An artificial line can be made from a number of lumped inductors and capacitors by arranging them alternately as shown in the diagram. The circuit can be analyzed in terms of an elementary \(T\) section, also shown. Show that in the low-frequency limit the characteristic impedance of the artificial line of arbitrary length is $$ Z_{0}=\left(\frac{L}{C}\right)^{1 / 2} $$ Artifici. ll liac Prob, \(8,1.12\) and that the speed of propagation, in sections per second, is $$ c^{\prime}=\frac{1}{(L C)^{\mathrm{t} / 3}} $$ Further show that the line does not transmit waves at frequencies greater than the culoff frequency $$ \omega_{c}=\frac{2}{(L C)^{1 / 2}} $$ How do \(Z_{4}\) and \(c^{\prime}\) vary with frequency near \(\omega_{c} ?\) What if each inductor has a small amount of stray capacitance in parallel with it? What if adjacent inductors are coupled by mutual inductance? What practical applications might such circuits have?

When discussing artificial transmission lines, an essential concept is the characteristic impedance, denoted as \(Z_0\). It is the inherent impedance that an infinite artificial transmission line would exhibit such that the line would not reflect any transmitted signals. In simpler terms, it acts as a perfect match between the source and the transmission medium, preventing any signal power from being reflected back to the source.

In the case of an artificial transmission line constructed from lumped elements, which is a simplification of a real transmission line, \(Z_0\) can be determined by analyzing a basic \(T\)-section of the line consisting of a series of inductors and capacitors. At low frequencies, the characteristic impedance is defined as the square root of the quotient of the inductance (\(L\)) over the capacitance (\(C\)), as shown in the exercise:
\[ Z_0 = \left(\frac{L}{C}\right)^{1/2} \]
This simplified representation does not account for more complex behaviors at higher frequencies or the effects of factors such as stray capacitance or mutual inductance between inductors. Adjustments must be considered to accurately describe \(Z_0\) under those varied conditions.

In the domain of electrical engineering, the speed of propagation, denoted \(c'\), is the rate at which an electrical signal travels through a transmission medium such as an artificial transmission line. For an idealized transmission line composed of a series of lumped inductors and capacitors, the speed is not limited by physical material but rather by the configuration and values of these components.

From the problem, we find that the speed of propagation in sections per second for an artificial line can be expressed as a function of the inductance (\(L\)) and capacitance (\(C\)) values of its basic \(T\)-section:
\[ c' = \frac{1}{(LC)^{1/3}} \]
This relationship implies that the speed with which the signal travels through the line is inversely related to both the inductance and the capacitance—higher values of either component result in a slower propagation speed. The speed of propagation is a critical factor for high-speed signal transmission and plays a significant role in the design and analysis of such artificial lines.

Another pivotal concept in the study of transmission lines is the cutoff frequency, often denoted as \(\omega_c\). It represents the maximum frequency at which a signal can be propagated through the line with minimal attenuation. Above this frequency, the transmission line starts to behave like a filter, diminishing the amplitude of signals or not allowing them to pass through at all.

As indicated in the exercise, the cutoff frequency of an artificial line made from lumped inductors and capacitors is obtained by equating the impedance of the inductors and capacitors and solving for the frequency. The resulting equation is:
\[ \omega_c = \frac{2}{(LC)^{1/2}} \]
This cutoff frequency demarcates the boundary between passband and stopband for an artificial line, a concept that is crucial when these lines are used to simulate real transmission line behaviors, or in filter design applications. Understanding \(\omega_c\) helps in predicting how a transmission line will react as the signal frequency approaches and surpasses this critical frequency.