# If a 1-Phase Supply is 230V, Why is 3-Phase 400V & Not 690V?

## When Single Phase Supply Voltage is 220V, Why is Three Phase 440V and Not 660V?

If there are three winding coils on the rotor of an alternator winded 120° degree apart from each other, there would be a 120° phase difference in the produced EMF in between three coils. This polyphase system is known as a three phase supply system. While there is only one produced voltage (between Phase & Neutral) in the single phase system.

There are 360° in one cycle. The three phase supply divided into three parts i.e. 360° per cycle divided by 3 is 120°. This way, the phase difference between any two phases or lines is 120°. On the other hand, single phase supply has only one AC sinusoidal wave i.e. full 360° in one complete cycle.

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Now, we know that the

**Voltage in Single Phase Supply System:**

Single Phase Voltage = **230V** (Phase to Neutral)

Phase Voltage (V_{PH}) = Line Voltage ÷ √3

= 400V ÷ √3 ≈ **230V**

Note: The Voltage between Phase and Neutral is known as **Phase Voltage** (V_{PH}).

**Voltage in a Three Phase Supply System:**

Three Phase Voltage = **400V** (Line to Line or Phase to Phase)

Line Voltage (V_{L}) = √3 x Phase Voltage

= √3 x 230V = 398 ≈ **400V**

Note: The Voltage between two lines (or phases) is known as **Line Voltage** (V_{L}).

Now come to the real question. The above calculations for both single phase and three phase supply system is based on the UK, EU and most other counties following the same supply voltage systems e.g. 230V in Single Phase, 50Hz of frequency (120, 208V (High Leg Delta), 240V, 277 & 480V etc. in US, Canada, 60Hz frequency) and 400V in Three Phase, 50Hz (208V, 240V & 480V etc in US & Canada, 60Hz). Some countries use the EU & UK version with little bit variations. For Example, In some of Asian countries e.g. India, Pakistan etc, the single phase voltage is 220V while the three phase voltage is 415-440V.

Based on the information, we got the question in email and page inbox multiple times from these regions as follow:

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### If the Single Phase Supply Voltage is 220V, Why is Three Phase 440V and Not 660V?

Well, the above question is a little bit tricky (& technically wrong) that’s why I have changed the article title to the correct one (e.g. 230V Single Phase & 400V Three Phase) . Beside this let’s simplify the question in technical as well as basic approach for laymen like me.

First of all, When they think about 230V single phase voltage, they simply multiply by 3 for three phase like addition of 230V by three times as follows:

= 230V + 230V + 230V = 690V

But this is not suitable in case of a three phase supply system. Because the produced EMF in three phase systems are not scalar quantities, but vector (more likely phasor which can be simplified by phasor diagrams) quantities having magnitude with direction. For example, the vector difference for two 230V lines is 400V.

In this case, these phasor or vector quantities having the phase difference can’t be added like in KCL and KVL which is for instantaneous value, not RMS or average values.

The vector addition based on the phasor diagram for 3-ϕ supply system, where each line has 230V, the vector difference is almost 400V for each phase (either Phase 1 & Phase 2, Phase 2 and Phase 3 or Phase 3 and Phase 1). This is due to the phase angle between two phases which is 120 degree and all the three phases change direction with respect time (in case of 230V, the sine wave changes it direction 50 times in a second due to 50Hz of frequency).

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The **Rule of Cosine** (basic** Law of Cosine in Trigonometry**) shows that in a three phase system, the value of voltage between any two phases is 400V instead of 660 or 690V. This is only applicable if there are three phases (e.g. three separate conductors apart 120 degrees from each other).

c^{2} = a^{2} + b^{2} – 2 ab cos N

c = √ (a^{2} + b^{2} – 2 ab cos N)

Putting the values

= √[230V^{2} + 230V^{2} – 2 x 230V x 230V x cos (120°)] = 398.37V ≈ 400V

From a different point of view, If we draw the sinusoidal waves of three phase power lines which are 120 degrees from the other phases, the graph clearly shows that we have only two Positive (+Ve) values at the same time and the third one is Negative (-Ve). We only calculate two out of three phases as all of them change the direction with respect to time. In Short, either Phase 1 and 2, Phase 2 and 3 or Phase 3 and 1 are positive while at the same time, either Phase 3, Phase 2 or Phase 1 are negative respectively. That’s why two phases in a three phase system have 400V instead of 600, 660 or 690V.

For example, phase difference is 400 if the phase angle is 120° between V_{R} and V_{Y} where the value of V_{R} and V_{Y} = 230V

2 x 230V Sin 120° ≈ 400V

In short, the single phase voltage can be calculated from three phase voltage and vice versa using the following basic formula.

Single Phase to Three Phase

- V
_{PH}x √3 = V_{L} - 230V x 1.732 ≈ 400V … (3ϕ)

Three Phase to Single Phase

- V
_{L}÷ √3 = V_{PH} - 400V ÷ 1.732 = 230V … (1ϕ)

Where; √3 = 1.732

**Concluded Points:**

- In the Single Phase system, the voltage level between Phase and Neutral is 230V
- In Three Phase System, the voltage level between any two phases out of three phases is 400V, Not 415, 440, 660 or 690V. This is only possible in case of ±% variation e.g. 400V±10% = 400V + 40V = 440V.

That is the case for Voltage only. Now what about Power. Well, it is a different scenario as follows.

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**If a 1-Phase Power is 230W, Will a 3-Phase Power be 400W or 690V?**

Power is additive (not like the voltage as mentioned above) and can be summed up like scalar addition e.g. Power in line 1 + line 2 + line 3.

**Power in Single Phase Supply= Voltage x Current**

- P = V x I

**Power in Three Phase Supply = 3 x Single Phase Power**

- P = √3 x V
_{L}x I_{L}x CosФ … Or - P = 3 x V
_{PH}x I_{PH}x CosФ

Where:

- P = Power in Watts
- V
_{L}= Line Voltage - I
_{L}= Line Current - V
_{PH}= Phase Voltage - I
_{PH}= Phase Current - CosФ = Power factor

In short, power in a 3-Phase circuit is P_{R} + P_{Y} + P_{B}. For example, if the power in Single Phase is 230W, the total power in Three Phase will be 3 x 230W = 690W.

As we cleared the basic concept about the confusing and most asked question, now you really know why the voltage level in three lines of power supply is 400V instead of 440, 460 or 490V.

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