A lossless transmission line has a characteristic impedance of Z0 and capacitance per unit length of C. The velocity of propagation of the travelling wave on the line is
Transmission Line Velocity Calculation
Understanding the behavior of waves on a transmission line is fundamental in electrical engineering. For a lossless transmission line, certain parameters dictate how fast an electromagnetic wave travels along it. This solution will explain how the velocity of propagation is related to the characteristic impedance and capacitance per unit length of such a line.
Lossless Transmission Line Basics
A lossless transmission line is an idealized model where there is no energy dissipation. This means its series resistance (R) and shunt conductance (G) are considered to be zero. The primary parameters for a lossless line are its series inductance per unit length (L) and shunt capacitance per unit length (C).
- Inductance (L): Represents the magnetic energy storage capability of the line.
- Capacitance (C): Represents the electric energy storage capability between the conductors.
- For a lossless line, we assume R = 0 and G = 0.
Characteristic Impedance (\(Z_0\))
The characteristic impedance (\(Z_0\)) of a transmission line is a crucial parameter that represents the impedance seen looking into an infinitely long line. For a lossless transmission line, the formula for characteristic impedance is given by:
\[ Z_0 = \sqrt{\frac{L}{C}} \]
Where:
- \(Z_0\) is the characteristic impedance (in Ohms).
- \(L\) is the inductance per unit length (in Henrys/meter).
- \(C\) is the capacitance per unit length (in Farads/meter).
Velocity of Wave Propagation (\(v\))
The velocity of propagation (\(v\)) refers to the speed at which an electromagnetic wave travels along the transmission line. For a lossless transmission line, this velocity is determined by the inductance and capacitance per unit length and is given by the formula:
\[ v = \frac{1}{\sqrt{LC}} \]
Where:
- \(v\) is the velocity of propagation (in meters/second).
- \(L\) is the inductance per unit length.
- \(C\) is the capacitance per unit length.
Deriving Velocity from \(Z_0\) and \(C\)
We are given the characteristic impedance \(Z_0\) and the capacitance per unit length \(C\), and we need to find the velocity of propagation \(v\). We can use the relationships we know to derive \(v\) in terms of \(Z_0\) and \(C\).
- We know the characteristic impedance \(Z_0\) for a lossless line is: \[ Z_0 = \sqrt{\frac{L}{C}} \]
- To isolate \(L\), we can square both sides of the equation: \[ Z_0^2 = \frac{L}{C} \]
- Now, solve for \(L\): \[ L = Z_0^2 C \]
- Next, we know the velocity of propagation \(v\) for a lossless line is: \[ v = \frac{1}{\sqrt{LC}} \]
- Substitute the expression for \(L\) from step 3 into the velocity equation: \[ v = \frac{1}{\sqrt{(Z_0^2 C)C}} \]
- Simplify the expression under the square root: \[ v = \frac{1}{\sqrt{Z_0^2 C^2}} \]
- Take the square root: \[ v = \frac{1}{Z_0 C} \]
Therefore, the velocity of propagation of the travelling wave on the lossless transmission line is \(\frac{1}{Z_0 C}\).
This derivation shows that if you know the characteristic impedance and the capacitance per unit length, you can directly calculate the speed at which signals travel along the transmission line.
Important Questions from Electromagnetic Wave Propagation
If the Polarization vector is given as N and the Direction of propagation is given as K then which one of the following relations is correct?
Ez ≠ 0 and Hz ≠ 0 represent which type of mode of propagation of microwaves?
For sky waves, following statements are given:
(A) n > 1, this shows 81 \(\rm\frac{N}{f^2}\) positive
(B) n > 1, show 81 \(\rm\frac{N}{f^2}\) Negative
(C) n < 1 shows 81 \(\rm\frac{N}{f^2}\) < 1
(D) v g x v p= c 2
(E) n = 0 shows 81 \(\rm\frac{N}{f^2}\) = 1, f = f c
Choose the correct answer from the options given below:
The phenomenon of microwave signals following the curvature of earth is
The relationship between wavelength and frequency of an electromagnetic wave
