Filters
We saw in the last section that the infinite ladder network of Fig. 22–20 absorbs energy continuously if it is driven at a frequency below a certain critical frequency $\sqrt{4/LC}$, which we will call the cutoff frequency $\omega_0$. We suggested that this effect could be understood in terms of a continuous transport of energy down the line. On the other hand, at high frequencies, for $w>\omega_0$, there is no continuous absorption of energy; we should then expect that perhaps the currents don’t “penetrate” very far down the line. Let’s see whether these ideas are right. Suppose we have the front end of the ladder connected to some ac generator and we ask what the voltage looks like at, say, the $754$th section of the ladder. Since the network is infinite, whatever happens to the voltage from one section to the next is always the same; so let’s just look at what happens when we go from some section, say the $n$th to the next. We will define the currents $I_n$ and voltages $V_n$ as shown in Fig. 22–21(a). We can get the voltage $V_{n+1}$ from $V_n$ by remembering that we can always replace the rest of the ladder after the $n$th section by its characteristic impedance $z_0$; then we need only analyze the circuit of Fig. 22–21(b). First, we notice that any $V_n$, since it is across $z_0$, must equal $I_nz_0$. Also, the difference between $V_n$ and $V_{n+1}$ is just $I_nz_1$: \begin{equation*} V_n-V_{n+1}=I_nz_1=V_n\,\frac{z_1}{z_0}. \end{equation*} So we get the ratio \begin{equation*} \frac{V_{n+1}}{V_n}=1-\frac{z_1}{z_0}=\frac{z_0-z_1}{z_0}. \end{equation*} We can call this ratio the propagation factor for one section of the  ladder; we’ll call it $\alpha$. It is, of course, the same for all sections: \begin{equation} \label{Eq:II:22:29} \alpha=\frac{z_0-z_1}{z_0}. \end{equation} The voltage after the $n$th section is then \begin{equation} \label{Eq:II:22:30} V_n=\alpha^n\emf. \end{equation} You can now find the voltage after $754$ sections; it is just $\alpha$ to the $754$th power times $\emf$.  Suppose we see what $\alpha$ is like for the $L$-$C$ ladder of Fig. 22–20(a). Using $z_0$ from Eq. (22.27), and $z_1=i\omega L$, we get \begin{equation} \label{Eq:II:22:31} \alpha=\frac{\sqrt{(L/C)-(\omega^2L^2/4)}-i(\omega L/2)} {\sqrt{(L/C)-(\omega^2L^2/4)}+i(\omega L/2)}. \end{equation} If the driving frequency is below the cutoff frequency $\omega_0=\sqrt{4/LC}$, the radical is a real number, and the magnitudes of the complex numbers in the numerator and denominator are equal. Therefore, the magnitude of $\alpha$ is one; we can write \begin{equation*} \alpha=e^{i\delta}, \end{equation*} which means that the magnitude of the voltage is the same at every section; only its phase changes. The phase change $\delta$ is, in fact, a negative number and represents the “delay” of the voltage as it passes along the network.  For frequencies above the cutoff frequency $\omega_0$ it is better to factor out an $i$ from the numerator and denominator of Eq. (22.31) and rewrite it as \begin{equation} \label{Eq:II:22:32} \alpha=\frac{\sqrt{(\omega^2L^2/4)-(L/C)}-(\omega L/2)} {\sqrt{(\omega^2L^2/4)-(L/C)}+(\omega L/2)}. \end{equation} The propagation factor $\alpha$ is now a real number, and a number less than one. That means that the voltage at any section is always less than the voltage at the preceding section by the factor $\alpha$. For any frequency above $\omega_0$, the voltage dies away rapidly as we go along the network. A plot of the absolute value of $\alpha$ as a function of frequency looks like the graph in Fig. 22–22.  We see that the behavior of $\alpha$, both above and below $\omega_0$, agrees with our interpretation that the network propagates energy for $\omega<\omega_0$ and blocks it for $\omega>\omega_0$. We say that the network “passes” low frequencies and “rejects” or “filters out” the high frequencies. Any network designed to have its characteristics vary in a prescribed way with frequency is called a “filter.” We have been analyzing a “low-pass filter.” You may be wondering why all this discussion of an infinite network which obviously cannot actually occur. The point is that the same characteristics are found in a finite network if we finish it off at the end with an impedance equal to the characteristic impedance $z_0$. Now in practice it is not possible to exactly reproduce the characteristic impedance with a few simple elements—like $R$’s, $L$’s, and $C$’s. But it is often possible to do so with a fair approximation for a certain range of frequencies. In this way one can make a finite filter network whose properties are very nearly the same as those for the infinite case. For instance, the $L$-$C$ ladder behaves much as we have described it if it is terminated in the pure resistance $R=\sqrt{L/C}$. If in our $L$-$C$ ladder we interchange the positions of the $L$’s and $C$’s, to make the ladder shown in Fig. 22–23(a), we can have a filter that propagates high frequencies and rejects low frequencies. It is easy to see what happens with this network by using the results we already have. You will notice that whenever we change an $L$ to a $C$ and vice versa, we also change every $i\omega$ to $1/i\omega$. So whatever happened at $\omega$ before will now happen at $1/\omega$. In particular, we can see how $\alpha$ will vary with frequency by using Fig. 22–22 and changing the label on the axis to $1/\omega$, as we have done in Fig. 22–23(b). The low-pass and high-pass filters we have described have various technical applications. An $L$-$C$ low-pass filter is often used as a “smoothing” filter in a dc power supply. If we want to manufacture dc power from an ac source, we begin with a rectifier which permits current to flow only in one direction. From the rectifier we get a series of pulses that look like the function $V(t)$ shown in Fig. 22–24, which is lousy dc, because it wobbles up and down. Suppose we would like a nice pure dc, such as a battery provides. We can come close to that by putting a low-pass filter between the rectifier and the load.  We know from Chapter 50 of Vol. I that the time function in Fig. 22–24 can be represented as a superposition of a constant voltage plus a sine wave, plus a higher-frequency sine wave, plus a still higher-frequency sine wave, etc.—by a Fourier series. If our filter is linear (if, as we have been assuming, the $L$’s and $C$’s don’t vary with the currents or voltages) then what comes out of the filter is the superposition of the outputs for each component at the input. If we arrange that the cutoff frequency $\omega_0$ of our filter is well below the lowest frequency in the function $V(t)$, the dc (for which $\omega=0$) goes through fine, but the amplitude of the first harmonic will be cut down a lot. And amplitudes of the higher harmonics will be cut down even more. So we can get the output as smooth as we wish, depending only on how many filter sections we are willing to buy. A high-pass filter is used if one wants to reject certain low frequencies. For instance, in a phonograph amplifier a high-pass filter may be used to let the music through, while keeping out the low-pitched rumbling from the motor of the turntable. It is also possible to make “band-pass” filters that reject frequencies below some frequency $\omega_1$ and above another frequency $\omega_2$ (greater than $\omega_1$), but pass the frequencies between $\omega_1$ and $\omega_2$. This can be done simply by putting together a high-pass and a low-pass filter, but it is more usually done by making a ladder in which the impedances $z_1$ and $z_2$ are more complicated—being each a combination of $L$’s and $C$’s. Such a band-pass filter might have a propagation constant like that shown in Fig. 22–25(a). It might be used, for example, in separating signals that occupy only an interval of frequencies, such as each of the many voice channels in a high-frequency telephone cable, or the modulated carrier of a radio transmission. We have seen in Chapter 25 of Vol. I that such filtering can also be done using the selectivity of an ordinary resonance curve, which we have drawn for comparison in Fig. 22–25(b). But the resonant filter is not as good for some purposes as the band-pass filter. You will remember (Chapter 48, Vol. I) that when a carrier of frequency $\omega_c$ is modulated with a “signal” frequency $\omega_s$, the total signal contains not only the carrier frequency but also the two side-band frequencies $\omega_c+\omega_s$ and $\omega_c-\omega_s$. With a resonant filter, these side-bands are always attenuated somewhat, and the attenuation is more, the higher the signal frequency, as you can see from the figure. So there is a poor “frequency response.” The higher musical tones don’t get through. But if the filtering is done with a band-pass filter designed so that the width $\omega_2-\omega_1$ is at least twice the highest signal frequency, the frequency response will be “flat” for the signals wanted. We want to make one more point about the ladder filter: the $L$-$C$ ladder of Fig. 22–20 is also an approximate representation of a transmission line. If we have a long conductor that runs parallel to another conductor—such as a wire in a coaxial cable, or a wire suspended above the earth—there will be some capacitance between the two conductors and also some inductance due to the magnetic field between them. If we imagine the line as broken up into small lengths $\Delta\ell$, each length will look like one section of the $L$-$C$ ladder with a series inductance $\Delta L$ and a shunt capacitance $\Delta C$. We can then use our results for the ladder filter. If we take the limit as $\Delta\ell$ goes to zero, we have a good description of the transmission line. Notice that as $\Delta\ell$ is made smaller and smaller, both $\Delta L$ and $\Delta C$ decrease, but in the same proportion, so that the ratio $\Delta L/\Delta C$ remains constant. So if we take the limit of Eq. (22.28) as $\Delta L$ and $\Delta C$ go to zero, we find that the characteristic impedance $z_0$ is a pure resistance whose magnitude is $\sqrt{\Delta L/\Delta C}$. We can also write the ratio $\Delta L/\Delta C$ as $L_0/C_0$, where $L_0$ and $C_0$ are the inductance and capacitance of a unit length of the line; then we have \begin{equation} \label{Eq:II:22:33} z_0=\sqrt{\frac{L_0}{C_0}}. \end{equation}  You will also notice that as $\Delta L$ and $\Delta C$ go to zero, the cutoff frequency $\omega_0=\sqrt{4/LC}$ goes to infinity. There is no cutoff frequency for an ideal transmission line.