18: Phasors and Transmission Lines - Imperial College … · Phasors and transmision lines 18:...

18: Phasors and TransmissionLines

18: Phasors andTransmission Lines• Phasors and transmisionlines

• Phasor Relationships

• Phasor Reflection

• Standing Waves

• Summary

• Merry Xmas

E1.1 Analysis of Circuits (2017-10116) Phasors and Transmission Lines: 18 – 1 / 7

Phasors and transmision lines




• Standing Waves

• Summary

• Merry Xmas


For a transmission line: v(t, x) = f(

t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(

f(t− xu)− g(t+ x

u))





• Standing Waves

• Summary

• Merry Xmas



t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(


u))

We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω





• Standing Waves

• Summary

• Merry Xmas



t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(


u))

We can use phasors to eliminate t from the equations if f() and g() aresinusoidal with the same ω: f(t) = A cos (ωt+ φ) ⇒ F = Aejφ.





• Standing Waves

• Summary

• Merry Xmas



t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(


u))


Then fx(t) = f(t− xu) = A cos

(

ω(

t− xu

)

+ φ)





• Standing Waves

• Summary

• Merry Xmas



t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(


u))



(

ω(

t− xu

)

+ φ)

⇒ Fx = Aej(−ωux+φ)





• Standing Waves

• Summary

• Merry Xmas



t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(


u))



(

ω(

t− xu

)

+ φ)

⇒ Fx = Aej(−ωux+φ)= Aejφe−j ω

ux





• Standing Waves

• Summary

• Merry Xmas



t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(


u))



(

ω(

t− xu

)

+ φ)


ux= F0e

−jkx

where the wavenumber is k , ωu

.





• Standing Waves

• Summary

• Merry Xmas



t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(


u))



(

ω(

t− xu

)

+ φ)


ux= F0e

−jkx


.Units: ω is “radians per second”, k is “radians per metre” (note k ∝ ω).





• Standing Waves

• Summary

• Merry Xmas



t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(


u))



(

ω(

t− xu

)

+ φ)


ux= F0e

−jkx



Similarly Gx = G0e+jkx.





• Standing Waves

• Summary

• Merry Xmas



t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(


u))



(

ω(

t− xu

)

+ φ)


ux= F0e

−jkx




Everything is time-invariant: phasors do not depend on t.





• Standing Waves

• Summary

• Merry Xmas



t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(


u))



(

ω(

t− xu

)

+ φ)


ux= F0e

−jkx





Nice things about sine waves:(1) a time delay is just a phase shift





• Standing Waves

• Summary

• Merry Xmas



t− xu

)

+ g(

t+ xu

)

and

i(t, x) = Z−10

(


u))



(

ω(

t− xu

)

+ φ)


ux= F0e

−jkx





Nice things about sine waves:(1) a time delay is just a phase shift(2) sum of delayed sine waves is another sine wave

Phasor Relationships


Time Domain Phasor Notes

f(t) = A cos (ωt+ φ) F = Aejφ F indep of t





fx(t) = f(

t− xu

)





fx(t) = f(

t− xu

)

= A cos(

ωt+ φ− ωux)





fx(t) = f(

t− xu

)

= A cos(

ωt+ φ− ωux)

Fx = Aej(φ−ωux)





fx(t) = f(

t− xu

)

= A cos(

ωt+ φ− ωux)

Fx = Aej(φ−ωux)

= Fe−jkx





fx(t) = f(

t− xu

)

= A cos(

ωt+ φ− ωux)

Fx = Aej(φ−ωux)

= Fe−jkx

|Fx| ≡ |F |indep of x





fx(t) = f(

t− xu

)

= A cos(

ωt+ φ− ωux)

Fx = Aej(φ−ωux)

= Fe−jkx


fy(t) = fx

(

t− (y−x)u

)

Fy = Fxe−jk(y−x)





fx(t) = f(

t− xu

)

= A cos(

ωt+ φ− ωux)

Fx = Aej(φ−ωux)

= Fe−jkx


fy(t) = fx

(

t− (y−x)u

)

Fy = Fxe−jk(y−x) Delayed by y−x

u





fx(t) = f(

t− xu

)

= A cos(

ωt+ φ− ωux)

Fx = Aej(φ−ωux)

= Fe−jkx


fy(t) = fx

(

t− (y−x)u

)


u

gy(t) = gx

(

t+ (y−x)u

)

Gy = Gxe+jk(y−x) Advanced by y−x

u





fx(t) = f(

t− xu

)

= A cos(

ωt+ φ− ωux)

Fx = Aej(φ−ωux)

= Fe−jkx


fy(t) = fx

(

t− (y−x)u

)


u

gy(t) = gx

(

t+ (y−x)u

)


u

vx(t) = fx(t) + gx(t) Vx = Fx +Gx





fx(t) = f(

t− xu

)

= A cos(

ωt+ φ− ωux)

Fx = Aej(φ−ωux)

= Fe−jkx


fy(t) = fx

(

t− (y−x)u

)


u

gy(t) = gx

(

t+ (y−x)u

)


u

vx(t) = fx(t) + gx(t) Vx = Fx +Gx

ix(t) =fx(t)−gx(t)

Z0Ix = Fx−Gx

Z0

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas


Phasors obey Ohm’s law: VL

IL= RL

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas



IL= RL = FL+GL

Z−10 (FL−GL)

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas



IL= RL = FL+GL

Z−10 (FL−GL)

So GL = ρLFL where ρL = RL−Z0

RL+Z0

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas



IL= RL = FL+GL

Z−10 (FL−GL)


RL+Z0

At any x, Gx

Fx= GLe−jk(L−x)

FLe+jk(L−x)

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas



IL= RL = FL+GL

Z−10 (FL−GL)


RL+Z0

At any x, Gx

Fx= GLe−jk(L−x)

FLe+jk(L−x) = ρLe−2jk(L−x)

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas



IL= RL = FL+GL

Z−10 (FL−GL)


RL+Z0

At any x, Gx

Fx= GLe−jk(L−x)


Ohm’s law at the load determines the ratio Gx

Fxeverywhere on the line.

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas



IL= RL = FL+GL

Z−10 (FL−GL)


RL+Z0

At any x, Gx

Fx= GLe−jk(L−x)




Note that∣

∣

∣

Gx

Fx

∣

∣

∣≡ |ρL| has the same value for all x.

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas



IL= RL = FL+GL

Z−10 (FL−GL)


RL+Z0

At any x, Gx

Fx= GLe−jk(L−x)




Note that∣

∣

∣

Gx

Fx

∣

∣


Vx = Fx +Gx

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas



IL= RL = FL+GL

Z−10 (FL−GL)


RL+Z0

At any x, Gx

Fx= GLe−jk(L−x)




Note that∣

∣

∣

Gx

Fx

∣

∣


Vx = Fx +Gx = Fx

(

1 + ρLe−2jk(L−x)

)

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas



IL= RL = FL+GL

Z−10 (FL−GL)


RL+Z0

At any x, Gx

Fx= GLe−jk(L−x)




Note that∣

∣

∣

Gx

Fx

∣

∣


Vx = Fx +Gx = Fx

(


)

Ix = Z−10 (Fx −Gx)

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas



IL= RL = FL+GL

Z−10 (FL−GL)


RL+Z0

At any x, Gx

Fx= GLe−jk(L−x)




Note that∣

∣

∣

Gx

Fx

∣

∣


Vx = Fx +Gx = Fx

(


)

Ix = Z−10 (Fx −Gx) = Z−1

0 Fx

(

1− ρLe−2jk(L−x)

)

Phasor Reflection




• Standing Waves

• Summary

• Merry Xmas



IL= RL = FL+GL

Z−10 (FL−GL)


RL+Z0

At any x, Gx

Fx= GLe−jk(L−x)




Note that∣

∣

∣

Gx

Fx

∣

∣


Vx = Fx +Gx = Fx

(


)

Ix = Z−10 (Fx −Gx) = Z−1

0 Fx

(

1− ρLe−2jk(L−x)

)

The exponent −2jk (L− x) is the phase delay from travelling from x to L

and back again (hence the factor 2).

Standing Waves


Standing Waves


Forward wave phasor: Fx = Fe−jkx

Standing Waves



Backward wave phasor: Gx = ρLFxe−2jk(L−x)

Standing Waves



Backward wave phasor: Gx = ρLFxe−2jk(L−x) = ρLFe−2jkLe+jkx

Standing Waves




Line Voltage phasor: Vx = Fx +Gx

Standing Waves




Line Voltage phasor: Vx = Fx +Gx = Fe−jkx(


)

Standing Waves






)

Line Voltage Amplitude: |Vx| = |F |∣

∣1 + ρLe−2jk(L−x)

∣

∣

Standing Waves






)



∣

∣ varies with x but not t

Standing Waves






)



∣


Max amplitude equals 1 + |ρL| at values of x where Fx and Gx are in phase. Thisoccurs every λ

2 away from L

Standing Waves






)



∣



2 away from L where λ is the wavelength, λ = 2πk

= uf

.

Standing Waves






)



∣




= uf

.

Min amplitude equals 1− |ρL| at values of x where Fx and Gx are out of phase.

Standing Waves






)



∣




= uf

.

Min amplitude equals 1− |ρL| at values of x where Fx and Gx are out of phase.

Standing waves arise whenever a periodic wave meets its reflection: e.g. ponds,musical instruments, microwave ovens.

Summary




• Standing Waves

• Summary

• Merry Xmas


• Use phasors if forward and backward waves are sinusoidal with thesame ω.

Summary




• Standing Waves

• Summary

• Merry Xmas


• Use phasors if forward and backward waves are sinusoidal with thesame ω.◦ fx(t) = f

(

t− xu

)

→ Fx = F0e−jkx

◦ gx(t) = g(

t+ xu

)

→ Gx = G0e+jkx

Summary




• Standing Waves

• Summary

• Merry Xmas



(

t− xu

)

→ Fx = F0e−jkx

◦ gx(t) = g(

t+ xu

)

→ Gx = G0e+jkx

⊲ k = ωu

is the wavenumber in “radians per metre”

Summary




• Standing Waves

• Summary

• Merry Xmas



(

t− xu

)

→ Fx = F0e−jkx

◦ gx(t) = g(

t+ xu

)

→ Gx = G0e+jkx

⊲ k = ωu


• Time delays ≃ phase shifts: Fy = Fxe−jk(y−x)

Summary




• Standing Waves

• Summary

• Merry Xmas



(

t− xu

)

→ Fx = F0e−jkx

◦ gx(t) = g(

t+ xu

)

→ Gx = G0e+jkx

⊲ k = ωu



• When a periodic wave meets its reflection you get a standing wave:

Summary




• Standing Waves

• Summary

• Merry Xmas



(

t− xu

)

→ Fx = F0e−jkx

◦ gx(t) = g(

t+ xu

)

→ Gx = G0e+jkx

⊲ k = ωu




◦ Oscillation amplitude varies with x: ∝∣


∣

∣

Summary




• Standing Waves

• Summary

• Merry Xmas



(

t− xu

)

→ Fx = F0e−jkx

◦ gx(t) = g(

t+ xu

)

→ Gx = G0e+jkx

⊲ k = ωu




◦ Oscillation amplitude varies with x: ∝∣


∣

∣

◦ Max amplitude of (1 + |ρL|) occurs every λ2

Merry Xmas


18: Phasors and Transmission Lines - Imperial College … · Phasors and transmision lines 18:...

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Transcript of 18: Phasors and Transmission Lines - Imperial College … · Phasors and transmision lines 18:...