Three-Phase Wye Transformer connection

I have a question related to a three-phase wye-connected secondary
> transformer. I know that if the phase to phase voltage is 208 on the
> secondary, the voltage from one phase to the neutral will be 120 volts. I
> also know that if you multiply 120 volts by 1.73 you get the phase to
phase
> voltage of 208. I also know that the 1.73 is the square root of 3.
>
> My question is this: Where does this 1.73 or square root of three come
> from? I know it has something to do with it being three-phase voltage.
Can
> you help? Thanks.

 

[cuky2000]

A balance three phase sinusoidal voltage to neutral wave can be represented on the frequency domain as three vectors(phasors)with center on the neutral point "n" with phase shifted 120 degree appart. The phase to phase voltage will be a vector conecting the end of any two phases vector (ei. AB, AC & CB).

Consider the vector Vab former as the sum of Van and Vbn. The rest is simple geometric problem with a triangle 120 degrees in n and 30 degrees between Van and Vab related as follow:

Cos(30o)=(Vab/2)/Van=SQRT(3)/2.

Thus Vab = 1.73Van.

In any circuit or basic power system book you could find plenty information in this subget.

 

[busbar]

To add to what cuky2000 posted, some understanding might be gained by drawing a “delta” equilateral triangle with the corners representing the phases of a balanced 3-phase system. The lengths of the sides correspond to relative magnitudes of three phase-to-phase voltages. Then draw three more lines inside; one from each corner to the geometric center of the triangle—shaped like the letter “Y”.

For balanced systems, that makes three identical (isosceles; not equilateral) triangles nested within the larger equilateral triangle.

For those inner triangles, the two sides opposite each 120° angle (also the large outer triangle sides) represent phase-to-phase voltage, and the three adjacent lines represent the phase-to-neutral voltages. The ratio of ø-ø to ø-n voltages corresponds to the tangent of 120°=1.732, (square root of 3.) This is literally the ratio of the opposite side length (ø-ø) to the adjacent side length (ø-n). Besides voltage, it shows up in current ratios and in various power/energy/impedance calculations for balanced 3ø circuits.

In the most basic sense, “delta” and “wye” represent the physical layout of transformer-secondary winding interconnections, as well as the nominal voltage of the windings; e.g., 4-wire 480Y/277V or 3-wire 480V∆.

It is a worthwhile, not-soon-forgotten exercise to wire up and power small transformers in various delta, wye and zigzag configurations and measure/record/plot voltage relationships. Transformers and polyphase circuits brought power delivery out of the electrical stone age.

Consult textbooks for gaining a more thorough understanding of transformer circuits and root-3 relationships. For general but limited information, see:

http://www.elec-toolbox.com/usefulinfo/xfmr-3ph.htm

and

http://www.cooperpower.com/Library/pdf/R201902.pdf

In the US, IEEE standards are extensively used. The granddaddy couple-of-thousand-page book on transformers is: Distribution, Power, and Regulating Transformers Standards Collection 1998 Edition; indexed at:

http://standards.ieee.org/catalog/distribution.html

Specifically, C57.105-1978 (R1999) IEEE Guide for Application of Transformer Connections in Three-Phase Distribution Systems; outlined at:

http://standards.ieee.org/reading/ieee/std_public/description/dtransformers/C57.105-1978_desc.html

And, C57.12.70-2001 IEEE Standard Terminal Markings and Connections for Distribution and Power Transformers; outlined at:

http://standards.ieee.org/reading/ieee/std_public/description/dtransformers/C57.12.70-1978_desc.html

Related general references are described at:

http://standards.ieee.org/colorbooks/index.html

 

[230842]

Three phase to neutral voltages are sinusoidal shaped waves in the instant form:

Phase A: ua(t) = 120.sqrt2.sin(wt) (Sqrt2 = Rate Amplitude/r.m.s. values)
Phase B: ub(t) = 120.sqrt2.sin(wt-2.pi/3) (2.pi/3 = 120º phase shift in radians)
Phase B: ub(t) = 120.sqrt2.sin(wt-4.pi/3)

Phase to phase voltage is the difference between two phase to neutral voltages. For instance:
Voltage A-B:
uab(t) = ua(t) - ub(t) = 120.sqrt2.sin(wt) + 120.sqrt2.sin(wt-2.pi/3+pi)

And using the following trigonometric properties:

A.sin(wt+a) + B.sin(wt+b) = C.sin(wt+c)
where:
C = sqrt[A^2 + B^2 + 2.A.B.cos(a-b)] and tan c =(A.sina+B.sinb)/(A.cosa+B.cosb)

uab(t) = Sqrt[120^2 + 120^2 + 2.120.120.cos(pi/6)]sin(wt + pi/6) =
= sqrt3.120.sqrt2.sin(wt + pi/6) = 208.sqrt2.sin(wt + pi/6)
Julian

 

[jbartos]

Suggestion: There are other threads in this Forum dealing with this topic. It may be worthwhile to search for them. E.g.
V=Vmax sin(wt +/- phase angle)
thread no. 238-10803