Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
introduction_to_digital_systems:calc_decimal_example [2021/09/15 02:16] tfischer |
introduction_to_digital_systems:calc_decimal_example [2021/09/15 03:45] (aktuell) tfischer |
||
---|---|---|---|
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} | ||
- | \text{value}: & | + | \color{black}{\text{numeral}:} |
- | \text{index}: | + | \color{white}{\text{index}: |
- | \text{place | + | \color{white}{\text{place |
- | & | + | \color{white}{} |
- | \text{digit}: & z_i & 2 & 6 & 5 & 8 | + | \color{white}{\text{digits |
- | \text{calc.}: & z_i \cdot B^i & 2000 & 600 & 50 | + | \color{white}{\text{place value}:} |
- | \text{result}: | + | \color{white}{\text{result}: |
\end{smallmatrix} | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
+ | <---- | ||
+ | |||
+ | ----> | ||
+ | 2. Write down the index for each position. \\ $\quad$ | ||
\begin{align*} | \begin{align*} | ||
- | value && | + | \begin{smallmatrix} |
- | index && i && 3 && 2 && 1 && 0 && -1 && -2 \\ | + | \color{black}{\text{numeral}: |
- | place value && B^i && 10^3 && 10^2 && 10^1 && 10^0 && 10^{-1} && 10^{-2} \\ | + | \color{blue }{\text{index}:} |
+ | \color{white}{\text{place factor}: | ||
+ | \color{white}{} | ||
+ | \color{white}{\text{digits | ||
+ | \color{white}{\text{place value}: | ||
+ | \color{white}{\text{result}: | ||
+ | \end{smallmatrix} | ||
\end{align*} | \end{align*} | ||
- | |||
- | | 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 | ||
<---- | <---- | ||
- | ----> | ||
- | | 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 | | + | 3. calculate the place factor |
- | | index | $i$ | + | |
- | | place value | $B^i$ | $\small{10^3}$ | + | |
- | | 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| | + | |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | |
- | <---- | + | |
- | ----> | + | \begin{align*} |
- | |aus (6)|$\color{blue}{I_o} = I_1 $ |$I_o$ ist damit bekannt, wenn $I_1$ bekannt ist| | + | \begin{smallmatrix} |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | \color{black}{\text{numeral}: |
+ | \color{black}{\text{index}: | ||
+ | \color{blue }{\text{place factor}: | ||
+ | \color{white}{} & \color{white}{} | ||
+ | \color{white}{\text{digits | ||
+ | \color{white}{\text{place value}: | ||
+ | \color{white}{\text{result}: | ||
+ | \end{smallmatrix} | ||
+ | \end{align*} | ||
<---- | <---- | ||
- | ----> | ||
- | |aus (7) und (3)|$I_1 - I_2 -\color{blue}{0} = 0 $ |$\quad$| | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
----> | ----> | ||
- | |$\quad$|$I_1 | + | 3. calculate the place factor |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | |
- | <---- | + | |
- | ----> | + | \begin{align*} |
- | |$\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)| | + | \begin{smallmatrix} |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | \color{black}{\text{numeral}: |
+ | \color{black}{\text{index}: | ||
+ | \color{blue }{\text{place factor}: | ||
+ | \color{blue }{} & \color{blue }{} & | ||
+ | \color{white}{\text{digits | ||
+ | \color{white}{\text{place value}: | ||
+ | \color{white}{\text{result}: | ||
+ | \end{smallmatrix} | ||
+ | \end{align*} | ||
<---- | <---- | ||
- | ----> | ||
- | | $\quad$ |$\frac{U_1}{R_1}= \frac{U_2}{R_2} = \frac{U_A}{R_1 + R_2}$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | <---- | ||
----> | ----> | ||
- | | (10)|$U_2= U_A\cdot\frac{R_2}{R_1+R_2}$ |Spannungsteilerformel| | + | 4. write down each digit of the numeral |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | |
- | <---- | + | |
- | ----> | + | \begin{align*} |
- | $II.\quad$ Betrachtung der Spannungsverstärkung | + | \begin{smallmatrix} |
+ | \color{black}{\text{numeral}: | ||
+ | \color{black}{\text{index}: | ||
+ | \color{black}{\text{place factor}: | ||
+ | \color{black}{} | ||
+ | \color{blue }{\text{digits | ||
+ | \color{white}{\text{place value}: | ||
+ | \color{white}{\text{result}: | ||
+ | \end{smallmatrix} | ||
+ | \end{align*} | ||
<---- | <---- | ||
- | ---->> | ||
- | |aus (0) |$\color{blue}{A_V}=\frac{U_A}{U_E}$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | ---->> | + | ----> |
- | | $\quad$ | + | 5. calculate the place value \\ $\quad$ |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | |
- | << | + | |
- | ---->> | + | \begin{align*} |
- | | $\quad$ |$A_V=\frac{U_A}{\color{blue}{U_2+U_D}}$ | $\quad$ | | + | \begin{smallmatrix} |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | \color{black}{\text{numeral}: |
- | <<---- | + | \color{black}{\text{index}: |
+ | \color{black}{\text{place factor}: | ||
+ | \color{black}{} | ||
+ | \color{black}{\text{digits | ||
+ | \color{blue }{\text{place value}: | ||
+ | \color{white}{\text{result}: | ||
+ | \end{smallmatrix} | ||
+ | \end{align*} | ||
+ | <---- | ||
- | ---->> | + | ----> |
- | | $\quad$ | + | 6.Add all place values |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | |
- | << | + | |
- | + | ||
- | ---->> | + | |
- | | $\quad$ | + | |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | |
- | << | + | |
- | ---->> | + | \begin{align*} |
- | | $\quad$ |$A_V=\frac{U_A}{U_A\cdot\frac{R_2}{R_1+R_2}+U_D}$ | $\quad$ | | + | \begin{smallmatrix} |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | \color{black}{\text{numeral}: |
- | << | + | \color{black}{\text{index}: |
- | + | \color{black}{\text{place factor}: | |
- | ---->> | + | \color{black}{} |
- | | $\quad$ | + | \color{black}{\text{digits |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | \color{black}{\text{place value}: |
- | << | + | \color{blue }{\text{result}: |
- | + | \end{smallmatrix} | |
- | ---->> | + | \end{align*} |
- | | $\quad$ |$A_V=\frac{U_A}{U_A\cdot\frac{R_2}{R_1+R_2}+\color{blue}{\frac{U_A}{A_D}}}$ | $\quad$ | | + | <---- |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | |
- | << | + | |
- | + | ||
- | ---->> | + | |
- | | $\quad$ | + | |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | |
- | << | + | |
- | + | ||
- | ---->> | + | |
- | | $\quad$ |$A_V=\frac{\color{blue}{U_A}}{\color{blue}{U_A}\cdot\frac{R_2}{R_1+R_2}+\frac{\color{blue}{U_A}}{A_D}}$ | Erweitern mit $\frac{1}{U_A}$ | | + | |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | |
- | << | + | |
- | + | ||
- | ---->> | + | |
- | | $\quad$ |$A_V=\frac{1}{\frac{R_2}{R_1+R_2}+\frac{1}{A_D}}$ | $\quad$ | | + | |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | |
- | << | + | |
- | + | ||
- | ---->> | + | |
- | | $\quad$ | + | |
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | + | |
- | <<---- | + | |
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << | ||
- | ---->> | ||
- | | $\quad$ | ||
- | |$\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$|$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$| | ||
- | << |