Chapter 2, Problem 1P

A transmission line of length l connects a load to a sinusoidal voltage source with an oscillation frequency f. Assuming that the velocity of wave propagation on the line is c, for which of the following situations is it reasonable to ignore the presence of the transmission line in the solution of the circuit:

 *(a)    l = 20 cm, f = 20 kHz

  (b)    l = 50 km, f = 60 Hz

 *(c)    l = 20 cm, f = 600 MHz

  (d)    l = 1 mm, f = 100 GHz

To determine

Whether it is reasonable to ignore the presence of the transmission line effects in the solution of the circuit in the given situation.

Answer to Problem 1P

The transmission line effects may be ignored in the given situation.

Explanation of Solution

Given data:

The length of the transmission line is $20\text{cm}$.

The frequency is $20\text{kHz}$.

Calculation:

The conversion of kilo Hertz into Hertz is done as,

$1\text{kHz}=1\times {10}^3\text{Hz}$

So, the conversion of $20\text{kHz}$ into $\text{Hz}$ is given as,

$20\text{kHz}=20\times {10}^3\text{Hz}$

The conversion of $\text{cm}$ into $\text{m}$ is done as,

$1\text{cm}=1\times {10}^{-2}\text{m}$

So, the conversion of $20\text{cm}$ into $\text{m}$ is given as,

$20\text{cm}=20\times {10}^{-2}\text{m}$

The conditions for the transmission line effects are as follows.

1. 1. If the ratio $\frac{l}{\lambda }$ is less than $0.01$, then the transmission line effects may be ignored.
2. 2. If the ratio $\frac{l}{\lambda }$ is equal to or greater than $0.01$, then the transmission line effects should be considered.

The wavelength is given as,

$\lambda =\frac{c}{f}$

Here,

$c$ is the velocity of light which has the standard value as $\left(c=3\times {10}^8\text{m/s}\right)$.

$f$ is the frequency.

$\lambda$ is the wavelength.

Substitute $3\times {10}^8\text{m/s}$ for $c$ and $20\times {10}^3\text{Hz}$ for $f$ in the above formula.

$\begin{array}{c}\lambda =\frac{3\times {10}^8\text{m/s}}{20\times {10}^3\text{Hz}}\\ =15000\text{m}\end{array}$

The ratio $\frac{l}{\lambda }$ is written as,

$\begin{array}{c}\frac{l}{\lambda }=\frac{20\times {10}^{-2}\text{m}}{15000\text{m}}\\ \frac{l}{\lambda }=1.33\times 10\end{array}$

So, the ratio $\frac{l}{\lambda }$ is less than $0.01$, then the transmission line effects may be ignored.

Conclusion:

Therefore, the transmission line effects may be ignored.

To determine

Whether it is reasonable to ignore the presence of the transmission line effects in the solution of the circuit in the given situation.

Answer to Problem 1P

The transmission line effects should be considered in the given situation.

Explanation of Solution

Given data:

The length of the transmission line is $50\text{km}$.

The frequency is $60\text{Hz}$.

Calculation:

The conversion of $\text{km}$ into $\text{m}$ is done as,

$1\text{km}=1\times {10}^3\text{m}$

So, the conversion of $50\text{km}$ into $\text{m}$ is given as,

$50\text{km}=50\times {10}^3\text{m}$

The wavelength is given as,

$\lambda =\frac{c}{f}$

Substitute $3\times {10}^8\text{m/s}$ for $c$ and $60\text{Hz}$ for $f$ in the above formula.

$\begin{array}{c}\lambda =\frac{3\times {10}^8\text{m/s}}{60\text{Hz}}\\ =5\times {10}^6\text{m}\end{array}$

The ratio $\frac{l}{\lambda }$ is written as,

$\begin{array}{c}\frac{l}{\lambda }=\frac{50\times {10}^3\text{m}}{5\times {10}^6\text{m}}\\ \frac{l}{\lambda }=0.01\end{array}$

So, the ratio $\frac{l}{\lambda }$ is equal to $0.01$, then the transmission line effects should be considered.

Conclusion:

Therefore, the transmission line effects should be considered.

To determine

Whether it is reasonable to ignore the presence of the transmission line effects in the solution of the circuit in the given situation.

Answer to Problem 1P

The transmission line effects should be considered.

Explanation of Solution

Given data:

The length of the transmission line is $20\text{cm}$.

The frequency is $600\text{MHz}$.

Calculation:

The conversion of Mega Hertz into Hertz is done as,

$1\text{MHz}=1\times {10}^6\text{Hz}$

So, the conversion of $600\text{MHz}$ into $\text{Hz}$ is given as,

$600\text{MHz}=600\times {10}^6\text{Hz}$

The conversion of $\text{cm}$ into $\text{m}$ is done as,

$1\text{cm}=1\times {10}^{-2}\text{m}$

So, the conversion of $20\text{cm}$ into $\text{m}$ is given as,

$20\text{cm}=20\times {10}^{-2}\text{m}$

The wavelength is given as,

$\lambda =\frac{c}{f}$

Substitute $3\times {10}^8\text{m/s}$ for $c$ and $600\times {10}^6\text{Hz}$ for $f$ in the above formula.

$\begin{array}{c}\lambda =\frac{3\times {10}^8\text{m/s}}{600\times {10}^6\text{Hz}}\\ =0.5\text{m}\end{array}$

The ratio $\frac{l}{\lambda }$ is written as,

$\begin{array}{c}\frac{l}{\lambda }=\frac{20\times {10}^{-2}\text{m}}{0.5\text{m}}\\ \frac{l}{\lambda }=0.4\end{array}$

So, the ratio $\frac{l}{\lambda }$ is greater than $0.01$, then the transmission line effects should be considered.

Conclusion:

Therefore, the transmission line effects should be considered.

To determine

Whether it is reasonable to ignore the presence of the transmission line effects in the solution of the circuit in the given situation.

Answer to Problem 1P

The transmission line effects should be considered.

Explanation of Solution

Given data:

The length of the transmission line is $1\text{mm}$.

The frequency is $100\text{GHz}$.

Calculation:

The conversion of Giga Hertz into Hertz is done as,

$1\text{GHz}=1\times {10}^9\text{Hz}$

So, the conversion of $100\text{GHz}$ into $\text{Hz}$ is given as,

$100\text{GHz}=100\times {10}^9\text{Hz}$

The conversion of $\text{mm}$ into $\text{m}$ is done as,

$1\text{mm}=1\times {10}^{-3}\text{m}$

The wavelength is given as,

$\lambda =\frac{c}{f}$

Substitute $3\times {10}^8\text{m/s}$ for $c$ and $100\times {10}^9\text{Hz}$ for $f$ in the above formula.

$\begin{array}{c}\lambda =\frac{3\times {10}^8\text{m/s}}{100\times {10}^9\text{Hz}}\\ =0.003\text{m}\end{array}$

The ratio $\frac{l}{\lambda }$ is written as,

$\begin{array}{c}\frac{l}{\lambda }=\frac{1\times {10}^{-3}\text{m}}{0.003\text{m}}\\ \frac{l}{\lambda }=0.33\end{array}$

So, the ratio $\frac{l}{\lambda }$ is greater than $0.01$, then the transmission line effects should be considered.

Conclusion:

Therefore, the transmission line effects should be considered.