# 2.8 Special Cases of the Lossless Line

We often encounter situations involving lossless transmission lines with particular terminations or lines whose lengths lead to particularly useful line properties. We now consider some of these special cases.

# 2.8.1 Short-Circuited Line

The transmission line shown in Fig. 2-19(a) is terminated in a short circuit, Z L=0. Consequently, the voltage reflection coefficient defined by Eq. (2.59) is Γ=−1, and the voltage standing-wave ratio given by Eq. (2.73) is S=∞. With z=−d and Γ=−1 in Eqs. (2.63a) and (2.63b), and Γ=−1 in Eq. (2.74), the voltage, current, and wave impedance on a short-circuited lossless transmission line are given by

V˜sc(d)=V 0+[e j β d−e−j β d]=2 j V 0+ sin β d,(2.83a)
I˜sc(d)=V 0+Z 0[e j β d+e−j β d]=2 V 0+Z 0 cos β d,(2.83b)
Z sc(d)=V˜sc(d)I˜sc(d)=j Z 0 tan β d.(2.83c)

The voltage V˜sc(d) is zero at the load (d=0), as it should be for a short circuit, and its amplitude varies as sin β d. In contrast, the current I˜sc(d) is a maximum at the load, and it varies as cos β d. Both quantities are displayed in Fig. 2-19 as a function of d.

# Figure 2-19 Transmission line terminated in a short circuit: (a) schematic representation, (b) normalized voltage on the line, (c) normalized current, and (d) normalized input impedance.

Figure 2 19-a Full Alternative Text
Figure 2 19-b-c Full Alternative Text

Denoting Z in sc as the input impedance of a short-circuited line of length l,

A plot of Z in sc/j Z 0 versus l is shown in Fig. 2-19(d). For the short-circuited line, if its length is less than λ/4, its impedance is equivalent to that of an inductor, and if it is between λ/4 and λ/2, it is equivalent to that of a capacitor.

In general, the input impedance Z in of a line terminated in an arbitrary load has a real part, called the input resistance R in, and an imaginary part, called the input reactance X in:

Z in=R in+j X in.(2.85)

In the case of the short-circuited lossless line, the input impedance is purely reactive (R in=0). If tan β l≥0, the line appears inductive to the source, acting like an equivalent inductor L eq whose impedance equals Z in sc. Thus,

j ω L eq=j Z 0 tan β l,if tan β l≥0,(2.86)

or

L eq=Z 0 tan β l ω(H).(2.87)

The minimum line length l that would result in an input impedance Z in sc equivalent to that of an inductor with inductance L eq is

l=1 β tan−1 ω L eq Z 0(m).(2.88)

Similarly, if tan β l≤0, the input impedance is capacitive, in which case the line acts like an equivalent capacitor with capacitance C eq such that

1 j ω C eq=j Z 0 tan β l,if tan β l≤0,(2.89)

or

C eq=−1 Z 0 ω tan β l(F).(2.90)

Since l is a positive number, the shortest length l for which tan β l≤0 corresponds to the range π/2≤β l≤π. Hence, the minimum line length l that would result in an input impedance Z in sc equivalent to that of a capacitor of capacitance C eq is

l=1 β π−tan−1 1 ω C eq Z 0(m).(2.91)

Such a practice is indeed common in the design of microwave circuits and high-speed integrated circuits, because making an actual capacitor or inductor often is much more difficult than fabricating a microstrip transmission line with a shorted end on a circuit board.

# Example 2-8: Equivalent Reactive Elements

Choose the length of a shorted 50 Ω lossless transmission line (Fig. 2-20) such that its input impedance at 2.25 GHz is identical to that of a capacitor with capacitance C eq=4 pF . The wave velocity on the line is 0.75 c.

# Figure 2-20 Shorted line as an equivalent capacitor (Example 2-8).

Figure 2 20 Full Alternative Text

Solution: We are given

u p=0.75 c=0.75×3×10 8=2.25×10 8 m/s,Z 0=50 Ω,f=2.25 GHz=2.25×10 9 Hz,C eq=4 pF=4×10−12 F.

The phase constant is

β=2 π λ=2 π f u p=2 π×2.25×10 9 2.25×10 8=62.8(rad/m).

From Eq. (2.89), it follows that

tan β l=−1 Z 0 ω C eq=−1 50×2 π×2.25×10 9×4×10−12=−0.354.

The tangent function is negative when its argument is in the second or fourth quadrants. The solution for the second quadrant is

β l 1=2.8 rad or l 1=2.8 β=2.8 62.8=4.46 cm,

and the solution for the fourth quadrant is

β l 2=5.94 rad or l 2=5.94 62.8=9.46 cm.

We also could have obtained the value of l 1 by applying Eq. (2.91). The length l 2 is greater than l 1 by exactly λ/2. In fact, any length l=4.46 cm+n λ/2, where n is a positive integer, also is a solution.

# 2.8.2 Open-Circuited Line

With Z L=∞, as illustrated in Fig. 2-21(a), we have Γ=1 and S=∞, and the voltage, current, and input impedance are given by

V˜oc(d)=V 0+[e j β d+e−j β d]=2 V 0+cos β d,(2.92a)
I˜oc(d)=V 0+Z 0[e j β d−e−j β d]=2 j V 0+Z 0 sin β d,(2.92b)

Plots of these quantities are displayed in Fig. 2-21 as a function of d for the voltage and current and as a function of l for the input impedance.

# Figure 2-21 Transmission line terminated in an open circuit: (a) schematic representation, (b) normalized voltage on the line, (c) normalized current, and (d) normalized input impedance.

Figure 2 21-a Full Alternative Text
Figure 2 21-b-c Full Alternative Text

# 2.8.3 Application of Short-Circuit/Open-Circuit Technique

A network analyzer is a radio-frequency (RF) instrument capable of measuring the impedance of any load connected to its input terminal. When used to measure (1) Z in sc, which is the input impedance of a lossless line terminated in a short circuit, and (2) Z in oc, which is the input impedance of the line when terminated in an open circuit, the combination of the two measurements can be used to determine the characteristic impedance of the line Z 0 and its phase constant β. Indeed, the product of the expressions given by Eqs. (2.84) and (2.93) gives

and the ratio of the same expressions leads to

Because of the π phase ambiguity associated with the tangent function, the length l should be less than or equal to λ/2 to provide an unambiguous result.

# Example 2-9: Measuring Z0 and β

Find Z 0 and β of a 57-cm long lossless transmission line whose input impedance was measured as Z in sc=j 40.42 Ω when terminated in a short circuit and as Z in oc=−j 121.24 Ω when terminated in an open circuit. From other measurements, we know that the line is between 3 and 3.25 wavelengths long.

Solution: From Eqs. (2.94) and (2.95),

Z 0=Z in sc Z in oc+=(j 40.42)(−j 121.24)=70 Ω,tan β l=−Z in sc Z in oc=1 3.

Since l is between 3 λ and 3.25 λ, β l=(2 π l/λ) is between 6 π radians and (13 π/2) radians. This places β l in the first quadrant (0 to π/2) radians. Hence, the only acceptable solution for tan β ℓ=1/3 is β l=π/6 radians. This value, however, does not include the 2 π multiples associated with the integer λ multiples of l. Hence, the true value of β l is

β l=6 π+π 6=19.4(rad),

in which case

β=19.4 0.57=34(rad/m).

# 2.8.4 Lines of Length l = nλ/2

If l=n λ/2, where n is an integer,

tan β l=tan 2 π/λ n λ/2=tan n π=0.

Consequently, Eq. (2.79) reduces to

which means that a half-wavelength line (or any integer multiple of λ/2) does not modify the load impedance.

# 2.8.5 Quarter-Wavelength Transformer

Another case of interest is when the length of the line is a quarter-wavelength (or λ/4+n λ/2, where n=0 or a positive integer), corresponding to β l=(2 π/λ)(λ/4)=π/2. From Eq. (2.79), the input impedance becomes

The utility of such a quarter-wave transformer is illustrated by Example 2-10.

# Example 2-10: λ/4 Transformer

A 50 Ω lossless transmission line is to be matched to a resistive load impedance with Z L=100 Ω via a quarter-wave section as shown in Fig. 2-22, thereby eliminating reflections along the feedline. Find the required characteristic impedance of the quarter-wave transformer.

# Figure 2-22 Configuration for Example 2-10.

Figure 2 22 Full Alternative Text

Solution: To eliminate reflections at terminal A A′, the input impedance Z in looking into the quarter-wave line should be equal to Z 01, which is the characteristic impedance of the feedline. Thus, Z in=50 Ω. From Eq. (2.97),

Z in=Z 02 2 Z L,

or

Z 02=Z in Z L=50×100=70.7 Ω.

Whereas this eliminates reflections on the feedline, it does not eliminate them on the λ/4 line. However, since the lines are lossless, all of the power incident on A A′ will end up getting transferred into the load Z L.

# 2.8.6 Matched Transmission Line: ZL = Z0

For a matched lossless transmission line with Z L=Z 0, (1) the input impedance Z in=Z 0 for all locations d on the line, (2) Γ=0, and (3) all of the incident power is delivered to the load, regardless of the line length l. A summary of the properties of standing waves is given in Table 2-4.

# Table 2-4 Properties of standing waves on a lossless transmission line.

Table 2 4 Full Alternative Text

Concept Question 2-10: What is the difference between the characteristic impedance Z 0 and the input impedance Z in? When are they the same?

Concept Question 2-11: What is a quarter-wave transformer? How can it be used?

Concept Question 2-12: A lossless transmission line of length l is terminated in a short circuit. If l<λ/4, is the input impedance inductive or capacitive?

Concept Question 2-13: What is the input impedance of an infinitely long line?

Concept Question 2-14: If the input impedance of a lossless line is inductive when terminated in a short circuit, will it be inductive or capacitive when the line is terminated in an open circuit?