[Circuit] AC Sinusoids 穩態分析

電路知識：AC Sinusoids 穩態分析
工具：Qucs

功能：穩態分析，轉換為 phasor domain 計算 sinusoids 的電壓電流

只適用 sinusoids，必需利用 fourier series 轉換並將 DC 成份去除才適用，務必小心使用

Phasor 基本公式

$$\begin{array}{rl}z=x+jy& =r(cos\varphi +jsin\varphi )=r{e}^{j\varphi }=r\phase{\varphi }\\ z^{\ast }=x-jy& =r(cos\varphi -jsin\varphi )=r{e}^{-j\varphi }=r\phase{-\varphi }\\ |z{|}^2& =z\times z^{\ast }=r^2\\ e^{\pm j\varphi }& =cos\varphi \pm jsin\varphi \\ cos\varphi & =\mathrm{Re}(e^{j\varphi })\\ sin\varphi & =\mathrm{Im}(e^{j\varphi })\end{array}$$

Phasor-domain 推導

因推論前提皆建立在 $cos(wt+\varphi )$ 上
故訊號皆需轉為 $cos(wt+\varphi )$ 表示，再轉過去 phasor domain，可利用 Fourier series
時域的表現，就是 $r\phase{\varphi }$ 實數的部分，也就是 $cos(wt+\varphi )$ 的部分

$$\begin{gathered} v(t)=V_{m}cos(wt+\varphi ) \\ \mathbf{V}=V_m{e}^{j\varphi }=V_m\phase{\varphi } \end{gathered}$$

$$\begin{array}{rl}v(t)& =V_{m}cos(wt+\varphi )\\ & =\mathrm{Re}(V_m{e}^{j(wt+\varphi )})\\ & =\mathrm{Re}(\mathbf{V}{e}^{jwt})\\ \mathbf{V}& =V_m{e}^{j\varphi }=V_m\phase{\varphi }\end{array}$$

電壓電流轉換表

$$\begin{array}{rl}{V}_{m}cos(wt+\varphi )& \iff V_m\phase{\varphi }\\ V_{m}sin(wt+\varphi )& \iff V_m\phase{\varphi -{90}^{\circ }}\\ I_{m}cos(wt+\varphi )& \iff I_m\phase{\varphi }\\ I_{m}sin(wt+\varphi )& \iff I_m\phase{\varphi -{90}^{\circ }}\end{array}$$

微分 in Phasor domain

$$\begin{gathered} v(t)=V_{m}cos(wt+\varphi ) \\ \frac{\mathrm{d}v}{\mathrm{d}t}\iff jw\mathbf{V} \end{gathered}$$

$$\begin{array}{rl}\frac{\mathrm{d}v}{\mathrm{d}t}& =-w{V}_{m}sin(wt+\varphi )\\ & =w{V}_{m}cos(wt+\varphi +{90}^{\circ })\\ & =\mathrm{Re}(w{V}_m{e}^{jwt}{e}^{j\varphi }{e}^{j{90}^{\circ }})\\ & =\mathrm{Re}(jw\mathbf{V}{e}^{jwt})\\ \frac{\mathrm{d}v}{\mathrm{d}t}& \iff jw\mathbf{V}\end{array}$$

積分 in Phasor domain

$$\begin{gathered} v(t)=V_{m}cos(wt+\varphi ) \\ \int vdt\iff \frac{\mathbf{V}}{jw} \end{gathered}$$

$$\begin{array}{rl}\int vdt& =\frac{1}{w}{V}_{m}sin(wt+\varphi )\\ & =\frac{1}{w}{V}_{m}cos(wt+\varphi -{90}^{\circ })\\ & =\mathrm{Re}(\frac{1}{w}{V}_m{e}^{jwt}{e}^{j\varphi }{e}^{-j{90}^{\circ }})\\ & =\mathrm{Re}(\frac{-j}{w}\mathbf{V}{e}^{jwt})\\ & =\mathrm{Re}(\frac{1}{jw}\mathbf{V}{e}^{jwt})\\ \int vdt& \iff \frac{\mathbf{V}}{jw}\end{array}$$

RLC in Phasor domain

元件	Time domain	Phasor domain	Impedance
$R$	$v(t)=Ri(t)$	$\mathbf{V}=R\mathbf{I}$	$\mathbf{Z}=R$
$L$	$v(t)=L\frac{\mathrm{d}i(t)}{\mathrm{d}t}$	$\mathbf{V}=jwL\mathbf{I}$	$\mathbf{Z}=jwL$
$C$	$i(t)=C\frac{\mathrm{d}v(t)}{\mathrm{d}t}$	$\mathbf{V}=\frac{\mathbf{I}}{jwC}$	$\mathbf{Z}=\frac{\mathbf{1}}{jwC}$

基本電路定律

歐姆定律
impedance $\mathbf{Z}$ 單位仍為 $\mathrm{\Omega }$
Admittance $\mathbf{Y}$ 單位仍為 $S$

$$\begin{gathered} \mathbf{V}\mathbf{=}\mathbf{ZI}\mathbf{=}\frac{\mathbf{I}}{\mathbf{Y}} \\ \mathbf{Z}=R+jX=\left|\mathbf{Z}\right|\phase{\theta } \end{gathered}$$

Kirchhoff 電路定律

KVL
$${\mathbf{V}}_1+{\mathbf{V}}_2+\cdots +{\mathbf{V}}_n=0$$

$$\begin{array}{rl}{v}_1+v_2+\cdots +v_n& =0\\ V_{m1}cos(wt+{\theta }_1)+V_{m2}cos(wt+{\theta }_2)+\cdots +V_{mn}cos(wt+{\theta }_n)& =0\\ \mathrm{Re}(V_{m1}{e}^{j{\theta }_1}{e}^{jwt})+\mathrm{Re}(V_{m2}{e}^{j{\theta }_2}{e}^{jwt})+\cdots +\mathrm{Re}(V_{mn}{e}^{j{\theta }_n}{e}^{jwt})& =0\\ \mathrm{Re}[({\mathbf{V}}_1+{\mathbf{V}}_2+\cdots +{\mathbf{V}}_n)e^{jwt}]& =0\\ \because e^{jwt}\ne 0\therefore {\mathbf{V}}_1+{\mathbf{V}}_2+\cdots +{\mathbf{V}}_n& =0\end{array}$$

KCL
$${\mathbf{I}}_1+{\mathbf{I}}_2+\cdots +{\mathbf{I}}_n=0$$

$$\begin{array}{rl}{i}_1+i_2+\cdots +i_n& =0\\ I_{m1}cos(wt+{\theta }_1)+I_{m2}cos(wt+{\theta }_2)+\cdots +I_{mn}cos(wt+{\theta }_n)& =0\\ \mathrm{Re}(I_{m1}{e}^{j{\theta }_1}{e}^{jwt})+\mathrm{Re}(I_{m2}{e}^{j{\theta }_2}{e}^{jwt})+\cdots +\mathrm{Re}(I_{mn}{e}^{j{\theta }_n}{e}^{jwt})& =0\\ \mathrm{Re}[({\mathbf{I}}_1+{\mathbf{I}}_2+\cdots +{\mathbf{I}}_n)e^{jwt}]& =0\\ \because e^{jwt}\ne 0\therefore {\mathbf{I}}_1+{\mathbf{I}}_2+\cdots +{\mathbf{I}}_n& =0\end{array}$$

串並聯等效 Impedance
因 KVL or KCL，故當 $i$, $v$ 為 Sinusoids，其元件上的跨壓必定也為 Sinusoids，才可互相抵消為 0

串聯
$${\mathbf{Z}}_{\mathbf{eq}}\mathbf{=}{\mathbf{Z}}_{\mathbf{1}}\mathbf{+}{\mathbf{Z}}_{\mathbf{2}}\mathbf{+}\cdots \mathbf{+}{\mathbf{Z}}_{\mathbf{N}}$$

$$\begin{array}{rl}0& =-v+v_1+v_2+\cdots +v_N\\ 0& =-V_{m}cos(wt+\theta )+V_{m1}cos(wt+{\theta }_1)+V_{m2}cos(wt+{\theta }_2)+\cdots +V_{mN}cos(wt+{\theta }_N)\\ 0& =\mathrm{Re}(-V_m{e}^{j\theta }{e}^{jwt})+\mathrm{Re}(V_{m1}{e}^{j{\theta }_1}{e}^{jwt})+\mathrm{Re}(V_{m2}{e}^{j{\theta }_2}{e}^{jwt})+\cdots +\mathrm{Re}(V_{mN}{e}^{j{\theta }_N}{e}^{jwt})\\ 0& =\mathrm{Re}[(-\mathbf{V}+{\mathbf{V}}_1+{\mathbf{V}}_2+\cdots +{\mathbf{V}}_N)e^{jwt}]\\ \mathbf{V}& ={\mathbf{V}}_{\mathbf{1}}\mathbf{+}{\mathbf{V}}_{\mathbf{2}}\mathbf{+}\cdots \mathbf{+}{\mathbf{V}}_{\mathbf{N}}\\ & =\mathbf{I}\mathbf{(}{\mathbf{Z}}_{\mathbf{1}}\mathbf{+}{\mathbf{Z}}_{\mathbf{2}}\mathbf{+}\cdots \mathbf{+}{\mathbf{Z}}_{\mathbf{N}}\mathbf{)}\\ {\mathbf{Z}}_{\mathbf{eq}}& =\frac{\mathbf{V}}{\mathbf{I}}={\mathbf{Z}}_{\mathbf{1}}\mathbf{+}{\mathbf{Z}}_{\mathbf{2}}\mathbf{+}\cdots \mathbf{+}{\mathbf{Z}}_{\mathbf{N}}\end{array}$$

並聯
$$\frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{eq}}}=\mathbf{(}\frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{1}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{2}}}\mathbf{+}\cdots \mathbf{+}\frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{N}}}\mathbf{)}$$

$$\begin{array}{rl}0& =-i+i_1+i_2+\cdots +i_N\\ 0& =-I_{m}cos(wt+\theta )+I_{m1}cos(wt+{\theta }_1)+I_{m2}cos(wt+{\theta }_2)+\cdots +I_{mN}cos(wt+{\theta }_N)\\ 0& =\mathrm{Re}(-I_m{e}^{j\theta }{e}^{jwt})+\mathrm{Re}(I_{m1}{e}^{j{\theta }_1}{e}^{jwt})+\mathrm{Re}(I_{m2}{e}^{j{\theta }_2}{e}^{jwt})+\cdots +\mathrm{Re}(I_{mN}{e}^{j{\theta }_N}{e}^{jwt})\\ 0& =\mathrm{Re}[(-\mathbf{I}+{\mathbf{I}}_1+{\mathbf{I}}_2+\cdots +{\mathbf{I}}_N)e^{jwt}]\\ \mathbf{I}& ={\mathbf{I}}_{\mathbf{1}}\mathbf{+}{\mathbf{I}}_{\mathbf{2}}\mathbf{+}\cdots \mathbf{+}{\mathbf{I}}_{\mathbf{N}}\\ & =\mathbf{V}\mathbf{(}\frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{1}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{2}}}\mathbf{+}\cdots \mathbf{+}\frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{N}}}\mathbf{)}\\ \frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{eq}}}& =\frac{\mathbf{I}}{\mathbf{V}}=\mathbf{(}\frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{1}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{2}}}\mathbf{+}\cdots \mathbf{+}\frac{\mathbf{1}}{{\mathbf{Z}}_{\mathbf{N}}}\mathbf{)}\end{array}$$

平均功率
詳細推導過程(含 DC)

$$\begin{array}{rl}\mathbf{V}& =V_m\phase{{\theta }_v}\\ \mathbf{I}& =I_m\phase{{\theta }_i}\\ P& =\frac{1}{2}\mathrm{Re}[{\mathbf{VI}}^{\mathbf{\ast }}]\end{array}$$

$$\begin{array}{rl}P& =\frac{1}{T}{\int }_0^{T}p(t)dt\\ & =\frac{1}{T}{\int }_0^{T}v(t)i(t)dt\\ & =\frac{1}{T}{\int }_0^T{V}_m{I}_{m}cos(wt+{\theta }_v)cos(wt+{\theta }_i)dt\\ & =\frac{1}{T}{\int }_0^T{V}_m{I}_m\frac{1}{2}[cos({\theta }_v-{\theta }_i)+cos(2wt+{\theta }_v+{\theta }_i)]dt\\ & =\frac{1}{2}{V}_m{I}_{m}cos({\theta }_v-{\theta }_i)\frac{1}{T}{\int }_0^{T}dt+\frac{1}{2}{V}_m{I}_m\frac{1}{T}\underset{0}{\underbrace{{\int }_0^{T}cos(2wt+{\theta }_v+{\theta }_i)dt}}\\ & =\frac{1}{2}{V}_m{I}_{m}cos({\theta }_v-{\theta }_i)\\ & =\frac{1}{2}\mathrm{Re}[{\mathbf{VI}}^{\mathbf{\ast }}]\end{array}$$

當 ${\theta }_v={\theta }_i$ ，也就是 $R$，為實功來源
$$P=\frac{1}{2}{V}_m{I}_m=\frac{1}{2}{I}_m^{2}R=\frac{1}{2}|\mathbf{I}{|}^{2}R$$ 當 ${\theta }_v-{\theta }_i=\pm {90}^{\circ }$ ，也就是 $L$ 或 $C$，為虛功來源
$$P=\frac{1}{2}{V}_m{I}_{m}cos{90}^{\circ }=0$$

Complex Power

全皆是平均功率來看

複數功率 (Complex Power) $=\mathbf{S}=P+jQ=\frac{1}{2}{\mathbf{VI}}^{\mathbf{\ast }}={\mathbf{V}}_{\mathbf{rms}}{\mathbf{I}}_{\mathbf{rms}}^{\mathbf{\ast }}=I_{rm{s}^2}\mathbf{Z}=\frac{V_{rms}^2}{{\mathbf{Z}}^{\mathbf{\ast }}}$
視在功率 (Apparent Power) $=S=|\mathbf{S}|=\frac{1}{2}{V}_m{I}_m=V_{rms}{I}_{rms}=\sqrt{P^2+Q^2}$
實功 (Real Power) $=P=\mathrm{Re}(\mathbf{S})=Scos({\theta }_v-{\theta }_i)$
虛功 (Reactive Power) $=Q=\mathrm{Im}(\mathbf{S})=Ssin({\theta }_v-{\theta }_i)$
功率因數 (Power Factor) $=\frac{P}{S}=cos({\theta }_v-{\theta }_i)$

$$\begin{array}{rl}\mathbf{S}& =\frac{1}{2}{\mathbf{VI}}^{\mathbf{\ast }}\\ & =\frac{1}{2}{V}_m{I}_m\phase{{\theta }_v-{\theta }_i}\\ & =V_{rms}{I}_{rms}\phase{{\theta }_v-{\theta }_i}\\ & ={\mathbf{V}}_{\mathbf{rms}}{\mathbf{I}}_{\mathbf{rms}}^{\mathbf{\ast }}\\ & ={\mathbf{ZI}}_{\mathbf{rms}}{\mathbf{I}}_{\mathbf{rms}}^{\mathbf{\ast }}=I_{rm{s}^2}\mathbf{Z}\\ & ={\mathbf{V}}_{\mathbf{rms}}\frac{{\mathbf{V}}_{\mathbf{rms}}^{\mathbf{\ast }}}{{\mathbf{Z}}^{\mathbf{\ast }}}=\frac{V_{rms}^2}{{\mathbf{Z}}^{\mathbf{\ast }}}\end{array}$$

參考

Alexander & Sadiku, “Fundamentals of Electric Circuits, Second Edition”, McGraw-Hill, New York, NY, 2004.