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fix LaTeX passthrough

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JamesFlare1212
2025-05-04 15:35:08 -04:00
parent 68ffcf4956
commit 6a996f1c09
5 changed files with 177 additions and 172 deletions
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148
content/zh-cn/posts/ecse-1010/lab01/index.md
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@@ -71,8 +71,8 @@ repost:
我们知道，欧姆定律、KCL 和 KVL 可以表示成以下公式：
$$
V = IR \\\
\sum I_{in} = \sum I_{out} \\\
V = IR \\
\sum I_{in} = \sum I_{out} \\
\sum V_n = 0
$$
@@ -82,11 +82,11 @@ $$
$$
\begin{align*}
V &= IR \\\
I &= \frac{V}{R} \\\
I_{total} &= \frac{5}{10K + \cfrac{1}{\frac{1}{1K} + \frac{1}{1K}} + 10K} \\\
I_{total} &= \frac{5}{10000 + 500 + 10000} \\\
I_{total} &= 0.000243902439 \\\
V &= IR \\
I &= \frac{V}{R} \\
I_{total} &= \frac{5}{10K + \cfrac{1}{\frac{1}{1K} + \frac{1}{1K}} + 10K} \\
I_{total} &= \frac{5}{10000 + 500 + 10000} \\
I_{total} &= 0.000243902439 \\
\end{align*}
$$
@@ -94,8 +94,8 @@ $$
$$
\begin{align*}
I(R2) = I(R3) &= I_{total} \times \frac{R2}{R2 + R3} \\\
I(R2) = I(R3) &= 0.000243902439 \times \frac{1000}{1000 + 1000} \\\
I(R2) = I(R3) &= I_{total} \times \frac{R2}{R2 + R3} \\
I(R2) = I(R3) &= 0.000243902439 \times \frac{1000}{1000 + 1000} \\
I(R2) = I(R3) &= 0.0001219512195
\end{align*}
$$
@@ -106,16 +106,16 @@ $$
$$
\begin{align*}
V(R1) = V(R4) &= V_{total} \times \frac{R1}{R1 + R2 \Vert R3 + R4} \\\
V(R1) = V(R4) &= 5 \times \frac{10000}{10000 + 500 + 10000} \\\
V(R1) = V(R4) &= V_{total} \times \frac{R1}{R1 + R2 \Vert R3 + R4} \\
V(R1) = V(R4) &= 5 \times \frac{10000}{10000 + 500 + 10000} \\
V(R1) = V(R4) &= 2.4390244
\end{align*}
$$
$$
\begin{align*}
V(R2) = V(R3) &= V_{total} - (V(R1) + V(R4)) \\\
V(R2) = V(R3) &= 5 - 2.4390244 - 2.4390244 \\\
V(R2) = V(R3) &= V_{total} - (V(R1) + V(R4)) \\
V(R2) = V(R3) &= 5 - 2.4390244 - 2.4390244 \\
V(R2) = V(R3) &= 0.1219512
\end{align*}
$$
@@ -185,24 +185,24 @@ V(n003): 2.43902 voltage
$$
\begin{align*}
V(R1) &= V(n001) - V(n002) \\\
V(R1) &= 5 - 2.56098 \\\
V(R1) &= V(n001) - V(n002) \\
V(R1) &= 5 - 2.56098 \\
V(R1) &= \boxed{2.43902}
\end{align*}
$$
$$
\begin{align*}
V(R2) = V(R3) &= V(n002) - V(n003) \\\
V(R2) = V(R3) &= 2.56098 - 2.43902 \\\
V(R2) = V(R3) &= V(n002) - V(n003) \\
V(R2) = V(R3) &= 2.56098 - 2.43902 \\
V(R2) = V(R3) &= \boxed{0.12196}
\end{align*}
$$
$$
\begin{align*}
V(R4) &= V(n003) - V(\text{GND}) \\\
V(R4) &= 2.43902 - 0 \\\
V(R4) &= V(n003) - V(\text{GND}) \\
V(R4) &= 2.43902 - 0 \\
V(R4) &= \boxed{2.43902}
\end{align*}
$$
@@ -238,8 +238,8 @@ I(V1): -0.000243902 device_current
$$
\begin{align*}
& \quad \thickspace I(R1) + I(R2) + I(R3) \\\
&= -0.000243902 + 0.000121951 + 0.000121951 \\\
& \quad \thickspace I(R1) + I(R2) + I(R3) \\
&= -0.000243902 + 0.000121951 + 0.000121951 \\
&= \boxed{0}
\end{align*}
$$
@@ -254,8 +254,8 @@ KCL 很可能为真。
$$
\begin{align*}
V(R1) &= 2.43902 \\\
V(R2) = V(R3) &= 0.12196 \\\
V(R1) &= 2.43902 \\
V(R2) = V(R3) &= 0.12196 \\
V(R4) &= 2.43902
\end{align*}
$$
@@ -264,9 +264,9 @@ $$
$$
\begin{align*}
& \quad \thickspace V(n001) - V(n002) - V(n003) \\\
&= 2.43902 - 0.12196 - 0.12196 \\\
&= 0 \\\
& \quad \thickspace V(n001) - V(n002) - V(n003) \\
&= 2.43902 - 0.12196 - 0.12196 \\
&= 0 \\
& \boxed{\text{True}}
\end{align*}
$$
@@ -290,24 +290,24 @@ V(R4) = 2.4616V
$$
\begin{align*}
V &= IR \\\
I &= \frac{V}{R} \\\
V &= IR \\
I &= \frac{V}{R} \\
I(R1) &= \frac{2.4963}{10000} = \boxed{0.00024963}
\end{align*}
$$
$$
\begin{align*}
V &= IR \\\
I &= \frac{V}{R} \\\
V &= IR \\
I &= \frac{V}{R} \\
I(R2) = I(R3) &= \frac{0.1665}{1000} = \boxed{0.0001665}
\end{align*}
$$
$$
\begin{align*}
V &= IR \\\
I &= \frac{V}{R} \\\
V &= IR \\
I &= \frac{V}{R} \\
I(R4) &= \frac{2.4616}{10000} = \boxed{0.00024616}
\end{align*}
$$
@@ -405,16 +405,16 @@ V(n002): 2.5 voltage
$$
\begin{align*}
V(R1) &= V(n001) - V(n002) \\\
V(R1) &= 5 - 2.5 \\\
V(R1) &= V(n001) - V(n002) \\
V(R1) &= 5 - 2.5 \\
V(R1) &= \boxed{2.5}
\end{align*}
$$
$$
\begin{align*}
V(R2) &= V(n002) - \text{GND} \\\
V(R2) &= 2.5 - 0 \\\
V(R2) &= V(n002) - \text{GND} \\
V(R2) &= 2.5 - 0 \\
V(R2) &= \boxed{2.5}
\end{align*}
$$
@@ -467,7 +467,7 @@ $$
$$
\begin{align*}
\frac{V_1}{V_2} &= \frac{R_1}{R_2} \\\
\frac{V_1}{V_2} &= \frac{R_1}{R_2} \\
\frac{V_1}{V_2} &= \frac{10K}{10K} = 1
\end{align*}
$$
@@ -478,8 +478,8 @@ $$
$$
\begin{align*}
I(R1) = I(R2) &= \frac{V}{R} \\\
I(R1) = I(R2) &= \frac{2.5}{10K} \\\
I(R1) = I(R2) &= \frac{V}{R} \\
I(R1) = I(R2) &= \frac{2.5}{10K} \\
I(R1) = I(R2) &= \boxed{0.00025}
\end{align*}
$$
@@ -537,16 +537,16 @@ $V(R2) = 2.5204V$
$$
\begin{align*}
I(R1) &= \frac{V}{R} \\\
I(R1) &= \frac{2.5539}{10K} \\\
I(R1) &= \frac{V}{R} \\
I(R1) &= \frac{2.5539}{10K} \\
I(R1) &= 0.00025539
\end{align*}
$$
$$
\begin{align*}
I(R2) &= \frac{V}{R} \\\
I(R2) &= \frac{2.5204}{10K} \\\
I(R2) &= \frac{V}{R} \\
I(R2) &= \frac{2.5204}{10K} \\
I(R2) &= 0.00025204
\end{align*}
$$
@@ -587,7 +587,7 @@ $$
$$
\begin{align*}
V(R1) = V(R2) &= 5 - 0 \\\
V(R1) = V(R2) &= 5 - 0 \\
V(R1) = V(R2) &= \boxed{5}
\end{align*}
$$
@@ -634,8 +634,8 @@ $$
$$
\begin{align*}
I_{total} &= \frac{5}{\cfrac{1}{\frac{1}{10K} + \frac{1}{10K}}} \\\
I_{total} &= \frac{5}{5K} \\\
I_{total} &= \frac{5}{\cfrac{1}{\frac{1}{10K} + \frac{1}{10K}}} \\
I_{total} &= \frac{5}{5K} \\
I_{total} &= \boxed{0.001}
\end{align*}
$$
@@ -686,7 +686,7 @@ V(n001): 5 voltage
$$
\begin{align*}
V(R1) = V(R2) &= 5 - 0 \\\
V(R1) = V(R2) &= 5 - 0 \\
V(R1) = V(R2) &= \boxed{5}
\end{align*}
$$
@@ -729,9 +729,9 @@ $$
$$
\begin{align*}
I_{total} &= I(R2) + I(R1) \\\
I_{total} &= 0.0005 + 0.0005 \\\
I_{total} &= 0.001 \\\
I_{total} &= I(R2) + I(R1) \\
I_{total} &= 0.0005 + 0.0005 \\
I_{total} &= 0.001 \\
& 0.001 = 0.001 \\; \boxed{\text{True}}
\end{align*}
$$
@@ -760,8 +760,8 @@ R2 = 10K
$$
\begin{align*}
I &= \frac{V}{R} \\\
I(R1) = I(R2) &= \frac{5.0305}{10000} \\\
I &= \frac{V}{R} \\
I(R1) = I(R2) &= \frac{5.0305}{10000} \\
I(R1) = I(R2) &= \boxed{0.0005305}
\end{align*}
$$
@@ -803,7 +803,7 @@ $$
$$
\begin{align*}
V(R1) = V(R2) &= 5 - 0 \\\
V(R1) = V(R2) &= 5 - 0 \\
V(R1) = V(R2) &= \boxed{5}
\end{align*}
$$
@@ -824,16 +824,16 @@ $$
$$
\begin{align*}
I(R1) &= \frac{V(R1)}{R1} \\\
I(R1) &= \frac{5}{10K} \\\
I(R1) &= \frac{V(R1)}{R1} \\
I(R1) &= \frac{5}{10K} \\
I(R1) &= \boxed{0.0005}
\end{align*}
$$
$$
\begin{align*}
I(R2) &= \frac{V(R2)}{R2} \\\
I(R2) &= \frac{5}{10K} \\\
I(R2) &= \frac{V(R2)}{R2} \\
I(R2) &= \frac{5}{10K} \\
I(R2) &= \boxed{0.0005}
\end{align*}
$$
@@ -842,10 +842,10 @@ $I(R1)$ 和 $I(R2)$ 的关系可以表示为：
$$
\begin{align*}
\frac{I(R1)}{I(R2)} &= \cfrac{\cfrac{V(R1)}{R1}}{\cfrac{V(R2)}{R2}} \\\
\because V(R1) &= V(R2) \\\
\therefore \frac{I(R1)}{I(R2)} &= \cfrac{\cfrac{\cancel{V(R1)}}{R1} \times \cfrac{1}{\cancel{V(R1)}}}{\cfrac{\cancel{V(R2)}}{R2} \times \cfrac{1}{\cancel{V(R2)}}} \\\
\frac{I(R1)}{I(R2)} &= \frac{\frac{1}{R1}}{\frac{1}{R2}} \\\
\frac{I(R1)}{I(R2)} &= \cfrac{\cfrac{V(R1)}{R1}}{\cfrac{V(R2)}{R2}} \\
\because V(R1) &= V(R2) \\
\therefore \frac{I(R1)}{I(R2)} &= \cfrac{\cfrac{\cancel{V(R1)}}{R1} \times \cfrac{1}{\cancel{V(R1)}}}{\cfrac{\cancel{V(R2)}}{R2} \times \cfrac{1}{\cancel{V(R2)}}} \\
\frac{I(R1)}{I(R2)} &= \frac{\frac{1}{R1}}{\frac{1}{R2}} \\
&\boxed{\frac{I(R1)}{I(R2)} = \frac{R2}{R1}}
\end{align*}
$$
@@ -860,9 +860,9 @@ $$
$$
\begin{align*}
I_{total} &= \frac{V_{total}}{R_{total}} \\\
I_{total} &= \frac{5}{\cfrac{1}{\cfrac{1}{10K} + \cfrac{1}{10K}}} \\\
I_{total} &= \frac{5}{5K} \\\
I_{total} &= \frac{V_{total}}{R_{total}} \\
I_{total} &= \frac{5}{\cfrac{1}{\cfrac{1}{10K} + \cfrac{1}{10K}}} \\
I_{total} &= \frac{5}{5K} \\
I_{total} &= 0.001
\end{align*}
$$
@@ -877,8 +877,8 @@ $$
$$
\begin{align*}
I(R1) = I(R2) &= I_{total} \times \frac {R1}{R1 + R2} \\\
I(R1) = I(R2) &= 0.001 \times \frac {10K}{10K + 10K} \\\
I(R1) = I(R2) &= I_{total} \times \frac {R1}{R1 + R2} \\
I(R1) = I(R2) &= 0.001 \times \frac {10K}{10K + 10K} \\
I(R1) = I(R2) &= \boxed{0.0005}
\end{align*}
$$
@@ -887,11 +887,11 @@ $$
$$
\begin{align*}
& \because R1 = R2 = 10K \\\
& \because I(R1) = I(R2) = 0.0005 \\\
& \because V(R1) = V(R2) = 5 \\\
& \because I_{total} = I(R1) + I(R2) = 0.001 \\\
& \because \frac{I(R1)}{I(R2)} = \frac{R2}{R1} \\\
& \because R1 = R2 = 10K \\
& \because I(R1) = I(R2) = 0.0005 \\
& \because V(R1) = V(R2) = 5 \\
& \because I_{total} = I(R1) + I(R2) = 0.001 \\
& \because \frac{I(R1)}{I(R2)} = \frac{R2}{R1} \\
& \therefore I(R1) = I(R2) = I_{total} \times \frac {R1}{R1 + R2}
\end{align*}
$$
@@ -952,7 +952,7 @@ V(n001): 5 voltage
$$
\begin{align*}
V(R1) = V(R2) &= 5 - 0 \\\
V(R1) = V(R2) &= 5 - 0 \\
V(R1) = V(R2) &= \boxed{5}
\end{align*}
$$
@@ -992,8 +992,8 @@ R2 = 10K
$$
\begin{align*}
I &= \frac{V}{R} \\\
I(R1) = I(R2) &= \frac{5.0305}{10000} \\\
I &= \frac{V}{R} \\
I(R1) = I(R2) &= \frac{5.0305}{10000} \\
I(R1) = I(R2) &= \boxed{0.0005305}
\end{align*}
$$