Input Impedance of TL

The input impedance of a transmission line is what the source sees when it looks into the line connected to a load. Unlike a simple resistor or lumped circuit element, the impedance presented at the input of a transmission line depends on the line's characteristic impedance Z0, the load impedance ZL, the electrical length of the line (beta*l), and the frequency. This makes transmission line input impedance one of the most analytically rich topics in GATE electromagnetics.

Core Concept Explanation

The input impedance of a transmission line terminated in load ZL is the ratio of total voltage to total current at the input terminals of the line. Because voltage and current on the line are superpositions of forward and backward waves, and because each wave accumulates phase as it travels the length l, the input impedance is a function of line length. Even a purely resistive load ZL can appear as a complex impedance at the input if the line length is not a multiple of lambda/2.

This property is both a challenge and a powerful tool. It is a challenge because the source impedance matching condition must account for line length. It is a tool because by choosing the right line length, any load impedance can be transformed to any desired input impedance. This is the operating principle of transmission line stubs and quarter-wave transformers, two techniques extensively tested in GATE.

For a lossless line, the general input impedance formula is: Zin = Z0 * (ZL + j*Z0*tan(beta*l)) / (Z0 + j*ZL*tan(beta*l)). Here beta = 2*pi/lambda is the phase constant, and l is the physical length of the line. The product beta*l is the electrical length in radians. When beta*l = pi/2, which corresponds to l = lambda/4, tan(beta*l) tends to infinity, and the formula simplifies to Zin = Z0^2 / ZL.

Mathematical Expression

The derivation of the input impedance formula begins by writing the voltage and current at any position z along the line in terms of the forward wave amplitude V+ and the load reflection coefficient Gamma_L. At position z = -l (measured from the load at z=0 toward the source), the total voltage and current are:

V(-l) = V+ * [exp(j*beta*l) + Gamma_L * exp(-j*beta*l)] and I(-l) = (V+/Z0) * [exp(j*beta*l) - Gamma_L * exp(-j*beta*l)]. Taking the ratio V(-l)/I(-l) and substituting Gamma_L = (ZL-Z0)/(ZL+Z0), then simplifying using Euler's formula gives the standard input impedance expression. This derivation is worth knowing conceptually even if the formula is directly applied in problems.

The formula Zin = Z0*(ZL + j*Z0*tan(beta*l))/(Z0 + j*ZL*tan(beta*l)) has several important special cases. When l = 0, Zin = ZL (trivially). When l = lambda/4, Zin = Z0^2/ZL. When l = lambda/2, Zin = ZL (the impedance repeats). When ZL = Z0, Zin = Z0 for all l. When ZL = 0 (short circuit), Zin = j*Z0*tan(beta*l), a pure reactance. When ZL is infinite (open circuit), Zin = -j*Z0*cot(beta*l), also a pure reactance.

Practical Understanding

The quarter-wave transformer is the most important application of input impedance transformation. By choosing a line of length lambda/4 with characteristic impedance Z0_t = sqrt(ZS * ZL), a source impedance ZS can be perfectly matched to a load impedance ZL. This is used in antenna feed networks, power dividers, and microwave amplifier matching networks. The quarter-wave transformer works perfectly only at one frequency (and odd multiples thereof), so it is inherently narrowband.

Transmission line stubs exploit the input impedance of shorted or open lines to synthesize reactive elements. A shorted stub of length less than lambda/4 looks like an inductor (positive reactance). A shorted stub of length between lambda/4 and lambda/2 looks like a capacitor (negative reactance). Similarly, an open stub can be capacitive or inductive depending on its length. Stubs are used for impedance matching and are a standard GATE topic.

Numerical Example

A lossless 50 ohm transmission line of length lambda/4 is terminated in a 25 ohm resistive load. Find the input impedance. This is the classic quarter-wave transformer GATE problem.

Given:
Z0 = 50 ohm
ZL = 25 ohm
l = lambda/4  so  beta*l = (2*pi/lambda)*(lambda/4) = pi/2

Why this formula applies:
At l = lambda/4, tan(beta*l) = tan(pi/2) = infinity
Use the quarter-wave special case formula

Formula:
Zin = Z0^2 / ZL  (quarter-wave transformer result)

Derivation from general formula:
Zin = Z0 * (ZL + j*Z0*tan(beta*l)) / (Z0 + j*ZL*tan(beta*l))
Divide numerator and denominator by tan(beta*l) as tan -> infinity:
Zin = Z0 * (ZL/tan + j*Z0) / (Z0/tan + j*ZL)
As tan -> infinity, ZL/tan -> 0 and Z0/tan -> 0:
Zin = Z0 * (j*Z0) / (j*ZL) = Z0^2 / ZL

Substitution:
Zin = (50)^2 / 25 = 2500 / 25

Calculation:
Zin = 100 ohm

Final Answer:
Zin = 100 ohm (purely resistive)
The line transforms 25 ohm to 100 ohm at its input.
If source impedance is 100 ohm, perfect power transfer achieved.

Exam Tip: The quarter-wave transformer formula Zin = Z0^2/ZL is frequently used in GATE. For a shorted stub (ZL=0), Zin = j*Z0*tan(beta*l): positive (inductive) for l less than lambda/4, and negative (capacitive) for lambda/4 less than l less than lambda/2. For an open stub (ZL=inf), Zin = -j*Z0*cot(beta*l). These signs are commonly confused in exams.

Mechanism: How Length Transforms Impedance

• The general formula Zin = Z0*(ZL + j*Z0*tan(beta*l))/(Z0 + j*ZL*tan(beta*l)) shows that line length l acts as a parameter that continuously rotates the impedance on the Smith chart.
• At l = lambda/2, beta*l = pi, tan = 0, and Zin = ZL. The input impedance equals the load impedance and the pattern repeats. This is because the wave makes a complete round trip of half the wavelength on each side.
• At l = lambda/4, Zin = Z0^2/ZL. This is the impedance inversion property of a quarter-wave line. A short circuit (ZL=0) becomes an open circuit at the input, and vice versa.
• A shorted stub of length l presents Zin = j*Z0*tan(beta*l). For l less than lambda/4, this is positive (inductive). For l between lambda/4 and lambda/2, tan is negative, so Zin is capacitive.
• An open stub presents Zin = -j*Z0*cot(beta*l). Its behavior is opposite to the shorted stub: capacitive for l less than lambda/4, inductive between lambda/4 and lambda/2.
• When ZL = Z0 (matched), the numerator and denominator of the general formula become identical and Zin = Z0 for any line length. This is why a matched termination makes the system independent of line length.

Quick Revision

• Zin = Z0*(ZL + j*Z0*tan(beta*l))/(Z0 + j*ZL*tan(beta*l)). Core formula for lossless terminated line.
• l = lambda/2: Zin = ZL. Impedance repeats every half wavelength.
• l = lambda/4: Zin = Z0^2/ZL. Quarter-wave transformer. Impedance inversion.
• Short stub: Zin = j*Z0*tan(beta*l). Inductive for l less than lambda/4. Capacitive for l between lambda/4 and lambda/2.
• Open stub: Zin = -j*Z0*cot(beta*l). Capacitive for l less than lambda/4. Inductive for l between lambda/4 and lambda/2.
• ZL = Z0: Zin = Z0 always. Line length has no effect on input impedance.
• GATE trap: Open and shorted stub behaviors are opposite. Shorted stub at l=lambda/4 looks like open circuit at input; open stub at l=lambda/4 looks like short circuit at input.