Circuit Theory/3-Phase Transmission

Polyphase Systems

A polyphase system is an AC circuit system that uses multiple sinusoidal inputs, with equal frequencies but different phase values, to transmit power to loads. Polyphase systems have a number of advantages over "single phase" systems: equal amounts of power can be transmitted using waves of smaller amplitude, polyphase motors and appliances run smoother than single-phase appliances, etc. One of the most popular polyphase implementations is 3-phase systems.

3-Phase Systems

3-phase systems use 3 sinusoidal forcing functions, each with the same magnitude and frequency, but 120^o apart from each other. 3-phase systems have a curious mathematical property:

$\cos (\omega t+0^{\circ })+\cos (\omega t+120^{\circ })+\cos (\omega t+240^{\circ })=0$

This means that if the waves have the same amplitude and frequency, those waves can cancel each other out entirely. This means that if a transmission system is appropriately balanced among 3 wires, no ground wire is required. Even if the loads aren't perfectly balanced, a ground wire can be supplied, with the expectation that most of the current will be cancelled out, and the ground wire will carry very little current back to the source.

In a 3-phase system, we define the 3 main transmission wires as a, b, and c. Then the respective voltages on these are defined as:

• line a: $v_a(t)=V_m\cos (\omega t)$
• line b: $v_b(t)=V_m\cos (\omega t+120^{\circ })$
• line c: $v_c(t)=V_m\cos (\omega t+240^{\circ })$

3-Phase Phasors

In phasor representation we can say:

• ${𝕍}_a=V_m\mathrm{\angle }{0}^{\circ }$
• ${𝕍}_b=V_m\mathrm{\angle }120^{\circ }$
• ${𝕍}_c=V_m\mathrm{\angle }240^{\circ }$

This is because all these functions have the same frequency, but differing phase values.

3-Phase Loads

Each load impedance should be equal to the others, to provide for a balanced system:

${\mathbb{Z}}_a={\mathbb{Z}}_b={\mathbb{Z}}_c$

We can define the currents on each

So that the return voltage should equal zero:

${𝕍}_{\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{n}}={\mathbb{Z}}_a{𝕀}_a+{\mathbb{Z}}_b{𝕀}_b+{\mathbb{Z}}_c{𝕀}_c=0$