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Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
introduction_to_digital_systems:calc_decimal_example [2021/09/15 02:33] tfischer |
introduction_to_digital_systems:calc_decimal_example [2021/09/15 03:45] (aktuell) tfischer |
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Zeile 2: | Zeile 2: | ||
----> | ----> | ||
- | $I.\quad$ | + | Calculation example for decimal value |
<---- | <---- | ||
----> | ----> | ||
+ | Idea: The numeral $2658.47$ is only the representation with the digits $[0..9]$, but what is the value behind it? | ||
+ | |||
+ | <---- | ||
+ | |||
+ | ----> | ||
+ | so lets start | ||
+ | <---- | ||
+ | |||
+ | ----> | ||
+ | 1. Put space between the digits \\ $\quad$ | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{black}{\text{value}}: | + | \color{black}{\text{numeral}:} & \color{black}{} |
- | \color{white}{\text{index}}: & \color{white}{i} | + | \color{white}{\text{index}: |
- | \color{white}{\text{place | + | \color{white}{\text{place |
\color{white}{} | \color{white}{} | ||
- | \color{white}{\text{digit}}: & \color{white}{z_i} | + | \color{white}{\text{digits |
- | \color{white}{\text{calc.}}: & \color{white}{z_i \cdot B^i} & \color{white}{2000} | + | \color{white}{\text{place value}:} |
- | \color{white}{\text{result}}: & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{2658.47} \\ | + | \color{white}{\text{result}: |
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
Zeile 20: | Zeile 31: | ||
----> | ----> | ||
+ | 2. Write down the index for each position. \\ $\quad$ | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{black}{\text{value}}: | + | \color{black}{\text{numeral}:} & \color{black}{} |
- | \color{white}{\text{index}}: & \color{white}{i} & \color{white}{3} & \color{white}{2} & \color{white}{1} & \color{white}{0 } & \color{white}{-1} & \color{white}{-2} \\ | + | \color{blue }{\text{index}:} & \color{blue }{i} & \color{blue }{3} & \color{blue }{2} & \color{blue }{1} & \color{blue }{0 } & \color{blue }{-1} & \color{blue }{-2} \\ |
- | \color{white}{\text{place | + | \color{white}{\text{place |
\color{white}{} | \color{white}{} | ||
- | \color{white}{\text{digit}}: & \color{white}{z_i} | + | \color{white}{\text{digits |
- | \color{white}{\text{calc.}}: & \color{white}{z_i \cdot B^i} & \color{white}{2000} | + | \color{white}{\text{place value}:} |
- | \color{white}{\text{result}}: & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{2658.47} \\ | + | \color{white}{\text{result}: |
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
<---- | <---- | ||
+ | |||
----> | ----> | ||
+ | 3. calculate the place factor \\ $\quad$ | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{black}{\text{value}}: | + | \color{black}{\text{numeral}:} & \color{black}{} |
- | \color{blue }{\text{index}}: & \color{blue }{i} & \color{blue }{3} & \color{blue }{2} & \color{blue }{1} & \color{blue }{0 } & \color{blue }{-1} & \color{blue }{-2} \\ | + | \color{black}{\text{index}: |
- | \color{white}{\text{place | + | \color{blue }{\text{place |
\color{white}{} | \color{white}{} | ||
- | \color{white}{\text{digit}}: & \color{white}{z_i} | + | \color{white}{\text{digits |
- | \color{white}{\text{calc.}}: & \color{white}{z_i \cdot B^i} & \color{white}{2000} | + | \color{white}{\text{place value}:} |
- | \color{white}{\text{result}}: & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{2658.47} \\ | + | \color{white}{\text{result}: |
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
Zeile 49: | Zeile 65: | ||
----> | ----> | ||
+ | 3. calculate the place factor \\ (= ..., thousands, hundreds, tens, ones, tenths, ...) | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{black}{\text{value}}: | + | \color{black}{\text{numeral}:} & \color{black}{} |
- | \color{black}{\text{index}}: & \color{black}{i} | + | \color{black}{\text{index}: |
- | \color{blue }{\text{place | + | \color{blue }{\text{place |
- | \color{white}{} & \color{white}{} & \color{white}{1000} | + | \color{blue }{} & \color{blue }{} & \color{blue }{1000} |
- | \color{white}{\text{digit}}: & \color{white}{z_i} | + | \color{white}{\text{digits |
- | \color{white}{\text{calc.}}: & \color{white}{z_i \cdot B^i} & \color{white}{2000} | + | \color{white}{\text{place value}:} |
- | \color{white}{\text{result}}: & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{2658.47} \\ | + | \color{white}{\text{result}: |
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
Zeile 64: | Zeile 82: | ||
----> | ----> | ||
+ | 4. write down each digit of the numeral | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{black}{\text{value}}: | + | \color{black}{\text{numeral}:} & \color{black}{} |
- | \color{black}{\text{index}}: & \color{black}{i} | + | \color{black}{\text{index}: |
- | \color{black}{\text{place | + | \color{black}{\text{place |
\color{black}{} | \color{black}{} | ||
- | \color{black}{\text{digit}}: & \color{black}{z_i} | + | \color{blue }{\text{digits |
- | \color{black}{\text{calc.}}: & \color{black}{z_i \cdot B^i} & \color{black}{2000} | + | \color{white}{\text{place value}:} |
- | \color{black}{\text{result}}: & \color{black}{\sum_i{ z_i \cdot B^i }} & & & \color{black}{2658.47} \\ | + | \color{white}{\text{result}: |
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
Zeile 79: | Zeile 99: | ||
----> | ----> | ||
+ | 5. calculate the place value \\ $\quad$ | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{blue }{\text{value}}: | + | \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8.} & \color{black}{4} & \color{black}{7} \\ |
- | \color{blue }{\text{index}}: & \color{blue }{i} & \color{blue }{3} & \color{blue }{2} & \color{blue }{1} & \color{blue }{0 } & \color{blue }{-1} & \color{blue }{-2} \\ | + | \color{black}{\text{index}: |
- | \color{blue }{\text{place | + | \color{black}{\text{place |
- | \color{blue }{} & \color{blue }{} & \color{blue }{1000} | + | \color{black}{} & \color{black}{} & \color{black}{1000} |
- | \color{blue }{\text{digit}}: | + | \color{black}{\text{digits |
- | \color{blue }{\text{calc.}}: & \color{blue }{z_i \cdot B^i} & \color{blue }{2000} | + | \color{blue }{\text{place value}:} |
- | \color{blue }{\text{result}}: & \color{blue }{\sum_i{ z_i \cdot B^i }} & & & \color{blue }{2658.47} \\ | + | \color{white}{\text{result}:} & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{2658.47} \\ |
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
Zeile 93: | Zeile 115: | ||
----> | ----> | ||
+ | 6.Add all place values | ||
+ | |||
\begin{align*} | \begin{align*} | ||
\begin{smallmatrix} | \begin{smallmatrix} | ||
- | \color{white}{\text{value}}: | + | \color{black}{\text{numeral}:} & \color{black}{} & \color{black}{2} & \color{black}{6} & \color{black}{5} & \color{black}{8.} & \color{black}{4} & \color{black}{7} \\ |
- | \color{white}{\text{index}}: & \color{white}{i} & \color{white}{3} & \color{white}{2} & \color{white}{1} & \color{white}{0 } & \color{white}{-1} & \color{white}{-2} \\ | + | \color{black}{\text{index}: |
- | \color{white}{\text{place | + | \color{black}{\text{place |
- | \color{white}{} & \color{white}{} & \color{white}{1000} | + | \color{black}{} & \color{black}{} & \color{black}{1000} |
- | \color{white}{\text{digit}}: & \color{white}{z_i} | + | \color{black}{\text{digits |
- | \color{white}{\text{calc.}}: & \color{white}{z_i \cdot B^i} & \color{white}{2000} | + | \color{black}{\text{place value}:} |
- | \color{white}{\text{result}}: & \color{white}{\sum_i{ z_i \cdot B^i }} & & & \color{white}{2658.47} \\ | + | \color{blue }{\text{result}: |
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
Zeile 107: | Zeile 131: | ||
- | ----> | ||
- | | value || 2 | 6 | 5 | 8 , | 4 | 7 | ||
- | | index | $i$ | 3 | 2 | 1 | 0 | -1 | -2 | | ||
- | | $\quad\quad$ \\ $\quad\quad$| $\quad\quad$| $\quad\quad$ | ||
- | | $\quad\quad$| $\quad\quad$ | ||
- | | $\quad\quad$| $\quad\quad$ | ||
- | | $\quad\quad$ \\ $\quad\quad$| $\quad\quad$ | ||
- | <---- | ||
- | |||
- | ----> | ||
- | | value || 2 | 6 | 5 | 8 , | 4 | 7 | ||
- | | index | $i$ | 3 | 2 | 1 | 0 | -1 | -2 | | ||
- | | place value | $B^i$ | $\small{10^3}$ \\ $\small{1000}$ | $\small{10^2}$ \\ $\small{100}$ | ||
- | | digit | $z_i$ | 2 | 6 | 5 | 8 | 4 | 7 | ||
- | | calc. | $z_i \cdot B^i$ | 2000 | 600 | 50 | 8 | 0.4 | 0.07 | | ||
- | | Result | ||
- | <---- | ||
- | ----> | ||
- | |aus (2+3)|$\color{blue}{I_p} = \color{blue}{I_m} = 0$ |$I_p$ und $I_m$ sind damit definiert| | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | |aus (6)|$\color{blue}{I_o} = I_1 $ |$I_o$ ist damit bekannt, wenn $I_1$ bekannt ist| | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | |aus (7) und (3)|$I_1 - I_2 -\color{blue}{0} = 0 $ |$\quad$| | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | |$\quad$|$I_1 = I_2 = I_o$ |$\quad$| | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | |$\quad$|$\color{blue}{I_1} = \color{blue}{I_2} = \color{blue}{I_o} $ |mit (8) und (9): $I_\boxed{}=\frac{U_\boxed{}}{R_\boxed{}}$ und (5)| | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | | $\quad$ |$\frac{U_1}{R_1}= \frac{U_2}{R_2} = \frac{U_A}{R_1 + R_2}$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | | (10)|$U_2= U_A\cdot\frac{R_2}{R_1+R_2}$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
- | |||
- | ----> | ||
- | $II.\quad$ Betrachtung der Spannungsverstärkung | ||
- | <---- | ||
- | |||
- | ---->> | ||
- | |aus (0) |$\color{blue}{A_V}=\frac{U_A}{U_E}$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | |||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << |