Summary

• Phasor Representation:

  $$v(t)=V\;\cos(\omega t +\phi),\;\;\;\;\;\; \dot{V}=V_{rms}\angle{\phi}$$ (436)

  $$i(t)=I\;\cos(\omega t +\psi),\;\;\;\;\;\; \dot{I}=I_{rms}\angle{\psi}$$ (437)

• KVL and KCL:

  $$\sum_k \dot{V}_k=0,\;\;\;\;\;\;\;\;\sum_k \dot{I}_k=0$$ (438)

• Ohm's Law:

  $$Z=\frac{\dot{V}}{\dot{I}}=\frac{V_{rms}}{I_{rms}}\angle{(\phi-\psi)}$$ (439)

• Impedance and Admittance:

  $$Z=R+jX,\;\;\;\;\;\;\; = +j\;$$ (440)

  $$Y=G+jB,\;\;\;\;\;\;\; = +j\;$$ (441)

  $$Y=\frac{1}{Z}=\frac{1}{R+jX}=\frac{R}{R^2+X^2}+j\;\frac{-X}{R^2+X^2}=G+jB$$ (442)

• Complete Response: The three components:
  • Initial value $f(0)$ for DC input or $f_0(t)$ for AC input.
  • steady state response $f(\infty)$ for DC input or $f_\infty(t)$ for AC input
  • Time constant $\tau$
  The complete response is:

  $$f(t)=f(\infty)+[f(0)-f(\infty)]\;e^{-t/\tau},\;\;\;\; \;\;\;\;f(t)=f_\infty(t)+[f_0)-f_\infty(0)]\;e^{-t/\tau}$$ (443)

  Find $\tau$:
  • remove C or L in question from circuit
  • short voltage sources and open current sources
  • find equivalent $R$
  • find $\tau=RC$ or $\tau=L/R$
• Series Resonance:

  $$Z=R+j\omega L+\frac{1}{j\omega C}=R+j(\omega L-\frac{1}{\omega C}) =R+jX$$ (444)

  if $\omega=\omega_n=1/\sqrt{LC}$, then $X=\omega L-1/\omega C=0$, and $Z=R$ is minimized.

  Quality factor:

  $$Q=\frac{\omega_n L}{R}=\frac{1}{\omega_n RC}=\frac{1}{R}\sqrt{\frac{L}{C}}$$ (445)

  $$\dot{V}_R=\dot{V},\;\;\;\;\dot{V}_L=jQ\dot{V},\;\;\;\;\;\dot{V}_C=-jQ\dot{V}$$ (446)

• Parallel Resonance:

  $$Y=G+j\omega C+\frac{1}{j\omega L}=G+j\left(\omega C-\frac{1}{\omega L}\right) =G+jB$$ (447)

  if $\omega=\omega_n=1/\sqrt{LC}$, then $B=\omega L-1/\omega C=0$, and $Y=G$ is minimized.

  Quality factor:

  $$Q=\frac{\omega_n C}{G}=\frac{1}{\omega_n LG}=\frac{1}{G}\sqrt{\frac{C}{L}}$$ (448)

  $$\dot{I}_R=\dot{I},\;\;\;\;\dot{I}_C=jQ\dot{I},\;\;\;\;\;\dot{I}_L=-jQ\dot{I}$$ (449)

• Band-pass filtering:

  Half-power point: frequencies $\omega_1<\omega_n$ and $\omega_2>\omega_n$ at which signal power reduced to half of the peak level at $\omega=\omega_n$. Correspondingly, the voltage (or current) is reduced to $1/\sqrt{2}=0.707$ of the peak level.

  Bandwidth:

  $$\triangle \omega=\omega_2-\omega_1=\frac{\omega_n}{Q}$$ (450)

• AC Power:
  • Phase difference between $v(t)$ and $i(t)$:

    $$\dot{I}=\frac{\dot{V}}{Z}=\frac{V_{rms}\angle 0}{\vert Z\vert\angle{\phi}} =I_{rms}\angle{-\phi}$$ (451)

  • Instantaneous power:

    $$p(t)=v(t) i(t)=2V_{rms}I_{rms}\cos(\omega t)\cos(\omega t-\phi)$$ (452)

  • Average power:

    $$P_{average}=\frac{1}{T}\int_T p(t) dt=V_{rms}I_{rms} \cos \phi$$ (453)

  • Apparent power: $S=V_{rms}I_{rms}$
  • Real power: $P=S \; \cos\phi=V_{rms} I_{rms} \cos \phi$
  • Reactive power: $Q=S \; \sin\phi=V_{rms} I_{rms} \sin \phi$
  • Power factor $0 < \lambda=\cos \phi < 1$
  • Shunt capacitor to improve power factor (close to 1)

• Two-port networks:
  • Z-model:

    $$\left[ \begin{array}{l} V_1 \\ V_2\end{array} \right] ={\bf Z}\left[ \begin{array}{l} I_1 \\ I_2\end{array} \right]$$ (454)

  • Y-model:

    $$\left[ \begin{array}{l} I_1 \\ I_2\end{array} \right] ={\bf A}\left[ \begin{array}{l} V_1 \\ V_2\end{array} \right]$$ (455)

  • A-model:

    $$\left[ \begin{array}{l} V_1 \\ I_1\end{array} \right] ={\bf Z}\left[ \begin{array}{l} V_1 \\ -I_2\end{array} \right]$$ (456)

  • H-model:

    $$\left[ \begin{array}{l} V_1 \\ I_2\end{array} \right] ={\bf H}\left[ \begin{array}{l} I_1 \\ V_2\end{array} \right]$$ (457)

  How to find these parameters? Open circuit to let $I=0$, or short circuit to let $V=0$.
• Principle of reciprocity: $Z_{12}=Z_{21}$, $Y_{12}=Y_{21}$.

• Conversion between Z-model and Y-model: ${\bf Z}^{-1}={\bf Y}$.

• Conversion between Z-model and T-network:

  $$Z_{11}=Z_1+Z_3,\;\;\;\;\;Z_{22}=Z_2+Z_3,\;\;\;\;\;Z_{12}=Z_{21}=Z_3$$ (458)

  $$Z_1=Z_{11}-Z_{12},\;\;\;\;Z_2=Z_{22}-Z_{21},\;\;\;\;Z_3=Z_{12}=Z_{21}$$ (459)

• Conversion between Y-model and Pi-network:

  $$Y_{11}=Y_1+Y_3,\;\;\;\;\;Y_{22}=Y_2+Y_3,\;\;\;\;\;Y_{12}=Y_{21}=-Y_3$$ (460)

  $$Y_1=Y_{11}+Y_{12},\;\;\;\;Y_2=Y_{22}+Y_{21},\;\;\;\;Y_3=-Y_{12}=-Y_{21}$$ (461)

• Ideal Transformer:

  $$\frac{V_2}{V_1}=\frac{I_2}{I_1}=\frac{n_1}{n_2},\;\;\;\;\; \frac{Z_2}{Z_1}=\frac{n^2_2}{n^2_1}$$ (462)