3091.40
3092.40
3093.40
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3095.40
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3099.40
3100.40
3101.40
3102.40
3103.40
3104.40
3105.40
3106.40
3107.40
3108.40
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3110.40
3111.40
3112.40
3113.40
3114.40
3115.40
3116.40
3117.40
3118.40
3119.40
3120.40
3121.40
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3128.40
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3130.40
3131.40
3132.40
3133.40
3134.40
3135.40
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3137.40
3138.40
3139.40
3140.40
3141.40
3142.40
3143.40
3144.40
3145.40
3146.40
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3148.40
3149.40
3150.40
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3198.40
3199.40
3200.40
3201.40
3202.40
3203.40
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3205.40
3206.40
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3209.40
3210.40
3211.40
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3218.40
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3220.40
3221.40
3222.40
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3225.40
3226.40
3227.40
3228.40
3229.40
3230.40
3231.40
3232.40
3233.40
3234.40
3235.40
3236.40
3237.40
3238.40
3239.40
3240.40
3241.40
3242.40
3243.40
3244.40
3245.40
3246.40
3247.40
3248.40
3249.40
3250.40
3251.40
3252.40
3253.40
3254.40
3255.40
3256.40
3257.40
3258.40
3259.40
3260.40
3261.40
3262.40
3263.40
3264.40
3265.40
3266.40
3267.40
3268.40
3269.40
3270.40
3271.40
3272.40
3273.40
3274.40
3275.40
3276.40
3277.40
3278.40
3279.40
3280.40
3281.40
3282.40
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3284.40
3285.40
3286.40
3287.40
3288.40
3289.40
3290.40
3291.40
3292.40
3293.40
3294.40
3295.40
3296.40
3297.40
3298.40
3299.40
3300.40
3301.40
3302.40
3303.40
3304.40
3305.40
3306.40
3307.40
3308.40
3309.40
3310.40
3311.40
3312.40
3313.40
3314.40
3315.40
3316.40
3317.40
3318.40
3319.40
3320.40
3321.40
3322.40
3323.40
3324.40
3325.40
3326.40
3327.40
3328.40
3329.40
3330.40
3331.40
3332.40
3333.40
3334.40
3335.40
3336.40
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3338.40
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3340.40
3341.40
3342.40
3343.40
3344.40
3345.40
3346.40
3347.40
3348.40
3349.40
3350.40
3351.40
3352.40
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3354.40
3355.40
3356.40
3357.40
3358.40
3359.40
3360.40
3361.40
3362.40
3363.40
3364.40
3365.40
3366.40
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3368.40
3369.40
3370.40
3371.40
3372.40
3373.40
3374.40
3375.40
3376.40
3377.40
3378.40
3379.40
3380.40
3381.40
3382.40
3383.40
3384.40
3385.40
3386.40
3387.40
3388.40
3389.40
3390.40
3391.40
3392.40
3393.40
3394.40
3395.40
3396.40
3397.40
3398.40
3399.40
3400.40
3401.40
3402.40
3403.40
3404.40
3405.40
3406.40
3407.40
3408.40
3409.40
3410.40
3411.40
3412.40
3413.40
3414.40
3415.40
3416.40
3417.40
3418.40
3419.40
3420.40
3421.40
3422.40
3423.40
3424.40
3425.40
3426.40
3427.40
3428.40
3429.40
3430.40
3431.40
3432.40
3433.40
3434.40
3435.40
3436.40
3437.40
3438.40
3439.40
3440.40
3441.40
3442.40
3443.40
3444.40
3445.40
3446.40
3447.40
3448.40
3449.40
3450.40
3451.40
3452.40
3453.40
3454.40
3455.40
3456.40
3457.40
3458.40
3459.40
3460.40
3461.40
3462.40
3463.40
3464.40
3465.40
3466.40
3467.40
3468.40
3469.40.00000E+00
Grfico1
3.9264077234
3.7362410336
3.4359839831
3.0353583534
2.5470111603
1.9861171078
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