Infinite Ladder Network and Its Applications

The following circuit is the well-known infinite ladder network, constructed by repeatedly adding a series impedance $Z_{\mathrm{s}}$ and a parallel impedance $Z_{\mathrm{p}}$.

The reader may question the usefulness of such a circuit, which appears to possess the sole purpose of nerd-sniping. While this is indeed its common usage, but it does have quite a few practical applications.

Physically, a discrete circuit has a finite number of elements and is governed by Ordinary Differential Equations, while a distributed circuit is continuous and is governed by Partial Differential Equations. An infinite discrete circuit, thus, is a way to approarch PDEs from ODEs. Even when a real circuit is not infinite, often it can still be seen as infinite within a sufficiently short time window.

Infinite networks are the models for some problems in materials science and soil conductivity measurements, for example. At the end of this article, one would see that it’s even the theoretical foundation of all radio communication systems and high-speed digital computers.

Input Impedance

First, consider the finite case of an invented-L section, with a single pair of series $Z_{\mathrm{s}}$ and parallel $Z_{\mathrm{p}}$. This is a voltage divider, its input impedance is given by:

$$Z_{\mathrm{unit}}=Z_{\mathrm{s}}+Z_{\mathrm{p}}$$

Viewing the infinite ladder from left to right, one can see that the next inverted-L section is connected at the mid-point of the previous section. Thus, a voltage divider is nested within another voltage divider, ad infinitum. The network’s input impedance corresponds to a continued fraction in the following form:

$$\begin{array}{r}{Z}_{\text{in}}(Z_{\text{s}},Z_{\text{p}})=Z_{\mathrm{s}}+\frac{1}{\frac{1}{Z_{\mathrm{p}}}+\frac{1}{Z_{\mathrm{s}}+\frac{1}{\frac{1}{Z_{\mathrm{p}}}+\cdots }}}\end{array}$$

To make this formula cleaner, define admittance $Y_{\mathrm{p}}=\frac{1}{Z_{\mathrm{p}}}$ rather than using the original impedance for shunt elements:

$$\begin{array}{r}{Z}_{\text{in}}(Z_{\text{s}},Y_{\text{p}})=Z_{\mathrm{s}}+\frac{1}{Y_{\mathrm{p}}+\frac{1}{Z_{\mathrm{s}}+\frac{1}{Y_{\mathrm{p}}+\cdots }}}\end{array}$$

To obtain the analytical expression for input impedance, mathematicians and physicists simply need to evaluate the limit of this infinite series. But most engineers are too impatient to do that, instead, an alternative method called “hand-waving” yields quick and dirty answers.

Let $Z_{\mathrm{in}}$ (labeled $Z_{\mathrm{it}}$) in the diagram be the input impedance of the original infinite ladder network, now we apply a small modification to this existing network: install a new invented-L section (removing one also works). This creates a new circuit with a finite voltage divider of $Z_{\mathrm{s}}$ and $Z_{\mathrm{p}}$, its center tap is connected to the modified infinite ladder network. Now, note that the infinite ladder network is infinite, its impedance does not change in spite of installing or removing a unit. As a result, its input impedance before and after modification are both $Z_{\text{in}}$ equally!

Thus:

$$\begin{array}{rl}{Z}_{\text{in}}& =Z_{\text{s}}+(Z_{\text{p}}\parallel Z_{\text{in}})\end{array}$$

After expansion:

$$\begin{array}{rl}{Z}_{\text{in}}& =Z_{\text{s}}+\frac{1}{\frac{1}{Z_{\text{p}}}+\frac{1}{Z_{\text{in}}}}\\ Z_{\text{in}}^2& =Z_{\text{s}}\cdot (Z_{\text{p}}+Z_{\text{in}})\end{array}$$

Solving this quadratic equation yields two roots:

$$\boxed{{\displaystyle \begin{array}{rl}{Z}_{\text{in+}}(Z_{\text{s}},Z_{\text{p}})& =\frac{Z_{\text{s}}+\sqrt{Z_{\text{s}}^2+4Z_{\text{s}}{Z}_{\text{p}}}}{2}\\ Z_{\text{in-}}(Z_{\text{s}},Z_{\text{p}})& =\frac{Z_{\text{s}}-\sqrt{Z_{\text{s}}^2+4Z_{\text{s}}{Z}_{\text{p}}}}{2}\end{array}}}$$

Usually, only the positive root is taken. The negative root requires active components with negative impendace, and is usually discarded as it’s not realizable with passive components.

If infinite networks don’t appear to be physically meaningful, a finite physical interpretation for $Z_{\mathrm{in}}$ also exists. In a finite network, connect a load in parallel at the right side and adjust its impedance iteratively. When the input impedance is exactly equal to the load impedance, this impedance is known as the iterative impedance ($Z_{\mathrm{it}}$). When a new network with the same iterative impedance is chained to an existing network, the input impedance remains constant. This is how the finite case can be generalized to the infinite case. Another related concept is image impedance, which is linked at the end of the article for interested readers.

Example 1: 1 Ω resistors

As a special case, if the ladder network is purely resistive and contains only 1 Ω resistors:

$$\begin{array}{rl}{Z}_{\text{in+}}(R,R){|}_{R=1}& =\frac{R+\sqrt{5R^2}}{2}\\ & =\phi R\\ & =\phi \end{array}$$

where $\phi$ is the golden ratio:

$$\begin{array}{r}\phi =\frac{1+\sqrt{5}}{2}\approx 1.618033988749\end{array}$$

At the same time, note that if one plugs 1 into the original continued fraction, it would immediately yield the definition of the golden ratio:

$$\begin{array}{rl}{Z}_{\text{in}}(1,1)& =1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots }}}\\ & =\phi \end{array}$$

In hindsight, the fact that the 1 Ω infinite ladder has $\phi$ Ω of input resistance, is obvious.

Example 2: $\sqrt{2}$ A current sink

Assume the ladder network is purely resistive again, set Zs = 2 Ω, Zp = 1/2 Ω, we have:

$$\begin{array}{r}{Z}_{\text{in+}}(2\mathrm{\Omega },\frac{1}{2}\mathrm{\Omega })=(1+\sqrt{2})\mathrm{\Omega }\end{array}$$

It’s not difficult to see that the resulted continued fraction contains a direct definition of $\sqrt{2}$, so the answer is again, obvious in hindsight.

$$\begin{array}{rl}{Z}_{\text{in+}}(2,\frac{1}{2})& =1+1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\cdots }}}\\ & =1+\sqrt{2}\end{array}$$

If one connects a 1 Ω resistor in parallel with this infinite ladder:

$$\begin{array}{rl}{Z}_{\text{total}}& =\frac{1}{1+\sqrt{2}}\parallel 1\\ & =\frac{1}{\frac{1}{1+\sqrt{2}}+1}\\ & =\frac{\sqrt{2}}{2}\mathrm{\Omega }\end{array}$$

After apply a voltage of 1 V, we obtain a $\sqrt{2}$ A current sink.

$$\begin{array}{r}I=\frac{U}{Z_{\text{total}}}=\frac{1}{1/\sqrt{2}}=\sqrt{2}\text{ A}\end{array}$$

This was my original motivation of researching this circuit, the nerd-sniping was working great…

Example 3: Reactive Ladder

If an infinite ladder network uses series inductors, parallel capacitors, and no resistors, what is its input impedance?

Assume that every component of this infinite ladder is infinitesimal, and keep using the analysis method called “hand-waving”, we decide that Zs has no contribution to the final result. Meanwhile, because Zs and Zp are inversely proportional to each other (inductor vs. capacitor), this ratio should remain constant.

This would simplify the impedance equation to:

$$\begin{array}{rl}{Z}_{\text{in+}}(Z_{\text{s}},Z_{\text{p}})& =\frac{Z_{\text{s}}+\sqrt{Z_{\text{s}}^2+4Z_{\text{s}}{Z}_{\text{p}}}}{2}\\ Z_{\text{in+}}(Z_{\text{s}},Z_{\text{p}})& \approx \frac{\cancel{Z_{\text{s}}}+\sqrt{\cancel{Z_{\text{s}}^2}+4Z_{\text{s}}{Z}_{\text{p}}}}{2}\approx \frac{\sqrt{4Z_{\mathrm{s}}{Z}_{\mathrm{p}}}}{2}\\ & \approx \sqrt{Z_{\text{s}}{Z}_{\mathrm{p}}}\end{array}$$

Plugging the impedances of inductors and capacitors into the equation, yields:

$$\begin{array}{rl}{Z}_{\text{in+}}(j\omega L,\frac{1}{j\omega C})& \approx \sqrt{Z_{\text{s}}{Z}_{\mathrm{p}}}=\sqrt{\frac{j\omega L}{j\omega C}}=\sqrt{\frac{L}{C}}\end{array}$$

This famous equation is the characteristic impedance of a lossless transmission line, of which its analysis is one of the most celebrated results in the history of electronics engineering.

In a data cable or signal line, the characteristic impedance represents the ratio between voltage and current at the moment a signal voltage is applied to the line. Transmission line theory is of paramount importance in electronics, as it’s the foundation of all radio communication and high-speed digital circuits. High-speed data interfaces such as USB, PCIe, DDR are all operating above 1 GHz. In this regime, the circuits are no longer lumped circuits that are solely described by voltages and currents, and are instead distributed circuits with electromagnetic waves. For circuits in these applications, one must follow the same strict guidelines for RF/microwave electronics. In other words, the principles of transmission lines.

A strange feature of the lossless transmission line is that, it’s a reactive circuit, but its impedance doesn’t change with frequency as the terms $\omega$ cancel out. An even more bizarre fact is that the imaginary unit $j$ is also canceled out, making the impedance a real number. So a line with $L:C=2:1$ has an input resistance of $\sqrt{2}$. This brings up the transmission line paradox: a circuit with only inductors and capacitors behaves like a resistor, isn’t it a violation of energy conservation? The answer is that the circuit only becomes reactive when an electromagnetic wave is reflected from the far end of the line. Before reflection occurs, within a short time window, it’s truly resistive. Another way is to say that information about what is located at the other end of the transmission line propagates no faster than the speed of light inside the dielectrics. In an infinite line, its length is infinite, so reflection never occurs. Conversely, one can simulate an infinite transmission line with a finite line by connecting a termination resistor matched to its characteristic impedance. This is the principle of impedance matching for high-speed digital circuits.

The analysis of transmission lines first appeared in the Telegrapher’s equations derived by Oliver Heaviside, one of the greatest physicists of the late 19th century. His results of the lossless LC and lossy RLCG transmission lines were improvements to Lord Kelvin’s earlier result of lossy RC transmission line, making the theory complete. (Kelvin’s result didn’t consider inductance, which becomes crucial for high-frequency signaling).

Conclusion

Here, with only an infinite ladder network and some industrial-grade hand-waving, we’ve derived the characteristic impedance of the lossless transmission line without using Telegrapher’s equations or Partial Differential Equations at all. Hence, I’ve shown that infinite networks are not merely math puzzles, but are also a practical tool for solving some engineering problems.

Appendix 0: See Also

• Wikipedia: Image Impedance

• Wikipedia: Iterative Impedance

• The Feynman Lectures on Physics, Vol. II, Chapter 22: AC Circuits, Section 6: A ladder network

Only a week after I finished writing this note using other EE textbooks as references, I realized that I completely missed its explanation in the all-time classic The Feynman Lectures on Physics, in Section 22–6: A ladder network. It could’ve saved me much time… To ensure future readers won’t miss Feynman like I did, I’ve added this citation in several Wikipedia articles.

• Lee, Thomas H. Planar microwave engineering: a practical guide to theory, measurement, and circuits. Cambridge university press, 2004.

Excellent textbook, both a fun read and a nice collection of tutorials on different aspects of RF and microwaves. The infinite ladder network analysis in this blog post is based on Chapter 2, Section 5: Driving-Point Impedance of Iterated Structures. The book also derives other transmission line properties via the same method.

• Bird, John. Electrical Circuit Theory and Technology. Revised second edition. Newnes, 2003.

Most textbooks derive the transmission line equations from Telegrapher’s Equations, this is one of the few books (along with Lee) that gives a derivation based on the infinite ladder network, see Part 3: Advanced Circuit Theory and Techonology, Section 44: Transmission Lines. Also see Section 41: Characteristic Impedance, which derives the characteristic impedance of a finite symmetrical T-network, which was reused in the later chapter to derive infinite transmission lines.

• Niknejad, Ali M. Electromagnetics for high-speed analog and digital communication circuits. Cambridge University Press, 2007.

This is yet another book that introduces the transmission line in terms of an infinite ladder network. See Chapter 9: Transmission line, Section 9.2: An Infinite Ladder Network. A preview of this section is available online.