Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 56

The time‐harmonic instantaneous voltage in the frequency domain, i.e. in the phasor form, is written as

(2.1.27)

where “Re” stands for the real part of the voltage phasor . The voltage phasor is given by the following expression:

(2.1.28)

The phasor is nothing but a polar form of a complex quantity. Likewise, the instantaneous current in the phasor form is

(2.1.29)

where current phasor is

(2.1.30)

The phasor is either a constant or a function of only the space variable. It is not a function of time t. The phasor is shown with a tilde (~) sign in this chapter. However, in the subsequent chapters, the tilde (~) sign is dropped. The phasor is used at a single frequency. Using the phasor notation, the voltage across R, L, and the current through C, G; given by equations (2.1.12)–(2.1.15) in the time domain, can be rewritten in the frequency‐domain:

(2.1.31)

In the above equations, the time derivative ∂/∂t is replaced by jω converting the expression from the time‐domain to frequency‐domain. Following the conversion process, the time‐domain coupled voltage‐current transmission line equation (2.1.20), is rewritten in the frequency‐domain:

(2.1.32)

On separation of the voltage and current variables, the following voltage and current wave equations are obtained in the frequency‐domain:

(2.1.33)

It is noted that the second‐order partial differential wave equations in the time‐domain are converted to the second‐order ordinary differential equations in the frequency‐domain. The factors at the right‐hand side of the above equations help to define a secondary parameter γ, known as the complex propagation constant of a transmission line:

(2.1.34)

where γ = α + jβ. The real part α(Np/m) of the complex propagation constant γ is called the attenuation constant and the imaginary part β (rad/sec) is the propagation constant of a lossy transmission line. The parameter β is also known as the phase‐shift constant or phase constant. On separating the real and imaginary parts of the above equation, the following expressions are obtained:

(2.1.35)

(2.1.36)

The attenuation constant α, and propagation constant β are given in terms of the primary line constants, R, L, C, G. Normally α and β are frequency‐dependent. Thus, the phase velocity of both the current and voltage waves, given by v_p = ω/β is frequency‐dependent. This kind of transmission line is known as the dispersive transmission line. A complex wave traveling on a lossy dispersive line gets distorted as each component of the complex wave travels with different phase velocity.

Using the complex propagation constant, the wave equation (2.1.33a and b) are rewritten as

(2.1.37)

The above homogeneous wave equations on a transmission line could be treated as a boundary value problem to get the voltage and current at any location on the line. If a transmission line is infinitely long and excited from one end, then the voltage and current waves on the line always move in the forward direction without any reflection. At any location on the line, the voltage and current are related by another secondary parameter called “the characteristic impedance” of a transmission line:

(2.1.38)

The characteristic impedance of a transmission line could be viewed as a mechanism that explains the wave propagation on a line. It recasts the Huygens' Principle of the secondary wave formation in terms of the characteristic impedance. The characteristic impedance could be called the Huygens' load. It is an unusual load impedance with a special property. It absorbs power from the source and itself becomes a secondary source for the further transmission of power in the form of wave motion. In this manner, the wave on a transmission line moves; as the characteristic impedance, i.e. Huygens' load, acts both as a load and also as a source of the wave motion. The process is similar to the Huygens' secondary source for the wavefront propagation. The concept of the Huygens' load is further extended to engineer the Huygens' metasurface with unique characteristics to control the reflected and transmitted (refracted) EM‐waves. It is discussed in subsection (22.5.2) of chapter 22.

The characteristic impedance, i.e. Huygens' load of a lossy line is a complex quantity. Its real part does not dissipate energy like the real part of a normal complex load. The imaginary part of Huygens's load indicates the presence of losses in a transmission line, whereas in the case of a normal complex load its imaginary part shows the storage of the reactive energy. For a lossless transmission line, Huygens' load is a real quantity that is nondissipative. Huygens adopted the secondary source model to explain the propagation of the light wave in the space [B.3, B.4]. The expression for the characteristic impedance of a line is obtained from equation (2.1.32).

(2.1.39)

In general, the characteristic impedance of a lossy transmission line is a complex quantity. However, for a lossless line, the lossy elements are zero, i.e. R = G = 0. It leads to the following expressions:

(2.1.40)

There is no attenuation in the propagating wave on a lossless line. If the line inductance L and the line capacitance C are frequency‐independent, the transmission line is nondispersive. The characteristic impedance is a real quantity. The line parameters such as the attenuation constant (α), propagation constant (β), and characteristic impedance (Z₀) are known as the secondary parameters of a transmission line. These secondary parameters are finally expressed in terms of the physical geometry and the electrical characteristics of material medium of a line. A microwave circuit designer is more interested in these secondary parameters as compared to the primary line constants, RLCG, of a transmission line. The secondary parameters are more conveniently measured and numerically computed for a large class of the transmission line structures. For any practical transmission line, the losses are always present on a line.