Ch12 Three-Phase Circuits

Terminologies




Three-phase voltages and source configurations


Y-connected 相电压是 $a,b,c$ 相之间的电压，线电压是到中性线的电压。


$\Delta-$connected 线电压和相电压相同。




Three-phase load configurations



电压是自然属性。



$a,b,c,n$ 在 source, $A,B,C,N$ 在 node.

Balanced Wye-Wye Connection


$\bold V_{ab}=\bold V_{an}-\bold V_{bn}$.

Expressed using $\bold V_{an}$.

\bold V_{ab}=\sqrt{3}\bold V_{an}\ang 30^\circ\\ \bold V_{bc}=\sqrt{3}\bold V_{bn}\ang 30^\circ=\sqrt{3}\bold V_{an}\ang -90^\circ\\ \bold V_{ca}=\sqrt{3}\bold V_{cn}\ang 30^\circ=\sqrt{3} \bold V_{an}\ang 150^\circ



需要 $\bold V_{an}=\bold V_{bn}=\bold V_{cn}$，$\bold Z_Y$ 相等，才能移除中性线。


Write KCL Equation.

$\sum_{cyc}\frac{\bold V_{an}-\bold V_n}{\bold Z_s}=0$


单相分析法。利用了相序顺序。


\bold I_a=\frac{\bold V_{an}}{\bold Z_Y} , \bold I_b=\bold I_a \ang -120^\circ ,\bold I_c=\bold I_a\ang 120^\circ



\bold I_A=\frac{440}{\sqrt{3}\ang 30}/(40\ang 30)

\bold V_{AB}=\sqrt{3} \cdot 220 \ang 62 ^\circ \\ v_{ab}={\color{red}\sqrt{2}} \cdot \sqrt{3} \cdot 220 \cos (\omega t+62^\circ)

Balanced Wye-Delta Connection



\bold I_{a}=\sqrt{3}\bold I_{AB}\ang -30^\circ\\ \bold I_{b}=\sqrt{3}\bold I_{BC}\ang -30^\circ=\sqrt{3}\bold I_{AB}\ang -150^\circ\\ \bold I_{c}=\sqrt{3}\bold I_{CA}\ang -30^\circ=\sqrt{3} \bold I_{AB}\ang 90^\circ




Balanced Delta-Delta Connection


Line voltages and phase voltages are the same.


Balanced Delta-Wye Connection

Power in a Balanced System

Unbalanced Three-Phase Systems

Applications