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title: EC Fin (6/13)

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# EC Fin (6/13)

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[TOC]

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## Important Electrical Units
| Quantity    | Q. Symbol | Unit    | U. Symbol      | Dimensions        |
| ----------- | --------- | ------- | -------------- | ----------------- |
| Current     | $I$       | Ampere  | $A$            | $C\ /\ s$         |
| Charge      | $Q$       | Coulomb | $C$            | $A \cdot s$       |
| Voltage     | $V$       | Volt    | $V$            | $J\ /\ C$         |
| Resistance  | $R$       | Ohm     | $\Omega$       | $V\ /\ A$         |
| Conductance | $G$       | 1 / Ohm | $1\ /\ \Omega$ | $1\ /\ \Omega$    |
| Energy      | $W$       | Joule   | $J$            |                   |
| Power       | $P$       | Watt    | $W$            | $J\ /\ s$         |
| Capacity    | $C$       | Farad   | $F$            | $C\ /\ V$         |
| Inductance  | $L$       | Henry   | $H$            | $V\ /\ (A\ /\ s)$ |


## Ch.9
- RC Circuits:
    - ![](https://i.imgur.com/CGu0vru.png =200x)
    - The current has the same shape as $V_R$.
    - Time Constant:
        $$\tau = RC,\ 1\ s = 1\ \Omega \times 1\ F$$
    - $$v = v_C = V_F + (V_i - V_F) e^{- \frac{t}{RC}}$$
    - Capacitive Reactance:
        $$X_C = \frac{1}{2 \pi f C},\ Z_C = \frac{1}{j \omega C} = \frac{1}{j 2 \pi f C}$$
    - Voltage and current are always $90^\circ$ out of phase
        ![](https://i.imgur.com/Fql881B.png =300x)
    - Reactive Power ($VAR$: Voltage-Ampere Reactive):
    - Power supply filtering


## Ch.10
### Series RC circuits
- Impedance:
    - ![](https://i.imgur.com/KOeCt5p.png =400x)
    - ![](https://i.imgur.com/zyOV5pc.png =150x)
    - $$Z = \sqrt{R^2 + X_C^2},\ \theta = \tan^{-1}{\left(\frac{X_C}{R}\right)}$$
- Ohm's Law:
    - $$V = IZ$$
    - $$V_Z = IZ,\ V_C = IX_C$$
    - ![](https://i.imgur.com/J02zDFb.png =150x)![](https://i.imgur.com/v8BmYgr.png =250x)
    - ![](https://i.imgur.com/ABK3Ol1.png =200x)![](https://i.imgur.com/XbHFYFq.png =400x)
- Variation of phase angle with frequency:
    - ![](https://i.imgur.com/GbgaUHa.png =200x)
- Power factor:
    - $$PF = \cos{\theta}$$
    - ![](https://i.imgur.com/5Ut2xRO.png =200x)
    - Apparent power consists of two components; a true power component, that does the work, and a reactive power component, that is simply power shuttled back and forth between source and load.
- Low-Pass Filter:
    - ![](https://i.imgur.com/FSL9BAD.png =550x)
    - $$V_{out} = \frac{Z}{R + Z} V_{in} \Rightarrow V_{out} = \frac{1/j\omega C}{R + 1/j\omega C} V_{in} \Rightarrow |V_{out}| = \frac{1}{\sqrt{(2\pi fRC)^2 + 1}} |V_{in}|$$
    - Cutoff Frequency (3dB or half-power frequency):
        - $$\frac{1}{\sqrt{(2\pi f_CRC)^2 + 1}} = \frac{1}{\sqrt{2}} \Rightarrow f_C = \frac{1}{2\pi RC}$$
- High-Pass Filter:
    - ![](https://i.imgur.com/UZjrEEE.png =550x)
    - $$V_{out} = \frac{R}{R + Z} V_{in} \Rightarrow V_{out} = \frac{R}{R + 1/j\omega C} V_{in} \Rightarrow |V_{out}| = \frac{2\pi fRC}{\sqrt{(2\pi fRC)^2 + 1}} |V_{in}|$$
    - Cutoff Frequency (3dB or half-power frequency):
        - $$\frac{2\pi f_CRC}{\sqrt{(2\pi f_CRC)^2 + 1}} = \frac{1}{\sqrt{2}} \Rightarrow f_C = \frac{1}{2\pi RC}$$

### Parallel RC circuits
- Conductance: $G$
    Capacitive Susceptance: $B_C$
    Admittance: $Y$
    $$G = \frac{1}{R},\ B_C = \frac{1}{X_C},\ Y = \frac{1}{Z}$$
- ![](https://i.imgur.com/QZSRE9Q.png =150x)
    $$Y = \sqrt{G^2 + B_C^2},\ \theta = \tan^{-1}{\left(\frac{B_C}{G}\right)}$$
- Ohm's Law:
    - $$V = IZ = \frac{I}{Y}$$
    - ![](https://i.imgur.com/NKBfU6Q.png =300x)
- Equivalent series RC circuits:
    - $$R_{(eq)} = Z\cos{\theta},\ X_{C(eq)} = Z\sin{\theta}$$


## Ch.11
- Inductors:
    - $$L = \frac{V}{I\ /\ t},\ 1\ H = \frac{1\ V}{1\ A\ /\ 1\ s}$$
    - $$L = \mu_0 \frac{N^2 A}{l}$$
- Faraday's Law:
    - The amount of voltage induced in a coil is directly proportional to the rate of change of the magnetic field with respect to the coil.
- Lenz's Law:
    - When the current through a coil changes and an induced voltage is created as a result of the changing magnetic field, the direction of the induced voltage is such that it always opposes the change in the current.
- Practical inductors: equivalent circuit:
        ![](https://i.imgur.com/O2lF9vR.png =200x)
- Series inductors:
        $$L_T = L_1 + ... + L_n$$
- Parallel inductors:
        $$L_T = \frac{1}{\frac{1}{L_1} + ... + \frac{1}{L_n}}$$
- RL Circuits:
    - ![](https://i.imgur.com/COSeXT2.png =200x)
    - The current has the same shape as $V_R$.
    - Time Constant:
        $$\tau = \frac{L}{R}$$
    - $$i = i_L = I_F + (I_i - I_F)e^{-\frac{t}{L/R}}$$
    - Inductive Reactance:
        $$X_L = 2\pi f L,\ Z_L = j \omega L = j2\pi f L$$
    - Voltage and current are always $90^\circ$ out of phase
        - ![](https://i.imgur.com/hlLlzD1.png =300x)
- Power in an inductor:
    - True Power:
        - $$P_{true} = I_{rms}^2 R_W$$
    - Reactive Power ($VAR$: Voltage-Ampere Reactive):
        - $$P_r = V_{rms} I_{rms}$$
- Quality Factor of a coil:
    - $$Q = \frac{I^2 X_L}{I^2 R_W}$$


## Ch.12
### Series RL circuits
- Impedance:
    - ![](https://i.imgur.com/XakWcpW.png =400x)
    - ![](https://i.imgur.com/m3XYy3j.png =150x)
    - $$Z = \sqrt{R^2 + X_L^2},\ \theta = \tan^{-1}{\left(\frac{X_L}{R}\right)}$$
- Ohm's Law:
    - $$V = IZ$$
    - ![](https://i.imgur.com/X1T8eCR.png =150x)![](https://i.imgur.com/VueViCo.png =200x)
    - ![](https://i.imgur.com/FacpQJJ.png =450x)
    - Variation of phase angle with frequency:
        - ![](https://i.imgur.com/na04vwx.png =200x)
- Power factor:
    - $$PF = \cos{\theta}$$
    - ![](https://i.imgur.com/MysG0lW.png =200x)
    - Apparent power consists of two components; a true power component, that does the work, and a reactive power component, that is simply power shuttled back and forth between source and load.
- High-Pass Filter:
    - ![](https://i.imgur.com/VRWH2Q5.png =550x)
    - $$V_{out} = \frac{Z}{R + Z}V_{in} \Rightarrow V_{out} = \frac{j\omega L}{R + j\omega L} V_{in} \Rightarrow |V_{out}| = \frac{2\pi fL/R}{\sqrt{(2\pi fL/R)^2 + 1}} |V_{in}|$$
    - Cutoff Frequency (3dB or half-power frequency):
        - $$\frac{2\pi fL/R}{\sqrt{(2\pi fL/R)^2 + 1}} = \frac{1}{\sqrt{2}} \Rightarrow f_C = \frac{1}{2\pi L/R}$$
- Low-Pass Filter:
    - ![](https://i.imgur.com/FbGkSGr.png =550x)
    - $$V_{out} = \frac{R}{R + Z}V_{in} \Rightarrow V_{out} = \frac{R}{R + j\omega L} V_{in} \Rightarrow |V_{out}| = \frac{1}{\sqrt{(2\pi fL/R)^2 + 1}} |V_{in}|$$
    - Cutoff Frequency (3dB or half-power frequency):
        - $$\frac{1}{\sqrt{(2\pi fL/R)^2 + 1}} = \frac{1}{\sqrt{2}} \Rightarrow f_C = \frac{1}{2\pi L/R}$$

### Parallel RL circuits
- Conductance: $G$
    Inductive Susceptance: $B_L$
    Admittance: $Y$
    $$G = \frac{1}{R},\ B_L = \frac{1}{X_L},\ Y = \frac{1}{Z}$$
- ![](https://i.imgur.com/yDJTPun.png =150x)
    $$Y = \sqrt{G^2 + B_L^2},\ \theta = \tan^{-1}{\left(\frac{B_L}{G}\right)}$$
- Ohm's Law:
    - $$V = IZ = \frac{I}{Y}$$
    - ![](https://i.imgur.com/5NQ5LbK.png =250x)


## Ch.13
- Impedance:
    - $$Z_R = \frac{v}{i} = R,\ Z_C = \frac{v}{i} = \frac{1}{j\omega C},\ Z_L = \frac{v}{i} = j\omega L,\ (j^{-1}= -j)$$

### Series RLC circuits
- Impedance:
    - ![](https://i.imgur.com/QoL9KTN.png =250x)![](https://i.imgur.com/BqYUPot.png =100x)
    - ![](https://i.imgur.com/6P1W2w4.png =300x)![](https://i.imgur.com/KqfAfnk.png =270x)
    - $$X_{tot} = |X_L - X_C|,\ Z_{tot} = \sqrt{R^2 + X_{tot}^2},\ \theta = \tan^{-1}{\left(\frac{X_{tot}}{R}\right)}$$
    - $$Z = R + j(\omega L - 1/\omega C)$$
- Voltages & Series resonance::
    - ![](https://i.imgur.com/X04k3Dr.png =300x)![](https://i.imgur.com/UiTH1xg.png =350x)
    - ![](https://i.imgur.com/8EU8MfO.png)
    - ![](https://i.imgur.com/5jS3uYs.png =x132)![](https://i.imgur.com/DqwZqAy.png =x132)
    - $$V_R = \frac{R}{R + 1/j\omega C + j\omega L}V_{in},\ V_C = \frac{1/j\omega C}{R + 1/j\omega C + j\omega L}V_{in},\ V_L = \frac{j\omega L}{R + 1/j\omega C + j\omega L}V_{in}$$
    - $$V_{CL} = V_C + V_L = \frac{1/j\omega C + j\omega L}{R + 1/j\omega C + j\omega L}V_{in} = \frac{j(\omega L - 1/\omega C)}{R + j(\omega L - 1/\omega C)}V_{in}$$
    - $$X_C = X_L \Leftrightarrow f_r = \frac{1}{2\pi\sqrt{LC}}$$
- Decibels & -3 dB frequency:
    - $$dB = 10\log{\left(\frac{P_{out}}{P_{in}}\right)},\ dB = 20\log{\left(\frac{V_{out}}{V_{in}}\right)},\ (P \propto V^2)$$
    - $$dB = 10\log{\left(\frac{1}{2}\right)} = 20\log{\left(\frac{1}{\sqrt{2}}\right)} =-3\ dB$$
- Band-pass filter:
    - ![](https://i.imgur.com/Oblieof.png =300x)![](https://i.imgur.com/sCkxE60.png =250x)
    - $$\left|\frac{V_{out}}{V_{in}}\right| = \left|\frac{R}{R + j(\omega L - 1/\omega C)}\right| = \frac{R}{\sqrt{R^2 + (\omega L - 1/\omega C)^2}} = \frac{1}{\sqrt{2}}$$
    - $$LC\omega^2 \pm RC\omega - 1 = 0,\ f = \frac{\pm RC + \sqrt{R^2C^2 + 4LC}}{4\pi LC},\ BW = \frac{R}{2\pi L}$$
    - Selectivity & Q factor:
        - $$Q = \frac{f_r}{BW} = \sqrt{\frac{L}{R^2C}},\ BW = \frac{f_r}{Q}$$
- Band-stop filter:
    - ![](https://i.imgur.com/Is6l4xG.png =300x)![](https://i.imgur.com/eJbZ98v.png =250x)
    - $$\left|\frac{V_{out}}{V_{in}}\right| = \left|\frac{j(\omega L - 1/\omega C)}{R + j(\omega L - 1/\omega C)}\right| = \frac{\omega L - 1/\omega C}{\sqrt{R^2 + (\omega L - 1/\omega C)^2}} = \frac{1}{\sqrt{2}}$$
    - $$LC\omega^2 \pm RC\omega - 1 = 0,\ f = \frac{\pm RC + \sqrt{R^2C^2 + 4LC}}{4\pi LC},\ BW = \frac{R}{2\pi L}$$

### Parallel RCL circuits
- Conductance: $G$
    Susceptance: $B$
    Admittance: $Y$
    $$G = \frac{1}{R},\ B = \frac{1}{X},\ Y = \frac{1}{Z}$$
- ==TODO==
- Band-pass filter:
    - ![](https://i.imgur.com/7L0pS7J.png =300x)![](https://i.imgur.com/Jwgolo4.png =250x)
    - $$\left|\frac{V_{out}}{V_{in}}\right| = \left|\frac{-j(1/(\omega L - 1/\omega C))}{R - j(1/(\omega L - 1/\omega C))}\right| = \frac{\omega L / (1 - \omega^2LC)}{\sqrt{R^2 + (\omega L / (1 - \omega^2LC))^2}} = \frac{1}{\sqrt{2}}$$
- Band-stop filter:
    - ![](https://i.imgur.com/5mjj3l7.png =300x)![](https://i.imgur.com/ZiOnNYe.png =250x)
    - $$\left|\frac{V_{out}}{V_{in}}\right| = \left|\frac{R}{R - j(1/(\omega L - 1/\omega C))}\right| = \frac{R}{\sqrt{R^2 + (\omega L / (1 - \omega^2LC))^2}} = \frac{1}{\sqrt{2}}$$


## Ch.14
- Mutual inductance:
    - $k$: the coefficient of coupling (dimensionless)
    - $$L_M = k\sqrt{L_1 L_2}$$
- Turns ratio:
    - ![](https://i.imgur.com/nWqi4P6.png =400x)
    - $N_{pri}$: number of primary windings
        $N_{sec}$: number of secondary windings
        $$n = \frac{N_{sec}}{N_{pri}}$$
    - Step-up transformer: $n > 1$
        Step-down transformer: $n < 1$
        Isolation transformer: $n = 1$
    - $$n = \frac{I_{pri}}{I_{sec}},\ P_{pri} = P_{sec},\ V_{pri}I_{pri} = V_{sec}I_{sec},\ \frac{V_{sec}}{V_{pri}} = \frac{I_{pri}}{I_{sec}}$$
    - Reflected resistance:
        $$R_{pri} = \frac{V_{pri}}{I_{pri}},\ R_L = \frac{V_{sec}}{I_{sec}},\ \frac{R_{pri}}{R_L} = (\frac{V_{pri}}{V_{sec}})(\frac{I_{sec}}{I_{pri}}) = (\frac{1}{n})(\frac{1}{n}) = \frac{1}{n^2} \Rightarrow R_{pri} = (\frac{1}{n})^2 R_L$$
- Transformer efficiency:
    - $$\eta = \left(\frac{P_{out}}{P_{in}}\right)100\%$$


## Ch.16
- The pn-junction diode:
    ![](https://i.imgur.com/6bpEXMc.png =350x)![](https://i.imgur.com/XYYiNae.png =300x)
- Diode characteristics:
    - Forward bias, Reverse bias, Breakdown:
        ![](https://i.imgur.com/VJOcfZf.png =300x)
    - For a silicon diode: $$V_B \approx 0.7$$
- Diode models:
    ![](https://i.imgur.com/Pl0MHGn.png =300x)
    - Ideal model (🔵): either an open or close switch
    - Practical model (🔴): + the barrier voltage in the approximation
    - Complete model (🟢): + the forward resistance of the diode
- Half-wave Rectifier:
    ![](https://i.imgur.com/oc0vit3.png =600x)
- Full-wave Rectifier:
    ![](https://i.imgur.com/isuutXK.png =550x)
    ![](https://i.imgur.com/1PBKjwx.png =550x)
- Bridge Rectifier:
    ![](https://i.imgur.com/bIdJWCn.png =550x)
- Peak inverse voltage (PIV):
    - Diodes must be able to withstand a reverse voltage when they are reverse biased. This is called the peak inverse voltage (PIV).


## Ch.17
- Bipolar junction transistors (BJTs):
    ![](https://i.imgur.com/q7MOqf4.png =330x)![](https://i.imgur.com/pegyxOt.png =330x)
- BJT bias:
    ![](https://i.imgur.com/imDN5zA.png)
- BJT currents:
    - $$I_E = I_C + I_B$$
    - $$I_C = \alpha_{DC} I_E,\ I_C = \beta_{DC} I_B,\ \alpha_{DC} \rightarrow 1$$
- Voltage-divider bias:
    - ![](https://i.imgur.com/I4Z0rQq.png =300x)![](https://i.imgur.com/Y5RS7YO.png =150x)
    - $$V_B \approx \left(\frac{R_2}{R_1 + R_2}\right) V_{CC}$$
    - $$V_E = V_B - V_{BE} \approx V_B - 0.7$$
    - $$I_E = \frac{V_E}{R_E},\ I_C = \left(\frac{\beta_{DC}}{\beta_{DC} + 1}\right) I_E$$
    - $$V_C = V_{CC} - I_C R_C$$
- Collector characteristic curves:
    ![](https://i.imgur.com/XmMAAvw.png)
    - Saturation region (left)
    - Active region (middle)
    - Breakdown region (right)
    - Cutoff region (bottom)
- Load lines:
    - ![](https://i.imgur.com/ZcM8wjc.png =250x)![](https://i.imgur.com/FvI9wHR.png =250x)
    - ![](https://i.imgur.com/0i1A7XD.png =250x)![](https://i.imgur.com/9GioLtK.png =250x)
- CE amplifier:
    - ![](https://i.imgur.com/1jRrPsz.png =450x)
    - ![](https://i.imgur.com/gdWXrn5.png =550x)
    - $$I_B << I_E,\ R_{in(tot)} = R_1 \mid\mid R_2 \mid\mid \beta_{ac} r_e$$
- CC amplifier:
    - ![](https://i.imgur.com/ree6DdM.png =350x)
    - $$R_{in(tot)} = R_1 \mid\mid R_2 \mid\mid \beta_{ac} (r_e + R_E)$$
- Class B amplifier:
    - ![](https://i.imgur.com/tzC9OmF.png =300x)
    - ![](https://i.imgur.com/9LwUjrF.png =450x)
- BJT as a switch (NOT gate):
    - ![](https://i.imgur.com/pm4yycp.png =550x)
    - $$I_B > \frac{I_{C(sat)}}{\beta_{DC}}$$


## Ch.18
- Op-amp (Operation amplifier):
    - ![](https://i.imgur.com/mmV4wt9.png =500x)
    - Ideal & practical op-amp
        - ![](https://i.imgur.com/y12Khnh.png =280x)![](https://i.imgur.com/TCcz9Pj.png =350x)
    - ![](https://i.imgur.com/WvGEucZ.png)
- Differential amplifier:
    - Single-ended mode: Only one signal applied.
    - Common mode: Two signals in phase applied.
    - Differential mode: Two signals out of phase applied.
- Common-Mode Rejection Ratio:
    $$CMRR = \frac{A_{v(d)}}{A_{cm}},\ CMRR = 20\log{\left(\frac{A_{v(d)}}{A_{cm}}\right)}$$
- Negative Feedback Network (closed-loop):
    - Non-inverting amplifier:
        - ![](https://i.imgur.com/JvzTw0t.png =300x)
        - $$V_{in} \approx V_f,\ $$
        - $$V_{out} = A_{cl(NI)} V_{in},\ A_{cl(NI)} = \frac{R_i + R_f}{R_i} = 1 + \frac{R_f}{R_i}$$
        - $$Z_{in(NI)} = (1 + A_{ol} B) Z_{in},\ Z_{out(NI)} = \frac{Z_{out}}{1 + A_{ol} B},\ B = \frac{1}{A_{cl(NI)}}$$
    - Inverting amplifier:
        - ![](https://i.imgur.com/m2xNoMU.png =300x)
        - $$V_{out} = A_{cl(I)} V_{in},\ A_{cl(I)} = -\frac{R_f}{R_i}$$
        - $$Z_{in(I)} = R_i,\ Z_{out(I)} = \frac{Z_{out}}{1 + A_{ol} B},\ B = \frac{1}{A_{cl(I)}}$$
    - Voltage-follower:
        - ![](https://i.imgur.com/U2wekad.png)
        - $$V_{out} = A_{cl} V_{in},\ A_{cl} = 1$$


## Ch.19
### Applications of amplifiers
- Comparators
    ![](https://i.imgur.com/6T1gncE.png =300x)
    $$\left\{\begin{array}{l} V_{out(max)},\ V_{+} - V_{-} > 0 \\ -V_{out(max)},\ \text{otherwise} \end{array}\right.$$
- Summing amplifier:
    ![](https://i.imgur.com/ZQk9kWx.png =300x)
    $$V_{out} = -R_f\sum^{n}_{i=1}{\left(\frac{V_{in(i)}}{R_i}\right)}$$
- Averaging amplifier:
    ![](https://i.imgur.com/pDOYYzB.png =300x)
    $$nR_f = R_{in},\ V_{out} = -\frac{R_f}{R_{in}}\sum^{n}_{i=1}{V_{in(i)}} = -\frac{1}{n}\sum^{n}_{i=1}{V_{in(i)}}$$
- Scaling adder:
    ![](https://i.imgur.com/BZC9uSn.png =300x)
    $$V_{out} = -\sum^{n}_{i=1}{\left(\frac{R_f}{R_i}\right)V_{in(i)}}$$
- Integrators:
    ![](https://i.imgur.com/nb3X5EO.png =200x)
    $$R_f >> R,\ \frac{\Delta V_{out}}{\Delta t} = -\frac{V_{in}}{R_i C}$$
- Differentiators:
    ![](https://i.imgur.com/bnPdhNG.png =200x)
    $$R_f >> R,\ V_{out} = -\left(\frac{\Delta V_C}{\Delta t}\right)R_f C$$



