Performance of Transmission Line & Travelling Wave Analysis For Electrical Engineering

where, β = Phase shift in red/mile

Key Points

• Surge impedance for the transmission line is 400Ω and for cable, it is 40 Ω.
• The phase angle of surge impedance (Z_C) of transmission lines is in range of 0^o to -15^o.
• If loading of the = SIL, then the power factor will be unity. Hence, maximum power can be transferred and the line can be called as the flat line.
• If loading of line < SIL, then power factor will be leading.
• If loading of line > SIL, then power factor will be lagging.

Wave Phenomenon

Refracted (transmitted) wave = Incident (forward) wave + Reflected wave

V_T = V_F + V_R

where, V_T = Refracted (transmitted) voltage wave

V_F = Incident or forward voltage wave

V_R = Reflected voltage

Transmission Line Termination

Case (1) Open-Ended Line

• Current becomes at load end zero. Due to this electromagnetic energ

Transmission Line

A transmission line is a set of conductors being run from one place to another supported on transmission towers. Therefore, such lines have four distributed parameters, series resistance, inductance shunt capacitance and conductance.

Transmission lines are typically made of conductive materials, such as copper or aluminum, that have low resistance to allow for efficient power transmission. They are used in both overhead and underground applications, depending on the specific requirements and environmental factors.

Short Transmission Line

The effect of capacitance is ignored in these lines, length of short transmission line is less than 80 km.

where, V_s = Sending end voltage

l_s = Sending end current

V_R = Receiving end voltage

I_R = Receiving end current

I_S = I_R

V_R +l(R+ jX) = V_s

Reglation % Regulation

Efficiency

where, P = Power received

Medium Transmission Line

The range of length of this transmission line is 80 km to 250 km. Medium transmission lines are modeled with lumped shunt admittance.

Nomial-π Representation

The lumped series impedance is placed in the middle while the shunt admittance is divided into equal parts and placed at the two ends.

Here,

For Voltage Regulation

The no-load receiving end voltage

∴ % regulation

% efficiency

Nominal-T Representation

The shunt admittance is placed in the middle and the series impedance is divided into equal parts and these parts are placed on either side of the shunt admittance.

Here,

For Voltage Regulation

The no-load receiving end voltage

% regulation

% efficiency

Note: Percentage regulation

Here, |V_S| = Sending end voltage |V_R.r₂| = Full load receiving end voltage.

Long Transmission Line

The length of long transmission line is more than 250 km.

The ABCD parameters of the long transmission line are,

where,  which is called the characteristic impedance

For lossless line R = 0; G = 0

π-Representation of a Long Transmission Line

In this, the series impedance is denoted by Z^j while the shunt admittance is denoted by Y’

ABCD, parameters are

where,

T-Representation of Long Transmission Line

Parameters are

and

 

Power Flow Through a Transmission Line

S_s = P_s + jθ_s = V_sI_s*

S_R = P_R + jθ_R

Let A = D = |A|<α°

B = |B| ∠β°

∴ V_s = AV_R + BI_R

The complex power VR IR at receiving end

The real active power at receiving end,

The reactive power at the receiving end,

For fixed value of VS and VR the receiving end real active power is maximum when

Surge Impedance

This is also known as characteristic impedance. It is impedance offered by the system under surge impedance loading.

Characteristic impedance

Here, Z = Series impedance per unit length

Y = Shunt admittance per unit length

For lossless line, R = 0, G = 0

It can be also given by 

where, Z_OC = Sending end impedance with receiving end open circuited 

Z_SC = Sending end impedance with receiving end short-circuited 

Propagation Constant

where, α = Attenuation constant

β = Phase constant

Surge Impedance Loading (SIL)

Surge Impedance Loading (SIL) of a line is the power delivered by a line to a purely resistive load equal to its surge impedance.

Wavelength

• y  converted into electrostatic energy 
• The potential at the open end becomes 2V.

where, I_T = Transmitted or refracted current wave

I_R = Reflected current wave

Case (2) Short Circuited Line

• When line is short-circuited at load end, the voltage at load end becomes zero and electrostatic energy is converted into electromagnetic energy 
• The current becomes 2l at the short end.

 

V_T = 0

V_R = -V_R

I_T = 2_/F

I_R = I_F

Case (3) Termination of Line with Resistor

Coefficient of refraction of voltage

Coefficient of reflection of voltage wave

Coefficient of refraction (transmit) of current wave 

Coefficient of refraction of current wave 

Note: When the terminated by a resistor equal to surge impedance

V_T =V_F; V_R = I_T = I_F; I_R = 0

The incident wave continuous as it is and there is no reflection.

Case (4) Line Terminated by an Inductor

Transmitted (refracted) voltage 

Reflected voltage wave 

Transmitted (refracted) current wave. 

Reflected current wave 

Where, Z_c = Surge impedance of the line.

Case (5) Line Terminated by a Capacitor

Transmitted (refracted) voltage wave

Reflected voltage wave

 

Transmitted (refracted) current wave 

Reflected current wave

 

Case (6) Parallel Reactive Termination

Refracted (transmitted) voltage wave,

Transmission Coefficient at T Junction (Forked Line)

Refraction coefficient 

Refracted (transmitted voltage wave)

 

Line Connected to Cable

Refracted voltage wave

 

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