The normalized impedance of a transmission line is given by expression :
\(\dfrac{|V_{max}|}{|I_{max}|\cdot Z_O}\)
Definition first. Normalisation on a transmission line always means dividing an impedance by the characteristic impedance of that line:
\(z=\dfrac{Z}{Z_O}\)
The result is a dimensionless number, which is why the Smith chart — drawn entirely in normalised coordinates — works for a 50 Ω line and a 300 Ω line alike.
Step 1 — identify the impedance in option 4. On a mismatched line the standing-wave pattern has voltage maxima where the incident and reflected waves add and current minima at the same points. The ratio of the peak voltage to the peak current is a genuine impedance:
\(Z_{max}=\dfrac{|V_{max}|}{|I_{max}|}\)
Step 2 — normalise it. Dividing by ZO gives precisely option 4:
\(z_{max}=\dfrac{|V_{max}|}{|I_{max}|\,Z_O}\)
This quantity is numerically equal to the standing wave ratio, \(S=\dfrac{1+|\Gamma|}{1-|\Gamma|}\), which is the classic result that the normalised impedance at a voltage maximum equals the VSWR.
Step 3 — eliminate the others, each of which is a real formula but for something else.
Option 1, \(Z_O/Z_{in}\): the ratio is the right idea but inverted. Dividing ZO by an impedance gives a normalised admittance, not a normalised impedance.
Option 2, \(\sqrt{Z_{SC}Z_{OC}}\): this is the standard laboratory measurement of the characteristic impedance ZO itself — take the input impedance with the far end shorted and again with it open and take the geometric mean. It has the units of ohms, so it cannot be a normalised (dimensionless) quantity.
Option 3, \(Z_O\cot\gamma L\): this is the input impedance of an open-circuited line of length L. Again it is in ohms, and again it is multiplied by ZO rather than divided by it.
The dimensional shortcut. Only one option is dimensionless, and normalisation must produce a pure number. That single observation settles the question before any transmission-line theory is used.
Hence, the normalized impedance is \(\dfrac{|V_{max}|}{|I_{max}|\cdot Z_O}\).
Similar Questions
Consider the following statements
A. Reflection coefficient is change in the magnitude of reflected wave with constant phase with respect to incident wave.
B. Transmission coefficient is change in the magnitude and phase of transmitted wave with respect to incident wave
C. On smith chart, \(\dfrac{\lambda}{2}\) distance is equivalent to 2π.
D. For distortion less line, RL = GC
E. Directivity of an antenna can be less than 1 (unity).
Choose the most appropriate answer from the options given below :
In a loss less transmission line of length 50 cm with L = 10 μH/m, C = 40 pF/m is operated at 25 MHz. Its electrical path length is
A lossless line has a characteristic impedance of 50 ohms. It is terminated in a load resistance of 75 ohms. The line is energised by a generator which has an output impedance of 50 ohms and an output voltage of 30 V (rms). The line is assumed to be 2.25 wavelength long. The input impedance and instantaneous load voltages are given :
(a) Zin = 50 ohms (b) VL (instantaneous) = 36 V (c) Zin = 33.33 ohms (d) VL (instantaneous) = 12 V
Which of the above are correct :
Arrange the below mentioned transmission lines in order of their increasing frequency handling capabilities :
(a) wave guide (b) Parallel wire (c) Rigid co-axial cables (d) Flexible co-axial cables
For a quarter wave ideal transmission line of characteristic impedance of 50 Ω and load impedance of 100 Ω, the input impedance of line will be :
The disadvantage of co-axial cable is/are :
(A) Support higher bandwidth than twisted pair cable
(B) Light weight
(C) Relatively expensive compared to twisted pair cable
(D) EMI resistant
(E) Doesn't at very high frequencyChoose the most appropriate answer from the options given below :
The propagation constant for uniform plane wave is given by the expression given below :
Important Questions from Transmission Lines
A characteristic impedance does NOT satisfy which of the following statements?
The dielectric constant of the material used in a transmission line is 2. What is the velocity factor of this line if its characteristic impedance is 300 Ω?
A transmission line of \(50{\rm{\;\Omega }}\) characteristic impedance is terminated with a \(\rm 100 \ Ω\) resistance. The minimum impedance measured on the line is equal to
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The input impedance of short circuited lossless transmission line quarter wavelength is
