-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathpr1.tex
114 lines (97 loc) · 3.96 KB
/
pr1.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
\section*{1.B}
{\Large Zadání:} \\
$U_1 = 95 \text{V} \; U_2=115 \text{V}$ \\
$R_1 = 650 \Omega \; R_2 = 730 \Omega \; R_3 = 340 \Omega \;
R_4 = 330 \Omega \; R_5 = 410 \Omega \; R_6 =830 \Omega$ \\
$R_7 = 340 \Omega \; R_8 = 220 \Omega$
$U_{R_8} = \: \text{?}$ \\
$I_{R_8} = \: \text{?}$ \\
\begin{figure}[H]
\vspace{-1.1cm}
\center\includegraphics[width=0.6\linewidth]{obr/1_1}
\end{figure}
{\Large Řešení metodou postupného zjednodušování}\\
\begin{figure}[H]
\center\includegraphics[width=0.6\linewidth]{obr/1_2}
\caption*{$R_3$ a $R_4$ jsou zapojeny paralelně a zdroje $U_1$ a $U_2$ sériově }
\end{figure}
\begin{gather*}
R_{34} = \frac{R_3 R_4}{R_3 + R_4} =\frac{340 \cdot 330}{340 + 330} \doteq 167.4627 \Omega \\
U_{12} = {U_1 + U_2} = {95 + 115} = 210 \text{V}
\end{gather*}
\begin{figure}[H]
\center\includegraphics[width=0.6\linewidth]{obr/1_3}
\caption*{$R_2$ a $R_{34}$ jsou zapojeny sériově}
\end{figure}
\begin{gather*}
R_{234} = R_2 + R_{34} = 730 + 167,4627 = 897,4627 \Omega
\end{gather*}
\begin{figure}[H]
\vspace{-1.1cm}
\center\includegraphics[width=0.6\linewidth]{obr/1_4}
\caption*{transfigurace - trojúhelník}
\end{figure}
\begin{gather*}
R_A = \frac{R_1 R_{234}}{R_1 + R_{234} + R_5} = \frac{650 \cdot 897,4627}{650 + 897,4627 + 410} = 298,0132 \Omega \\
R_B = \frac{R_1 R_5}{R_1 + R_{234} + R_5} = \frac{650 \cdot 410}{650 + 897,4627 + 410} = 136,1456 \Omega \\
R_C = \frac{R_5 R_{234}}{R_1 + R_{234} + R_5} = \frac{410 \cdot 897,4627}{650 + 897,4627 + 410} = 187,9779 \Omega
\end{gather*}
\begin{figure}[H]
\center\includegraphics[width=0.6\linewidth]{obr/1_5}
\caption*{$R_B$ a $R_7$ jsou zapojeny sériově stejně jako $R_C$ a $R_6$}
\end{figure}
\begin{gather*}
R_{B7} = R_B + R_7 = 136,1456 + 340 = 476,1456 \Omega \\
R_{C6} = R_C + R_6 =187,9779 + 830 = 1017,9779 \Omega
\end{gather*}
\begin{figure}[H]
\center\includegraphics[width=0.6\linewidth]{obr/1_6}
\caption*{ $R_{B7}$ a $R_{C6}$ jsou zapojeny paralelně}
\end{figure}
\begin{gather*}
R_{BC67} = \frac{R_{B7} R_{C6}}{R_{B7} + R_{C6}} = \frac{476,1456 \cdot 1017,9779}{476,1456 + 1017,9779} = 324,4081 \Omega
\end{gather*}
\begin{figure}[H]
\center\includegraphics[width=0.3\linewidth]{obr/1_7}
\caption*{$R_A$ a $R_{BC67}$ a $R_8$ jsou zapojeny sériově - získáváme $R_{EKV}$}
\end{figure}
\begin{gather*}
R_{EKV} = R_{A} + R_{BC67} + R_8 = 298,01232 + 324,4081 + 220 = 842,4213 \Omega
\end{gather*}
Celkový proud $I$:
\begin{gather*}
I = \frac{U_{12}}{R_{EKV}} = \frac{210}{842,4213} \doteq 0,2493 \text{A}
\end{gather*}
Začneme zpětně počítat napětí a proudy, až dojdeme k $U_{R_6}$ a $I_{R_6}$:
\begin{figure}[H]
\center\includegraphics[width=0.6\linewidth]{obr/1_8}
\caption*{Spočítáme si napětí na jednotlivých odporech}
\end{figure}
\begin{gather*}
U_{R_A} = {I R_A} = {0,2493\cdot 298,0132} \doteq 74,2947 \text{V} \\
U_{R_{BC67}} = {I R_{BC67}} = {0,2493 \cdot 324,4081} \doteq 80,8749 \text{V} \\
U_{R_8} = {I R_8} = {0,2493 \cdot 220} = 54,846 \text{V} \\
\\
\text{Provedeme kontrolu pomocí II. Kirchhoffova zákona.} \\
U_{R_A} + U_{R_{BC67}} + U_{R_8} - U_{12} = 0 \\
\end{gather*}
\begin{figure}[H]
\center\includegraphics[width=0.6\linewidth]{obr/1_9}
\caption*{Spočítáme si proud ve větvích s $R_{B7}$ a $R_{C6}$}
\end{figure}
\begin{gather*}
I_{R_{B7}} = \frac{U_{R_{BC67}}{R_{B7}}} = \frac{80,8749}{476,1456} \doteq 0,1699 \text{A} \\
I_{R_{C6}} = \frac{U_{R_{BC67}}{R_{C6}}} = \frac{80,8749}{1017,9779} \doteq 0,0794 \text{A} \\
\\
\text{Provedeme kontrolu pomocí I. Kirchhoffova zákona.} \\
I_{R_{B7}} + I_{R_{C6}} - I = 0
\end{gather*}
\begin{figure}[H]
\center\includegraphics[width=0.6\linewidth]{obr/1_10}
\caption*{Spočítáme si napětí a proud u $R_6$}
\end{figure}
\begin{gather*}
I_{R_{C6}} = I_{R_6} = 0,0794 \text{A} \\
U_{R_6} = {I_{R_6} R_6} = {0,0794 \cdot 830} = 65,902 \text{V} \\
\end{gather*}
\newpage