# EC Mid Past (4/18)

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[TOC]

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## 2021
### 1.
#### (1)
<!-- TODO -->
$V_{S1} = V_1 + V_2 + V_4$
$V_{S2} = - V_2 + V_3$
$V_{S1} + V_{S2} = V_1 + V_3 + V_4$
$I_1 = I_4$
$I_1 = I_2 + I_3$
$I_2 + I_3 = I_4$
==**ANS**==: $V_{S1} + V_{S2} = V_1 + V_3 + V_4,\ I_2 + I_3 = I_4$

#### (2)
$R_{(S1)} = R_1 + R_2 \mid\mid R_3 + R_4$
==**ANS**==: $R_1 + R_2 \mid\mid R_3 + R_4$

#### (3)
$R_{(S1)} = R_1 + R_2 + R_4$
==**ANS**==: $R_1 + R_2 + R_4$

#### (4)
<!-- TODO -->
$R_{(S1)} = R_1 + R_2 \mid\mid R_3 + R_4$
$I_{4(S1)} = \frac{V_{S1}}{R_{(S1)}} = \frac{V_{S1}}{R_1 + R_2 \mid\mid R_3 + R_4}$
$V_{4(S1)} = I_{4(S1)} R_4 = \frac{V_{S1} R_4}{R_1 + R_2 \mid\mid R_3 + R_4}$
$R_{(S2)} = (R_1 + R_4) \mid\mid R_2 + R_3$
$V_{4(S2)} = V_{S2} \times \frac{(R_1 + R_4) \mid\mid R_2}{(R_1 + R_4) \mid\mid R_2 + R_3} \times \frac{R_4}{R_1 + R_4}$
$I_{4(S2)} = \frac{V_{4(S2)}}{R_4} = V_{S2} \times \frac{(R_1 + R_4) \mid\mid R_2}{(R_1 + R_4) \mid\mid R_2 + R_3} \times \frac{1}{R_1 + R_4}$
$I_4 = I_{4(S1)} + I_{4(S2)} = \frac{V_{S1}}{R_1 + R_2 \mid\mid R_3 + R_4} + V_{S2} \times \frac{(R_1 + R_4) \mid\mid R_2}{(R_1 + R_4) \mid\mid R_2 + R_3} \times \frac{1}{R_1 + R_4}$
$V_4 = V_{4(S1)} + V_{4(S2)} = \frac{V_{S1} R_4}{R_1 + R_2 \mid\mid R_3 + R_4} + V_{S2} \times \frac{(R_1 + R_4) \mid\mid R_2}{(R_1 + R_4) \mid\mid R_2 + R_3} \times \frac{R_4}{R_1 + R_4}$
==**ANS**==: $V_4 = \frac{V_{S1} R_4}{R_1 + R_2 \mid\mid R_3 + R_4} + V_{S2} \times \frac{(R_1 + R_4) \mid\mid R_2}{(R_1 + R_4) \mid\mid R_2 + R_3} \times \frac{R_4}{R_1 + R_4}, \\
I_4 = \frac{V_{S1}}{R_1 + R_2 \mid\mid R_3 + R_4} + V_{S2} \times \frac{(R_1 + R_4) \mid\mid R_2}{(R_1 + R_4) \mid\mid R_2 + R_3} \times \frac{1}{R_1 + R_4}$

### 2.
#### (1)
$V_{AB} = V_A - V_B = 12 \times (\frac{2\ (k\Omega)}{R_X + 2\ (k\Omega)} - \frac{1\ (k\Omega)}{1 + 3\ (k\Omega)})\ (V)$
$V_{AB} = 0\ (V) \Leftrightarrow R_X = 6\ (k\Omega)$
$V_{AB} = -1\ (V) \Leftrightarrow R_X = 10\ (k\Omega)$
$V_{AB} = 1\ (V) \Leftrightarrow R_X = 4\ (k\Omega)$
==**ANS**==: $V_{AB} = 0\ (V) \Leftrightarrow R_X = 6\ (k\Omega),\ V_{AB} = -1\ (V) \Leftrightarrow R_X = 10\ (k\Omega),\ V_{AB} = 1\ (V) \Leftrightarrow R_X = 4\ (k\Omega)$

#### (2)
<!-- TODO -->
$R_X(t) = 6 - 0.005 \times(t - 25)\ (k\Omega)$
$V_{AB}(t) = 12 \times (\frac{2\ (k\Omega)}{2 + R_X(t)\ (k\Omega)} - \frac{1\ (k\Omega)}{1 + 3\ (k\Omega)})\ (V) = 12 \times (\frac{2\ (k\Omega)}{6 - 0.005 \times(t - 25) + 2\ (k\Omega)} - \frac{1\ (k\Omega)}{1 + 3\ (k\Omega)})\ (V) \\
\qquad\quad = 12 \times (\frac{2}{-0.005 \times t + 8.125} - \frac{1}{4})\ (V)$
==**ANS**==:

#### (3)
$R_X = 6\ (k\Omega)$
$R_{TH} = R_X \mid\mid 2 + 3 \mid\mid 1\ (k\Omega) = 6 \mid\mid 2 + 3 \mid\mid 1\ (k\Omega) = (1/6 + 1/2)^{-1} + (1/3 + 1/1)^{-1}\ (k\Omega) \\
\qquad \approx 1.5 + 0.75\ (k\Omega) = 2.25\ (k\Omega)$
$V_{TH} = 0$
==**ANS**==: $V_{TH} = 0,\ R_{TH} \approx 2.25\ (k\Omega)$

### 3.
$C_T = (1/10 + 1/(10 + 10) + 1/5)^{-1}\ (nF) \approx 2.86\ (nF)$
$X_C$ ==OUT==
$Z_C$ ==OUT==
==**ANS**==: $C_T \approx 2.86\ (nF),\ X_C: (OUT),\ Z_C: (OUT)$

### 4.
#### (1)
==OUT==

#### (2)
==OUT==

#### (3)
==OUT==

## 2020
### 1.
#### (1)
$V_S = V_1 - V_3 + V_4$
$I_2 + I_3 + I_4 = 0$
==**ANS**==: $V_S = V_1 - V_3 + V_4,\ I_2 + I_3 + I_4 = 0$

#### (2)
$R = R_1 + R_5 \mid\mid (R_4 + R_2 \mid\mid R_3) = 0.8 + 3 \mid\mid (1 + 2 \mid\mid 2)\ (k\Omega) = 0.8 + 3 \mid\mid 2\ (k\Omega) = 0.8 + 1.2\ (k\Omega) \\
\quad = 2\ (k\Omega)$
==**ANS**==: $2\ (k\Omega)$

#### (3)
<!-- TODO -->
$V_S = 2\ (V)$
$I_2 = \frac{V_S}{R} \times \frac{(R_4 + R_2 \mid\mid R_3)^{-1}}{(R_4 + R_2 \mid\mid R_3)^{-1} + R_5^{-1}} \times \frac{R_2^{-1}}{R_2^{-1} + R_3^{-1}} =\frac{2\ (V)}{2\ (k\Omega)} \times (0.5 \times 1.2) \times \frac{0.5}{1} = 0.3\ (mA)$
$V_2 = I_2 R_2 = 0.3\ (mA) \times 2\ (k\Omega) = 0.6\ (V)$
$P_2 = V_2 I_2 = 0.3\ (mA) \times 0.6\ (V) = 0.18\ (mW)$
==**ANS**==: $I_2 \approx 0.3\ (mA),\ V_2 \approx 0.6\ (V),\ P_2 \approx 0.18\ (mW)$

### 2.
$R_{(S:\ 4V)} = R_1 \mid\mid R_3 + R_2 = 1 \mid\mid 3 + 1\ (k\Omega) = 0.75 + 1\ (k\Omega) = 1.75\ (k\Omega)$
$I_{1(S:\ 4V)} = \frac{4\ (V)}{R_{(S:\ 4V)}} \times \frac{R_1^{-1}}{R_1^{-1} + R_3^{-1}} = \frac{4\ (V)}{1.75\ (k\Omega)} \times 0.75 \approx 1.71\ (mA)$
$R_{(S:\ 5V)} = R_1 \mid\mid R_2 + R_3 = 1 \mid\mid 1 + 3\ (k\Omega) = 3.5\ (k\Omega)$
$I_{1(S:\ 5V)} = \frac{5\ (V)}{R_{(S:\ 5V)}} \times \frac{R_1^{-1}}{R_1^{-1} + R_2^{-1}} = \frac{5\ (V)}{3.5\ (k\Omega)} \times \frac{1}{2} \approx 0.71\ (mA)$
$I_1 = I_{1(S:\ 4V)} + I_{1(S:\ 5V)} \approx 1.71 + 0.71\ (mA) = 2.42\ (mA)$
$V_1 = I_1 R_1 = 2.42 \times 1\ (V) = 2.42\ (V)$
==**ANS**==: $V_1 \approx 2.42\ (V),\ I_1 \approx 2.42\ (mA)$

### 3.
$R_{TH} = R_1 \mid\mid R_2 + R_3 = 1 \mid\mid 1 + 2\ (k\Omega) = 2.5\ (k\Omega)$
$V_{TH} = V \frac{R_1}{R_1 + R_2} = 2\ (V)$
==**ANS**==: $R_{TH} = 2.5\ (k\Omega),\ V_{TH} = 2\ (V)$

### 4.
#### (1)
$V_{AB} = 9 \times (\frac{2\ (k\Omega)}{R_X + 2\ (k\Omega)} - \frac{1\ (k\Omega)}{2 + 1\ (k\Omega)})\ (V) = 9 \times (\frac{2\ (k\Omega)}{R_X + 2\ (k\Omega)} - \frac{1}{3})\ (V)$
$V_{AB} = 0\ (V) \Leftrightarrow R_X = 4\ (k\Omega)$
$V_{AB} = -1\ (V) \Leftrightarrow R_X = 7\ (k\Omega)$
$V_{AB} = 1\ (V) \Leftrightarrow R_X = 2.5\ (k\Omega)$
==**ANS**==: $V_{AB} = 0\ (V) \Leftrightarrow R_X = 4\ (k\Omega),\ V_{AB} = -1\ (V) \Leftrightarrow R_X = 7\ (k\Omega),\ V_{AB} = 1\ (V) \Leftrightarrow R_X = 2.5\ (k\Omega)$

#### (2)
$R_X(t) = 4 - 0.02 \times (t - 25)\ (k\Omega)$
$V_{AB}(t) = 9 \times (\frac{2\ (k\Omega)}{R_X + 2\ (k\Omega)} - \frac{1}{3})\ (V) = 9 \times (\frac{2}{2 + 4 - 0.02 \times (t - 25)} - \frac{1}{3})\ (V) = 9 \times (\frac{2}{-0.02 \times t + 6.5} - \frac{1}{3})\ (V)$
==**ANS**==:

### 5.
$C_T = (1/10 + 1/(10 + 10) + 1/5)^{-1}\ (nF) \approx 2.86\ (nF)$
$X_C$ ==OUT==
$Z_C$ ==OUT==
==**ANS**==: $C_T \approx 2.86\ (nF),\ X_C: (OUT),\ Z_C: (OUT)$

### 6.
#### (1)
==OUT==

#### (2)
==OUT==

#### (3)
==OUT==

## 2019
### 1.
#### (1)
$V_1 = V_3 + V_7 - V_4$
==**ANS**==: $V_1 = V_3 + V_7 - V_4$

#### (2)
$I_2 + I_4 + I_5 + I_7 = 0$
==**ANS**==: $I_2 + I_4 + I_5 + I_7 = 0$

#### (3)
<!-- TODO -->
$R_{(S1)} = R_3 \mid\mid (R_1 \mid\mid R_6 + R_4 \mid\mid R_5) + R_2 = 2 \mid\mid (2 \mid\mid 3 + 1 \mid\mid 1) + 2\ (k\Omega) \approx 2.92\ (k\Omega)$
$I_{3(S1)} = \frac{V_1}{R_{(S1)}} \times \frac{(R_1 \mid\mid R_6 + R_4 \mid\mid R_5)^{-1}}{(R_1 \mid\mid R_6 + R_4 \mid\mid R_5)^{-1} + R_3^{-1}} \times \frac{R_1^{-1}}{R_1^{-1} + R_6^{-1}} \approx \frac{10\ (V)}{2.92\ (k\Omega)} \times (\frac{10}{17} \times \frac{34}{37}) \times (\frac{1}{2} \times \frac{6}{5}) \approx 1.11\ (mA)$
$R_{(S2)} = R_5 \mid\mid (R_2 \mid\mid R_3 + R_1 \mid\mid R_6) + R_4 = 1 \mid\mid (2 \mid\mid 2 + 2 \mid\mid 3) + 1\ (k\Omega) \approx 1.69\ (k\Omega)$
$I_{3(S2)} = \frac{V_2}{R_{(S2)}} \times \frac{(R_2 \mid\mid R_3 + R_1 \mid\mid R_6)^{-1}}{(R_2 \mid\mid R_3 + R_1 \mid\mid R_6)^{-1} + R_5^{-1}} \times \frac{R_1^{-1}}{R_1^{-1} + R_6^{-1}} \approx \frac{20\ (V)}{1.69\ (k\Omega)} \times (\frac{5}{11} \times \frac{11}{16}) \times (\frac{1}{2} \times \frac{6}{5}) \approx 2.22\ (mA)$
$I_3 = I_{3(S1)} + I_{3(S2)} \approx 1.11 + 2.22\ (mA) = 3.33\ (mA)$
$V_3 = I_3 R_1 \approx 3.33 \times 2\ (V) = 6.66\ (V)$
==**ANS**==: $I_3 \approx 3.33\ (mA),\ V_3 \approx 6.66\ (V)$

#### (4)
$R_{TH} = R_2 \mid\mid R_3 + R_1 \mid\mid R_6 = 2 \mid\mid 2 + 2 \mid\mid 3\ (k\Omega) = 2.2\ (k\Omega)$
$V_{TH} = V_1 \times \frac{R_3}{R_3 + R_2} = 10 \times \frac{2}{4}\ (V) = 5\ (V)$
==**ANS**==: $R_{TH} = 2.2\ (k\Omega),\ V_{TH} = 5\ (V)$

### 2.
#### (1)
$V_{AB} = 12 \times (\frac{1\ (k\Omega)}{R_X + 1\ (k\Omega)} - \frac{4\ (k\Omega)}{2 + 4\ (k\Omega)})\ (V) = 12 \times (\frac{1\ (k\Omega)}{R_X + 1\ (k\Omega)} - \frac{2}{3})\ (V)$
$V_{AB} = 0\ (V) \Leftrightarrow R_X = 0.5\ (k\Omega)$
$V_{AB} = -1\ (V) \Leftrightarrow R_X = 0.7\ (k\Omega)$
$V_{AB} = 1\ (V) \Leftrightarrow R_X = 0.3\ (k\Omega)$
$V_{AB} = -2\ (V) \Leftrightarrow R_X = 1\ (k\Omega)$
$V_{AB} = 2\ (V) \Leftrightarrow R_X = 0.2\ (k\Omega)$
==**ANS**==: $V_{AB} = 0\ (V) \Leftrightarrow R_X = 0.5\ (k\Omega),\ V_{AB} = -1\ (V) \Leftrightarrow R_X = 0.7\ (k\Omega),\ V_{AB} = 1\ (V) \Leftrightarrow R_X = 0.3\ (k\Omega),\\
V_{AB} = -2\ (V) \Leftrightarrow R_X = 1\ (k\Omega),\ V_{AB} = 2\ (V) \Leftrightarrow R_X = 0.2\ (k\Omega)$

#### (2)
$R_X(t) = 0.5 - 0.005 \times (t - 25)\ (k\Omega)$
$V_{AB}(t) = 12 \times (\frac{1}{1 + 0.5 - 0.005 \times (t - 25)} - \frac{2}{3})\ (V) = 12 \times (\frac{1}{-0.005 \times t + 1.625} - \frac{2}{3})\ (V)$
$V_{AB} = 0\ (V) \Leftrightarrow t = 25\ (^\circ C)$
$V_{AB} = -1\ (V) \Leftrightarrow t = -17.8\ (^\circ C)$
$V_{AB} = 1\ (V) \Leftrightarrow t = 58.3\ (^\circ C)$
$V_{AB} = -2\ (V) \Leftrightarrow t = -75\ (^\circ C)$
$V_{AB} = 2\ (V) \Leftrightarrow t = 85\ (^\circ C)$
==**ANS**==: $V_{AB} = 0\ (V) \Leftrightarrow t = 25\ (^\circ C),\ V_{AB} = -1\ (V) \Leftrightarrow t = -17.8\ (^\circ C),\ V_{AB} = 1\ (V) \Leftrightarrow t = 58.3\ (^\circ C),\\
V_{AB} = -2\ (V) \Leftrightarrow t = -75\ (^\circ C),\ V_{AB} = 2\ (V) \Leftrightarrow t = 85\ (^\circ C)$

### 3.
$C_{T(On)} = (1/20 + 1/(10 + (1/20 + 1/(2 + 3))^{-1}) + 1/20)^{-1} \approx 5.83\ (nF)$
$C_{T(Off)} = (1/20 + 1/10 + 1/20)^{-1} = 5\ (nF)$
$X_C$ ==OUT==
$Z_C$ ==OUT==
==**ANS**==: $C_{T(On)}\approx 5.83\ (nF),\ C_{T(Off)} = 5\ (nF),\ X_C: (OUT),\ Z_C: (OUT)$

### 4.
#### (1)
==OUT==

#### (2)
==OUT==

#### (3)
==OUT==

#### (4)
==OUT==

#### (5)
==OUT==

## 2018
### 1. (20m)
#### (1)
$V_1 = V_3 + V_6 - V_2 - V_4$
==**ANS**==: $V_1 = V_3 + V_6 - V_2 - V_4$

#### (2)
$I_2 + I_4 + I_5 + I_7 = 0$
==**ANS**==: $I_2 + I_4 + I_5 + I_7 = 0$

#### (3)
$R_{(S1)} = R_3 \mid\mid (R_1 + R_4 \mid\mid R_5) + R_2$
$I_{3(S1)} = \frac{V_1}{R_{(S1)}} \times \frac{(R_1 + R_4 \mid\mid R_5)^{-1}}{(R_1 + R_4 \mid\mid R_5)^{-1} + R_3^{-1}}$
$R_{(S2)} = R_5 \mid\mid (R_2 \mid\mid R_3 + R_1) + R_4$
$I_{3(S2)} = \frac{V_2}{R_{(S2)}} \times \frac{(R_2 \mid\mid R_3 + R_1)^{-1}}{(R_2 \mid\mid R_3 + R_1)^{-1} + R_5^{-1}}$
$I_3 = I_{3(S1)} + I_{3(S2)} = \frac{V_1}{R_{(S1)}} \times \frac{(R_1 + R_4 \mid\mid R_5)^{-1}}{(R_1 + R_4 \mid\mid R_5)^{-1} + R_3^{-1}} + \frac{V_2}{R_{(S2)}} \times \frac{(R_2 \mid\mid R_3 + R_1)^{-1}}{(R_2 \mid\mid R_3 + R_1)^{-1} + R_5^{-1}}$
$V_3 = I_3 R_1 = R_1 \times (\frac{V_1}{R_{(S1)}} \times \frac{(R_1 + R_4 \mid\mid R_5)^{-1}}{(R_1 + R_4 \mid\mid R_5)^{-1} + R_3^{-1}} + \frac{V_2}{R_{(S2)}} \times \frac{(R_2 \mid\mid R_3 + R_1)^{-1}}{(R_2 \mid\mid R_3 + R_1)^{-1} + R_5^{-1}})$
==**ANS**==: $I_3 = \frac{V_1}{R_{(S1)}} \times \frac{(R_1 + R_4 \mid\mid R_5)^{-1}}{(R_1 + R_4 \mid\mid R_5)^{-1} + R_3^{-1}} + \frac{V_2}{R_{(S2)}} \times \frac{(R_2 \mid\mid R_3 + R_1)^{-1}}{(R_2 \mid\mid R_3 + R_1)^{-1} + R_5^{-1}},\\
V_3 = R_1 \times (\frac{V_1}{R_{(S1)}} \times \frac{(R_1 + R_4 \mid\mid R_5)^{-1}}{(R_1 + R_4 \mid\mid R_5)^{-1} + R_3^{-1}} + \frac{V_2}{R_{(S2)}} \times \frac{(R_2 \mid\mid R_3 + R_1)^{-1}}{(R_2 \mid\mid R_3 + R_1)^{-1} + R_5^{-1}})$

#### (4)
$R_{TH} = R_1 + R_2 \mid\mid R_3$
$V_{TH} = V_1 \times \frac{R_3}{R_2 + R_3}$
==**ANS**==: $R_{TH} = R_1 + R_2 \mid\mid R_3,\ V_{TH} = V_1 \times \frac{R_3}{R_2 + R_3}$

### 2. (30m)
#### (1)
$V_{AB} = 10 \times (\frac{1\ (k\Omega)}{R_X + 1\ (k\Omega)} - \frac{1\ (k\Omega)}{1 + 1\ (k\Omega)})\ (V) = 10 \times (\frac{1\ (k\Omega)}{R_X + 1\ (k\Omega)} - \frac{1}{2})\ (V)$
$V_{AB} = 0\ (V) \Leftrightarrow R_X = 1\ (k\Omega)$
$V_{AB} = -1\ (V) \Leftrightarrow R_X = 1.5\ (k\Omega)$
$V_{AB} = 1\ (V) \Leftrightarrow R_X = 0.7\ (k\Omega)$
$V_{AB} = -2\ (V) \Leftrightarrow R_X = 2.3\ (k\Omega)$
$V_{AB} = 2\ (V) \Leftrightarrow R_X = 0.4\ (k\Omega)$
==**ANS**==: $V_{AB} = 0\ (V) \Leftrightarrow R_X = 1\ (k\Omega),\ V_{AB} = -1\ (V) \Leftrightarrow R_X = 1.5\ (k\Omega),\ V_{AB} = 1\ (V) \Leftrightarrow R_X = 0.7\ (k\Omega),\\
V_{AB} = -2\ (V) \Leftrightarrow R_X = 2.3\ (k\Omega),\ V_{AB} = 2\ (V) \Leftrightarrow R_X = 0.4\ (k\Omega)$

#### (2)
$R_X(t) = 1 - 0.01 \times (t - 25)\ (k\Omega)$
$V_{AB}(t) = 10 \times (\frac{1}{1 - 0.01 \times (t - 25) + 1} - \frac{1}{2})\ (V) = 10 \times (\frac{1}{-0.01 \times t + 2.25} - \frac{1}{2})\ (V)$
$t = -100 \times (\frac{10}{V_{AB} + 5} - 2.25)\ (^\circ C)$
$V_{AB} = 0\ (V) \Leftrightarrow t = 25\ (^\circ C)$
$V_{AB} = -1\ (V) \Leftrightarrow t = -25\ (^\circ C)$
$V_{AB} = 1\ (V) \Leftrightarrow t = 58\ (^\circ C)$
$V_{AB} = -2\ (V) \Leftrightarrow t = -108\ (^\circ C)$
$V_{AB} = 2\ (V) \Leftrightarrow t = 83\ (^\circ C)$
==**ANS**==: $V_{AB} = 0\ (V) \Leftrightarrow t = 25\ (^\circ C),\ V_{AB} = -1\ (V) \Leftrightarrow t = -25\ (^\circ C),\ V_{AB} = 1\ (V) \Leftrightarrow t = 58\ (^\circ C),\\
V_{AB} = -2\ (V) \Leftrightarrow t = -108\ (^\circ C),\
V_{AB} = 2\ (V) \Leftrightarrow t = 83\ (^\circ C)$

### 3. (5m/)
$C_T = (1/(10 + 20) + 1/(200 + 300) + 1/2 + 1/3)^{-1}\ (nF) \approx 1.15\ (nF)$
$X_C$ ==OUT==
$Z_C$ ==OUT==
==**ANS**==: $C_T\approx 1.15\ (nF),\ X_C: (OUT),\ Z_C: (OUT)$

### 4.
#### (1)
==OUT==

#### (2)
==OUT==

#### (3)
==OUT==

#### (4)
==OUT==