Example problem with "bird" transmission line

[DK PE]

There was an earlier post on this topic so anyone interested, try this out... otherwise ignore.

A 3Φ, 60 Hz transmission line is arranged in a triangle fashion with 12' each between the top conductor and the lower left and lower right conductors, and 18' between the lower two conductors. Each conductor is ACSR Drake.

Determine the per-unit series reactance of a 35 mile line using a 1.5MVA, 69kV base.

I will post answer sometime over weekend.

 

[R2KBA]

Question: Do we assume the lines are transposed and therefore ignore separate calculations for mutual inductances?

ETA: Ah, I see what you did. I hope to have an answer for you shortly.

ETA: I got 25.09 ohms which comes to 0.0079 pu.

Even though I don't think this type of problem will be on the exam, I'm really happy you posted this because earlier I was thinking that the formulas that gave answers in H/m meant "Henries per mile" which is totally wrong! It is "Henries per meter." I guess I didn't expect to see a formula where the input distances are in feet and the output is in SI units. But since it is a ratio I suppose I could convert ft to meters if I really wanted to. Grainger/Stevenson helped me out with this one.

 

[EEVA PE]

There was an earlier post on this topic so anyone interested, try this out... otherwise ignore.

A 3Φ, 60 Hz transmission line is arranged in a triangle fashion with 12' each between the top conductor and the lower left and lower right conductors, and 18' between the lower two conductors. Each conductor is ACSR Drake.

Determine the per-unit series reactance of a 35 mile line using a 1.5MVA, 69kV base.

I will post answer sometime over weekend.

Radius of conductor is missing from needed data.

Inductance/Phase = 2x10-7 ln(D/r')

r' = e-1/4r = ,7788r

D = ((18)(12)(12))1/3 = 13.7 feet

Inductance/phase = 2x10-7 ln (13.7/.7788r)

Zbase =( vbase)^2/Sbase = (69x10^3)^2 /1.5x10 ^6 = 3174

reactance per unit =(( 2x10^-7 (13.7/.7788r)) / 3174) X 35 miles X 3

So radius is .0462 feet (D = 1.108 inches per spec)

Inductance/phase = 2x10-7 ln (13.7/.7788(.0462) = 1.1884 x10^-6 H/m

Inductance all phases and 35 miles = 3 X 35 X 1.884 x 10^-6 = 1.24782 x 10^-4

PU = 1.24782 x 10^-4 / 3174 = 3.93 x10-3 pu

 

[DK PE]

.

Radius of conductor is missing from needed data.

Zbase = Sbase/sq3XVbase = 1.5x10 ^6/sq3 X 69K = 12.55

Conductor properties were given: Each conductor is ACSR Drake.

You may want to double-check your base impedance equation...

 

[EEVA PE]

.

Radius of conductor is missing from needed data.

Zbase = Sbase/sq3XVbase = 1.5x10 ^6/sq3 X 69K = 12.55

Conductor properties were given: Each conductor is ACSR Drake.

You may want to double-check your base impedance equation...

I used the wrong formula. Made changes above. I was able to re-edit my original.

 

[DK PE]

Determine the per-unit series reactance per conductor of a 35 mile line using a 1.5MVA, 69kV base.

Inductance/phase = 2x10-7 ln (13.7/.7788r)

Zbase =( vbase)^2/Sbase = (69x10^3)^2 /1.5x10 ^6 = 3174

reactance per unit =(( 2x10^-7 (13.7/.7788r)) / 3174) X 35 miles X 3

So radius is .0462 feet (D = 1.108 inches per spec)

Inductance/phase = 2x10-7 ln (13.7/.7788(.0462) = 1.1884 x10^-6 H/m

Inductance all phases and 35 miles = 3 X 35 X 1.884 x 10^-6 = 1.24782 x 10^-4

PU = 1.24782 x 10^-4 / 3174 = 3.93 x10-3 pu

Better check your relationship one more time... in order to get per unit, you should be dividing reactance in Ohms by base impedance in Ohms. It looks like you are dividing H/m by ohms?

I also should have been more clear in question... for a one line model you would need the series reactance per conductor...

 

[EEVA PE]

Determine the per-unit series reactance per conductor of a 35 mile line using a 1.5MVA, 69kV base.

Inductance/phase = 2x10-7 ln (13.7/.7788r)

Zbase =( vbase)^2/Sbase = (69x10^3)^2 /1.5x10 ^6 = 3174

reactance per unit =(( 2x10^-7 (13.7/.7788r)) / 3174) X 35 miles X 3

So radius is .0462 feet (D = 1.108 inches per spec)

Inductance/phase = 2x10-7 ln (13.7/.7788(.0462) = 1.1884 x10^-6 H/m

Inductance all phases and 35 miles = 3 X 35 X 1.884 x 10^-6 = 1.24782 x 10^-4

PU = 1.24782 x 10^-4 / 3174 = 3.93 x10-3 pu

Better check your relationship one more time... in order to get per unit, you should be dividing reactance in Ohms by base impedance in Ohms. It looks like you are dividing H/m by ohms?

I also should have been more clear in question... for a one line model you would need the series reactance per conductor...

Inductance/Phase = 2x10-7 ln(D/r')

r' = e-1/4r = 0.7788r

D = ((18)(12)(12))^1/3 = 13.7 feet

Inductance/phase = 2x10-7 ln (13.7/.7788r)

So radius is .0462 feet (D = 1.108 inches per spec)

inductance/phase for 35 miles =(( 2x10^-7 ln (13.7/.7788(.0462)) X 35 miles )

Inductance= 4.1595x10^-5 H/m

Zbase =( vbase)^2/Sbase = (69x10^3)^2 /1.5x10 ^6 = 3174

Reactance per phase is X = jwl = (377)( 4.1595x10^-5) = 0.01568 ohms

PU reactance per phase = 0.01568 / 3174 = 4.94 x 10^-6

I was a little sloppy earlier.

 

[DK PE]

Inductance/phase = 2x10-7 ln (13.7/.7788r)

So radius is .0462 feet (D = 1.108 inches per spec)

inductance/phase for 35 miles =(( 2x10^-7 ln (13.7/.7788(.0462)) X 35 miles )

Inductance= 4.1595x10^-5 H/m

Zbase =( vbase)^2/Sbase = (69x10^3)^2 /1.5x10 ^6 = 3174

Reactance per phase is X = jwl = (377)( 4.1595x10^-5) = 0.01568 ohms

PU reactance per phase = 0.01568 / 3174 = 4.94 x 10^-6

I was a little sloppy earlier.

Hope you don't think I'm picking on you.. . but better check whether H/m is per meter or per mile... think there is still is a units problem.

See R2BKA's post above.

 

[EEVA PE]

Inductance/phase = 2x10-7 ln (13.7/.7788r)

So radius is .0462 feet (D = 1.108 inches per spec)

inductance/phase for 35 miles =(( 2x10^-7 ln (13.7/.7788(.0462)) X 35 miles )

Inductance= 4.1595x10^-5 H/m

Zbase =( vbase)^2/Sbase = (69x10^3)^2 /1.5x10 ^6 = 3174

Reactance per phase is X = jwl = (377)( 4.1595x10^-5) = 0.01568 ohms

PU reactance per phase = 0.01568 / 3174 = 4.94 x 10^-6

I was a little sloppy earlier.

Hope you don't think I'm picking on you.. . but better check whether H/m is per meter or per mile... think there is still is a units problem.

See R2BKA's post above.

Better to know m = meters, not miles before the exam. Thanks.

Inductance/phase = 2x10-7 ln (13.7/.7788r)

So radius is .0462 feet (D = 1.108 inches per spec)

inductance/phase for 35 miles =(( 2x10^-7 ln (13.7/.7788(.0462)) X 35 miles X (3.281ft/meter) X (mile/5280 ft) )

Inductance= 2.5847 x 10^-8 H per 35 miles

Zbase =( vbase)^2/Sbase = (69x10^3)^2 /1.5x10 ^6 = 3174

Reactance per phase is X = jwl = (377)( 2.5847x10^-8) = 9.744 X 10^-6 ohms

PU reactance per phase = 9.744 X 10^-6 / 3174 = 3.07 x 10^-9

 

[EEVA PE]

Inductance/phase = 2x10-7 ln (13.7/.7788r)

So radius is .0462 feet (D = 1.108 inches per spec)

inductance/phase for 35 miles =(( 2x10^-7 ln (13.7/.7788(.0462)) X 35 miles )

Inductance= 4.1595x10^-5 H/m

Zbase =( vbase)^2/Sbase = (69x10^3)^2 /1.5x10 ^6 = 3174

Reactance per phase is X = jwl = (377)( 4.1595x10^-5) = 0.01568 ohms

PU reactance per phase = 0.01568 / 3174 = 4.94 x 10^-6

I was a little sloppy earlier.

Hope you don't think I'm picking on you.. . but better check whether H/m is per meter or per mile... think there is still is a units problem.

See R2BKA's post above.

Better to know m = meters, not miles before the exam. Thanks.

Inductance/phase = 2x10-7 ln (13.7/.7788r)

So radius is .0462 feet (D = 1.108 inches per spec)

inductance/phase for 35 miles =(( 2x10^-7 ln (13.7/.7788(.0462)) X 35 miles X (3.281ft/meter) X (mile/5280 ft) )

Inductance= 2.5847 x 10^-8 H per 35 miles

Zbase =( vbase)^2/Sbase = (69x10^3)^2 /1.5x10 ^6 = 3174

Reactance per phase is X = jwl = (377)( 2.5847x10^-8) = 9.744 X 10^-6 ohms

PU reactance per phase = 9.744 X 10^-6 / 3174 = 3.07 x 10^-9

Looks like I got some of the conversions backwards, correction below. Thanks DK_PE

Inductance/phase = 2x10-7 ln (13.7/.7788r)

So radius is .0462 feet (D = 1.108 inches per spec)

inductance/phase for 35 miles =(( 2x10^-7 ln (13.7/.7788(.0462)) X 35 miles X (meter/3.281ft) X (5280ft/mile) )

Inductance= 0.0669 H per 35 miles

Zbase =( vbase)^2/Sbase = (69x10^3)^2 /1.5x10 ^6 = 3174

Reactance per phase is X = jwl = (377)( 0.0669) = 25.22 ohms

PU reactance per phase = 25.22 / 3174 = .0079

 

[DK PE]

Congrats, both of you arrived at correct series reactance per conductor of 0.0079 pu and it sounds like also learned that the equation you were using for transmission line inductance has units of Henries/meter, not Henries/mile. Good luck on the exam.

 

[Mahmoud]

there is a much easier method to solve this problem Table A.3 and A.4 of Grainger and Stevenson book indicate that the inductance of the Drake cable/Mi (1ft spacing at 60hz ) is 0.399ohm /mi

D=(12*12*18)^(1/3)=13.73ft

Table A.4 @ 14 ft spacing is =0.3202ohm/mi

Total ohm = 0.399+0.3202=0.7192ohm/mi

for ohms for 35 mile = 25.172 ohm

Zbase =( vbase)^2/Sbase = (69x10^3)^2 /1.5x10 ^6 = 3174

PU reactance per phase = 25.17 / 3174 = 0.0079

 

[R2KBA]

Thanks DK PE and others for your assistance.