# Electronic Circuits Homework 3
> 劉子雍．108502523
> 資訊工程學系三年級 A 班

## 1.
$T = \frac{10\ (\mu s)}{5} = 2\ (\mu s)$
==**ANS**==: $2\ (\mu s)$

## 2.
### (a)
$V_P = 12\ (V)$
$V_{rms} = \frac{1}{\sqrt{2}} \times V_P \approx 0.707 \times 12\ (V) = 8.48\ (V)$
==**ANS**==: $V_{rms} \approx 8.48\ (V)$

### (b)
$V_{PP} = 2 \times V_P = 2 \times 12\ (V) = 24\ (V)$
==**ANS**==: $V_{PP} = 24\ (V)$

### \(c\)
$V_{h.c.avg} = \frac{2}{\pi} \times V_P \approx 0.637 \times 12\ (V) = 7.64\ (V)$
==**ANS**==: $V_{h.c.avg} \approx 7.64\ (V)$

## 3.
### (a)
$V_P = 10\ (V)$
$I_P = \frac{10\ (V)}{1.0\ (k\Omega)} = 10\ (mA)$
$I_{rms} = \frac{1}{\sqrt{2}} \times I_P \approx 0.707 \times 10\ (mA) = 7.07\ (mA)$
==**ANS**==: $I_{rms} \approx 7.07\ (mA)$

### (b)
<!-- TODO -->
<!-- $I_{avg} = \frac{2}{\pi} \times I_P \approx 0.637 \times 10\ (mA) = 6.37\ (mA)$ -->
$I_{avg} = 0\ (mA)$
==**ANS**==: $I_{avg} = 0\ (mA)$

### \(c\)
==**ANS**==: $I_P = 10\ (mA)$

### (d)
$I_{PP} = 2 \times I_P = 2 \times 10\ (mA) = 20\ (mA)$
==**ANS**==: $I_{PP} = 20\ (mA)$

### (e)
$i = I_P = 10\ (mA)$
==**ANS**==: $i = 10\ (mA)$

## 4.
$V_{rms} = 120\ (V)$
$V_{1(rms)} = V_{rms} - V_{4(rms)} = 120 - 65\ (V) = 55\ (V)$
$V_{2(rms)} = V_{4(rms)} - V_{3(rms)} = 65 - 30\ (V) = 35\ (V)$
$V_{1(h.c.avg)} = V_{1(rms)} \times \sqrt{2} \times \frac{2}{\pi} = 55 \times \sqrt{2} \times \frac{2}{\pi}\ (V) \approx 49.52\ (V)$
$V_{2(h.c.avg)} = V_{2(rms)} \times \sqrt{2} \times \frac{2}{\pi} = 35 \times \sqrt{2} \times \frac{2}{\pi}\ (V) \approx 31.51\ (V)$
==**ANS**==: $V_{1(h.c.avg)} \approx 49.52\ (V),\ V_{2(h.c.avg)} \approx 31.51\ (V)$

## 5.
### (a)
$C = \frac{Q}{V} = \frac{50\ (\mu C)}{10\ (V)} = 5\ (\mu F)$
==**ANS**==: $5\ (\mu F)$

### (b)
$Q = CV = 0.001\ (\mu F) \times 1\ (kV) = 1\ (\mu C)$
==**ANS**==: $1\ (\mu C)$

### \(c\)
$V = \frac{Q}{C} = \frac{2\ (mC)}{200\ (\mu F)} = 10\ (V)$
==**ANS**==: $10\ (V)$

## 6.
$W = \frac{1}{2} C V^2$
$C = 2 \times \frac{W}{V^2} =2 \times \frac{10\ (mJ)}{100 \times 100\ (V^2)} = 0.002\ (mF) = 2\ (\mu F)$
==**ANS**==: $2\ (\mu F)$

## 7.
### (a)
$C_T = (1/1 + 1/2.2)^{-1}\ (\mu F) \approx 0.69\ (\mu F)$
==**ANS**==: $0.69\ (\mu F)$

### (b)
$C_T = (1/100 + 1/560 + 1/390)^{-1}\ (pF) \approx 69.69\ (pF)$
==**ANS**==: $69.69\ (pF)$

### \(c\)
$C_T = (1/10 + 1/4.7 + 1/47 + 1/22)^{-1}\ (\mu F) \approx 2.64\ (\mu F)$
==**ANS**==: $2.64\ (\mu F)$

## 8.
<!-- TODO -->
$V_1 = \frac{Q}{C_1} = \frac{10\ (\mu C)}{4.7\ (\mu F)} \approx 2.13\ (V)$
$V_2 = \frac{Q}{C_2} = \frac{10\ (\mu C)}{1\ (\mu F)} = 10\ (V)$
$V_3 = \frac{Q}{C_3} = \frac{10\ (\mu C)}{2.2\ (\mu F)} \approx 4.54\ (V)$
$V_4 = \frac{Q}{C_4} = \frac{10\ (\mu C)}{10\ (\mu F)} = 1\ (V)$
==**ANS**==: $V_1 \approx 2.13\ (V),\ V_2 = 10\ (V),\ V_3 \approx 4.54\ (V),\ V_4 = 1\ (V)$

## 9.
### (a)
$C_T = 47 + 10 + 1000\ (pF) = 1057\ (pF)$
==**ANS**==: $1057\ (pF)$

### (b)
$C_T = 0.1 + 0.01 + 0.001 + 0.01\ (\mu F) = 0.121\ (\mu F)$
==**ANS**==: $0.121\ (\mu F)$

## 10.
<!-- TODO -->
### (a)
$\tau = RC = 100\ (\Omega) \times 1\ (\mu F) = 100\ (\mu s)$
==**ANS**==: $100\ (\mu s)$

### (b)
$\tau = RC = 10\ (M\Omega) \times 56\ (pF) = 560\ (\mu s)$
==**ANS**==: $560\ (\mu s)$

### \(c\)
$\tau = RC = 4.7\ (k\Omega) \times 0.0047\ (\mu F) = 4.7\ (k\Omega) \times 4.7\ (nF) = 22.09\ (\mu s)$
==**ANS**==: $22.08\ (\mu s)$

### (d)
$\tau = RC = 1.5\ (M\Omega) \times 0.01\ (\mu F) = 0.015\ (s) = 15\ (ms)$
==**ANS**==: $15\ (ms)$

## 11.
### (a)
$\tau = RC = 10\ (k\Omega) \times 0.001\ (\mu F) = 10\ (k\Omega) \times 1\ (n F) = 10\ (\mu s)$
$v_C(t) = V_F + (V_i - V_F) e^{- \frac{t}{RC}} = 15 + (0 - 15) e^{- \frac{t}{10\ (\mu s)}}\ (V) = 15 \times (1 - e^{- \frac{t}{10\ (\mu s)}})\ (V)$
$v_C(10\ (\mu s)) = 15 \times (1 - e^{-1})\ (V) \approx 9.48\ (V)$
==**ANS**==: $9.48\ (V)$

### (b)
$v_C(20\ (\mu s)) = 15 \times (1 - e^{-2})\ (V) \approx 12.97\ (V)$
==**ANS**==: $12.97\ (V)$

### \(c\)
$v_C(30\ (\mu s)) = 15 \times (1 - e^{-3})\ (V) \approx 14.25\ (V)$
==**ANS**==: $14.25\ (V)$

### (d)
$v_C(40\ (\mu s)) = 15 \times (1 - e^{-4})\ (V) \approx 14.72\ (V)$
==**ANS**==: $14.72\ (V)$

### (e)
$v_C(50\ (\mu s)) = 15 \times (1 - e^{-5})\ (V) \approx 14.90\ (V)$
==**ANS**==: $14.90\ (V)$

## 12.
<!-- TODO -->
### (a)
$X_C(f) = \frac{1}{2 \pi f C} = \frac{1}{2 \pi f \times 0.047\ (\mu F)} = \frac{1}{94 \pi f}\ (G\Omega) \approx 3.389 \times f\ (M\Omega) = 3389 \times f\ (k\Omega)$
$X_C(10\ (Hz)) \approx \frac{3389}{10}\ (k\Omega) = 338.9\ (k\Omega)$
==**ANS**==: $X_C \approx 338.9\ (k\Omega)$

### (b)
$X_C(250\ (Hz)) \approx \frac{3390}{250}\ (k\Omega) \approx 13.56\ (k\Omega)$
==**ANS**==: $X_C \approx 13.56\ (k\Omega)$

### \(c\)
$X_C(5\ (kHz)) \approx \frac{3389}{5}\ (\Omega) = 677.8\ (k\Omega)$
==**ANS**==: $X_C \approx 677.8\ (k\Omega)$

#### (d)
$X_C(100\ (kHz)) \approx \frac{3389}{100}\ (\Omega) = 33.89\ (k\Omega)$
==**ANS**==: $X_C \approx 33.89\ (k\Omega)$

## 13.
<!-- TODO -->
### (a)
$X_C = \frac{1}{2 \pi f C} = \frac{1}{2 \pi \times 1 \times 0.047}\ (k\Omega) \approx 3.39\ (k\Omega)$
==**ANS**==: $X_C \approx 3.39\ (k\Omega)$

### (b)
$C_T = 10 + 15\ (\mu F) = 25\ (\mu F)$
$X_C = \frac{1}{2 \pi f C_T} = \frac{1}{2 \pi \times 1 \times 25}\ (M\Omega) \approx 6.37\ (k\Omega)$
==**ANS**==: $X_C \approx 6.37\ (k\Omega)$

### \(c\)
$C_T = (1/1 + 1/1)^{-1}\ (\mu F) = 0.5\ (\mu F)$
$X_C = \frac{1}{2 \pi f C_T} = \frac{1}{2 \pi \times 60 \times 0.5}\ (M\Omega) \approx 5.30\ (k\Omega)$
==**ANS**==: $X_C \approx 5.30\ (k\Omega)$

## 14.
### (a)
$X_C = \frac{1}{2 \pi f C} \Leftrightarrow f = \frac{1}{2 \pi C X_C}$
$f = \frac{1}{2 \pi C X_C} = \frac{1}{2 \pi \times 0.047 \times 100}\ (MHz) \approx 33.86\ (kHz)$
$f = \frac{1}{2 \pi C X_C} = \frac{1}{2 \pi \times 0.047 \times 1}\ (kHz) \approx 3.39\ (kHz)$
==**ANS**==: $33.86\ (kHz),\ 3.39\ (kHz)$

### (b)
$f = \frac{1}{2 \pi C_T X_C} = \frac{1}{2 \pi \times 25 \times 100}\ (MHz) \approx 63.7\ (Hz)$
$f = \frac{1}{2 \pi C_T X_C} = \frac{1}{2 \pi \times 25 \times 1}\ (kHz) \approx 6.37\ (Hz)$
==**ANS**==: $63.7\ (Hz),\ 6.37\ (Hz)$

### \(c\)
$f = \frac{1}{2 \pi C_T X_C} = \frac{1}{2 \pi \times 0.5 \times 100}\ (MHz) \approx 3.18\ (kHz)$
$f = \frac{1}{2 \pi C_T X_C} = \frac{1}{2 \pi \times 0.5 \times 1}\ (kHz) \approx 318\ (Hz)$
==**ANS**==: $3.18\ (kHz),\ 318\ (Hz)$

## 15.
<!-- TODO -->
### (a)
$R_T = 100 + 47\ (k\Omega) = 147\ (k\Omega)$
$C_T = (1/10 + 1/22)^{-1}\ (nF) = 6.875\ (nF)$
$X_C = \frac{1}{2\pi fC_T} = \frac{1}{2\pi \times 100 \times 6.875}\ (G\Omega) \approx 231.50\ (k\Omega)$
$Z = \sqrt{R_T^2 + X_C^2} \approx \sqrt{147^2 + 231.50^2}\ (k\Omega) \approx 274.23\ (k\Omega)$
$\theta = \tan^{-1}{\left(\frac{X_C}{R_T}\right)} \approx \tan^{-1}{\left(\frac{231.50}{147}\right)} \approx 57.58^\circ$
==**ANS**==: $Z \approx 274.23\ (k\Omega),\ \theta \approx 57.58^\circ$

### (b)
$C_T = 560 + 560\ (pF) = 1.12\ (nF)$
$X_C = \frac{1}{2\pi fC_T} = \frac{1}{2\pi \times 20 \times 1.12}\ (M\Omega) \approx 7.10\ (k\Omega)$
$Z = \sqrt{R^2 + X_C^2} \approx \sqrt{10^2 + 7.10^2}\ (k\Omega) \approx 12.26\ (k\Omega)$
$\theta = \tan^{-1}{\left(\frac{X_C}{R}\right)} \approx \tan^{-1}{\left(\frac{7.10}{10}\right)} \approx 35.37^\circ$
==**ANS**==: $Z \approx 12.26\ (k\Omega),\ \theta \approx 35.37^\circ$

## 16.
### (a)
$X_C(f) = \frac{1}{2\pi fC} = \frac{1}{2\pi f \times 2.2}\ (G\Omega) \approx 72343.15 \times \frac{1}{f}\ (k\Omega)$
$Z(f) = \sqrt{R^2 + X_C^2(t)} = \sqrt{56^2 + (\frac{72343.15}{f})^2}\ (k\Omega)$
$Z(100\ (Hz)) = \sqrt{56^2 + (\frac{72343.15}{100})^2}\ (k\Omega) \approx 725.60\ (k\Omega)$
==**ANS**==: $725.60\ (k\Omega)$

### (b)
$Z(500\ (Hz)) = \sqrt{56^2 + (\frac{72343.15}{500})^2}\ (k\Omega) \approx 155.14\ (k\Omega)$
==**ANS**==: $155.14\ (k\Omega)$

### \(c\)
$Z(1.0\ (kHz)) = \sqrt{56^2 + (\frac{72343.15}{1000})^2}\ (k\Omega) \approx 91.48\ (k\Omega)$
==**ANS**==: $91.48\ (k\Omega)$

### (d)
$Z(2.5\ (kHz)) = \sqrt{56^2 + (\frac{72343.15}{2500})^2}\ (k\Omega) \approx 63.03\ (k\Omega)$
==**ANS**==: $63.03\ (k\Omega)$

## 17.
<!-- TODO -->
$X_C = \frac{1}{2\pi fC} = \frac{1}{2\pi \times 2 \times 0.22}\ (k\Omega) \approx 361.72\ (\Omega)$
$Y = \sqrt{G^2 + B_C^2} \Leftrightarrow Z = \left(\sqrt{R^{-2} + X_C^{-2}}\right)^{-1} \approx \left(\sqrt{750^{-2} + 361.72^{-2}}\right)^{-1}\ (\Omega) \approx 325.81\ (\Omega)$
$\theta = \tan^{-1}{\left(\frac{B_C}{G}\right)} = \tan^{-1}{\left(\frac{R}{X_C}\right)} \approx \tan^{-1}{\left(\frac{750}{361.72}\right)} \approx 64.25^\circ$
==**ANS**==: $Z \approx 325.81\ (\Omega),\ \theta \approx 64.25^\circ$

## 18.
<!-- TODO -->
$X_{C_1} = \frac{1}{2\pi fC_1} = \frac{1}{2\pi \times 50 \times 0.047}\ (k\Omega) \approx 67.72\ (\Omega)$
$I_{C_1} = \frac{V_S}{X_{C_1}} \approx \frac{8}{67.72}\ (A) \approx 118.13\ (mA)$
$X_{C_2} = \frac{1}{2\pi fC_2} = \frac{1}{2\pi \times 50 \times 0.022}\ (k\Omega) \approx 144.69\ (\Omega)$
$I_{C_2} = \frac{V_S}{X_{C_2}} \approx \frac{8}{144.69}\ (A) \approx 55.29\ (mA)$
$I_{R_1} = \frac{V_S}{R_1} = \frac{8}{220}\ (A) \approx 36.36\ (mA)$
$I_{R_2} = \frac{V_S}{R_2} = \frac{8}{180}\ (A) \approx 44.44\ (mA)$
$I_{tot} = \sqrt{(I_{R_1} + I_{R_2})^2 + (I_{C_1} + I_{C_2})^2} \\
\quad \approx \sqrt{(36.36 + 44.44)^2 + (118.13 + 55.29)^2}\ (mA) \approx 191.32\ (mA)$
$C_T = C_1 + C_2 = 47 + 22\ (nF) = 69\ (nF)$
$R_T = (1/R_1 + 1/R_2)^{-1} = (1/220 + 1/180)^{-1}\ (\Omega) = 99\ (\Omega)$
$\theta = \tan^{-1}{\left(\frac{B_G}{G}\right)} = \tan^{-1}{\left(\frac{R_T}{X_C}\right)} = \tan^{-1}{(2\pi fR_TC_T)} = \tan^{-1}{(2\pi \times 50 \times 99 \times 69 \times 10^{-6})} \approx 65.02^\circ$
==**ANS**==: $I_{C_1} \approx 118.13\ (mA),\ I_{C_2} \approx 55.29\ (mA),\ I_{R_1} \approx 36.36\ (mA),\ I_{R_2} \approx 44.44\ (mA),\ I_{tot} \approx 191.32\ (mA),\\
\theta \approx 65.02^\circ$

## 19.
<!-- TODO -->
$X_{C_1} = \frac{1}{2\pi f C_1} = \frac{1}{2\pi \times 15 \times 0.1}\ (k\Omega) \approx 106.10\ (\Omega)$
$Z_{C_1} = X_{C_1} \approx 106.10\ (\Omega)$
$X_{C_2,C_3} = \frac{1}{2\pi f(C_2 + C_3)} = \frac{1}{2\pi \times 15 \times(0.047 + 0.22)}\ (k\Omega) \approx 39.73\ (\Omega)$
$Z_{R_1,R_2,C_2,C_3} = \left(\sqrt{(R_1 + R_2)^{-2} + X_{C_2,C_3}^{-2}}\right)^{-1} \approx \left(\sqrt{(330 + 180)^{-2} + 39.73^{-2}}\right)^{-1}\ (\Omega) \approx 39.61\ (\Omega)$
$V_{C_1} = V_S \times \frac{Z_{C_1}}{Z_{C_1} + Z_{R_1,R_2,C_2,C_3}} \approx 12 \times \frac{106.10}{106.10 + 39.61}\ (V) \approx 8.74\ (V)$
$V_{C_2} = V_{C_3} = V_S - V_{C_1} \approx 12 - 8.74\ (V) = 3.26\ (V)$
$V_{R_1} = V_{C_2} \times \frac{R_1}{R_1 + R_2} \approx 3.26 \times \frac{330}{330 + 180}\ (V) \approx 2.11\ (V)$
$V_{R_2} = V_{C_2} - V_{R_1} \approx 3.26 - 2.11 = 1.15\ (V)$
$\theta_{R_1,R_2,C_2,C_3} = \tan^{-1}{\left(\frac{B_{C_2,C_3}}{G_{R_1,R_2}}\right)} = \tan^{-1}{\left(\frac{R_1 + R_2}{X_{C_2,C_3}}\right)} \approx \tan^{-1}{\left(\frac{510}{39.73}\right)} \approx 85.55^\circ$
$R_{eq} = Z_{R_1,R_2,C_2,C_3}\cos{(\theta_{R_1,R_2,C_2,C_3})} \approx 39.61 \times \cos{(85.55^\circ)}\ (\Omega) \approx 3.07\ (\Omega)$
$X_{C(eq)} = Z_{R_1,R_2,C_2,C_3}\sin{(\theta_{R_1,R_2,C_2,C_3})} \approx 39.61 \times \sin{(85.55^\circ)}\ (\Omega) \approx 39.49\ (\Omega)$
$\theta = \tan^{-1}{\left(\frac{X_C}{R}\right)} = \tan^{-1}{\left(\frac{X_{C_1} + X_{C(eq)}}{R_{(eq)}}\right)} \approx \tan^{-1}{\left(\frac{106.10 + 39.49}{3.07}\right)} \approx 88.79^\circ$
==**ANS**==: $V_{C_1} \approx 8.74\ (V),\ V_{C_2} = V_{C_3} \approx 3.26\ (V),\ V_{R_1} \approx 2.11\ (V),\ V_{R_2} \approx 1.15\ (V),\ \theta \approx 88.79^\circ$

## 20.
<!-- TODO -->
$V_{out} = \frac{R}{R + Z_C} V_{in} = \frac{R}{R + 1/j\omega C} V_{in} \Rightarrow \| V_{out} \| = \frac{R}{\sqrt{R^2 + (1/2\pi fC)^2}} \| V_{in} \| = \frac{2\pi fRC}{\sqrt{(2\pi fRC)^2 + 1}} \| V_{in} \|$
$\left\| \frac{V_{out}}{V_{in}} \right\|(f) = \frac{2\pi fRC}{\sqrt{(2\pi fRC)^2 + 1}} = \frac{2\pi f \times 1 \times 0.1}{\sqrt{(2\pi f \times 1 \times 0.1)^2 + (10^3)^2}} = \frac{0.628 \times f}{\sqrt{(0.628 \times f)^2 + 10^6}}$
==**ANS**==: $y(x) = \frac{0.628 x}{\sqrt{(0.628 x)^2 + 10^6}}$, $y(x)$: The frequency response curve, $x$: Frequency (Hz)
| $x$ | $0$ | $1000$ | $2000$ | $3000$ | $4000$ | $5000$ | $6000$ | $7000$ | $8000$ | $9000$ | $10000$ |
|:---:|:---:|:-------:|:-------:|:-------:|:-------:|:-------:|:-------:|:-------:|:-------:|:-------:|:-------:|
| $y$ | $0$ | $0.532$ | $0.783$ | $0.883$ | $0.929$ | $0.953$ | $0.967$ | $0.975$ | $0.981$ | $0.985$ | $0.988$ |


<style>
.markdown-body table th, .markdown-body table td
{
padding: 6px 11px; !important
}
</style>