Input Impedance of Open and Shorted Lossless Lines

Open-Circuited Line Less than λ/4 in Length. It is assumed that the line is lossless and that the characteristic impedance is pure resistance. The voltage E₀ and the current I₀ at the open-circuited receiving end are in phase. These have been taken as a convenient base in Fig. 10. The initial voltage wave E_is and the initial current wave I_is at the sending end will be displaced φ degress (less than 90° for the line under consideration) as shown. At the distant open end the current component is reflected with a change in phase of 180° and then travels toward the sending end, becoming I_rs. The angles φ and φ' are equal because as much time is required to travel in one direction as the other. The voltage component is reflected without a change in phase, and the reflected component E_rs at the sending end has the position shown. Of course φ" and φ are equal.

The voltage at the sending end is composed of two voltages, the initial component E_is is just entering the line, and the reflected component E_rs is just arriving back from the distant open end. The actual sending-end voltage will accordingly be the vector sum of E_is and E_rs and becomes E_s of Fig. 10. Two current components exist at the sending end, the initial component I_is is just entering the line, and I_rs is just arriving back from the distant open end. The resultant current I_s at the sending end is the vector sum of these two components.

Thus, the sending-end voltage E_s and the sending-end current I_s for a lossless line less than 1/4λ. in length are 90° out of phase, with the current leading the voltage. The input impedance is, therefore, a value of pure capacitive reactance, and the line is equivalent to a capacitor. The numerical value of the reactance or the capacitance can be determined by equation 58 or by equation 61 discussed on page 206.

Open-Circuited Line Greater than λ/4 in Length. If an open-circuited lossless line has a length greater than λ/4, but less than λ/2, Fig. 10 can be extended to cover the conditions. Vectors E_is and I_is will lead vectors E₀ and I₀ by an angle φ greater than 90° because the line length is greater than λ/4. Also, vector I_rs lags vector -I₀ by an angle φ' that is greater than 90°. Furthermore, angle φ" exceeds 90°. If these changes are made on Fig. 10, E_is will be in the second quadrant, E_rs will be in the third quadrant, and the resultant voltage E_s will be opposite in direction to that shown in Fig. 10. Vector I_s will be in the same position, and hence the sending-end voltage E_s will lead the sending-end current I_s by 90°. The input impedance is, therefore, a value of pure inductive reactance, and the line is equivalent to an inductor. The numerical value of the reactance or the inductance can be determined by equation 58 or by equation 61 on page 206.

Figure 10. Vector diagram for an open-circuited lossless line less than λ/4 in length. The input impedance is pure capacitive reactance because Is leads E& by 90°.

Short-Circuited Line Less than λ/4 in Length, A vector diagram for this line is shown in Fig. 11. Vectors E₀,I₀, E_is and I_is are located as previously explained. But, since the line is short circuited, the voltage component is reversed at reflection as shown by -E₀. The reflected component at the sending end is E_rs, and lags -E₀ by the angle φ', and the sending-end voltage is the resultant, E_s. The reflected current component at the sending end is I_rs and lags I₀ by the angle φ". The sending-end current is the resultant I_S, which lags the sending-end voltage E_s by 90°. The input impedance is, therefore, a value of pure inductive reactance, and the line is equivalent to an inductor. The numerical value of the reactance or the inductance can be determined by equation 58 or by equation 65 on page 207.

Short-Circuited Line Greater than λ/4 in Length. If a short-circuited lossless line has a length greater than λ/4, but less than λ/2, Fig. 11 can be extended to give the input-impedance relations. The angles are greater than 90°, and because of this the resultant sending-end current vector will be opposite in direction from that of Fig. 11 and the resultant sending-end voltage vector will remain in the position shown. Thus, the current leads the voltage by 90°, the input impedance is a value of pure capacitive reactance, and the line is equivalent to a capacitor. The numerical value of the reactance or the capacitance can be determined by equation 58 or by equation 65 on page 207.

The relationships discussed in this section are summarized in Fig. 12. In studying these diagrams, line lengths should be measured from the distant end. For instance, an open-circuited line f A in length would have a value of reactance that was "low," and the angle would be lagging, indicating inductive reactance.

Figure 11. Vector diagram for a short-circuited lossless line less than λ/4 in length. The input impedance is pure inductive reactance because Is lags Es by 90°.

Figure 12. Input-impedance characteristics of open-circuited and short-circuited lossless lines of various lengths. For instance, for an open-circuited line slightly less than λ/4 in length, the input impedance is a low value of capacitive reactance (leading), but, if the line is slightly greater than λ/4, the input impedance is a low value of inductive reactance. For the short-circuited line slightly less than λ/4, the input impedance is a large value of inductive reactance (lagging), but, if the line is slightly greater than λ4, the input impedance is a large value of capacitive reactance.

Open-circuited and short-circuited lossless lines often are called resonant lines because their input impedances vary between zero and infinity (Fig. 12). Such lines are extensively used to simulate (Fig. 13) capacitors, inductors, series resonant circuits, and parallel resonant circuits.