Home > Articles

This chapter is from the book

7.18 An n-Section Lumped-Circuit Model

The ideal transmission-line circuit element is a distributed element that very accurately predicts the measured performance of real interconnects. Figure 7-37 shows the comparison of the measured and simulated impedance of a 1-inch-long transmission line in the frequency domain. We see the excellent agreement even up to 5 GHz, the bandwidth of the measurement.

Figure 7-37

Figure 7-37 Measured (circles) and simulated (line) impedance of a 1-inch-long, 50-Ohm transmission line. The model is an ideal, lossless transmission line. The agreement is excellent up to the bandwidth of the measurement.

We can approximate this ideal model with a combination of LC lumped-circuit sections. How do we know how many LC sections to use for a given level of accuracy? What happens if we use too few sections?

These questions can be explored using a simulation tool like SPICE. We will first work in the frequency domain to evaluate the impedance by looking into the front end of a transmission line and then interpret this result in the time domain.

In the frequency domain, we can ask What is the impedance when looking into the front end of a transmission line, with the far end open? In this example, we will use a 50-Ohm line that is 6 inches long with a dielectric constant of 4. Its time delay, TD, is 1 nsec.

The total capacitance is given by Ctotal = TD/Z0 = 1 nsec/50 Ohms = 20 pF. The total loop inductance is given by Ltotal = Z0 × TD = 50 Ohms × 1 nsec = 50 nH.

The simplest approximation for a transmission line is a single LC model, with the L and C values as the total values of the transmission line. This is the simplest lumped-circuit model for an ideal transmission line.

Figure 7-38 shows the predicted impedance for an ideal, distributed transmission line and the calculated impedance of a single-section LC lumped-circuit model using these values. In the low-frequency range, the LC model matches the performance really well. The bandwidth of this model is about 100 MHz. The limitation to the bandwidth occurs because, in fact, this ideal transmission line does not have all its capacitance in one place. Instead, it is distributed down the length, and between each capacitor is some loop inductance associated with the length of the sections. However, it is clear from this comparison that a transmission line, open at the far end, will look exactly like an ideal capacitor at low frequency.

Figure 7-38

Figure 7-38 Simulated impedance of an ideal transmission line (circles) and simulated impedance of a single-section LC lumped-circuit model (line). The agreement is excellent up to bandwidth of about 100 MHz.

The impedance of the ideal transmission line shows the resonance peaks occurring when the frequency matches another half wavelength that can fit in the length of the transmission line. The peak resonant frequencies, fres, are given by:


fres = frequency for the peaks in the impedance

m = number of the peak, also the number of half waves that fit in the transmission line

TD = time delay of the transmission line

f0 = frequency at which one complete wave fits in the transmission line

The first resonance for m = 1 is for 1 × 1 GHz/2 = 0.5 GHz. Here, just one half wave will fit evenly in the length of the transmission line, with a TD of 1 nsec. The second resonance for m = 2 is at 2 × 1 GHz/2 = 1 GHz. Here, exactly one wave will fit in the transmission line. These standing wave patterns are shown in Figure 7-39.

Figure 7-39

Figure 7-39 Voltage waves on the transmission line. Resonances occur when an additional half wavelength can fit in the line.

The bandwidth of the single-section LC model is about one-quarter the frequency of the first resonance, up to about 125 MHz. We can improve the bandwidth of the model by breaking the transmission line into more sections. If we break it up in two sections, each section can be modeled as an identical LC model, and the values of each L and C would be Ltotal/2 and Ctotal/2. The predicted impedance of this two-section LC model compared with the ideal T line is shown in Figure 7-40. The bandwidth of this model is roughly at half the first resonance peak. This is a frequency of about 250 MHz.

Figure 7-40

Figure 7-40 Simulated impedance of an ideal transmission line (circles) and simulated impedance of a one-section and two-section LC lumped-circuit model (lines).

We can further increase the bandwidth of this lumped-circuit model by breaking up the length into more LC sections. Figure 7-41 shows the comparison with an ideal transmission line and using 16 different LC sections, with each L and C being Ltotal/16 and Ctotal/16. As we increase the number of sections, we are able to better approximate, to a higher bandwidth, the impedance behavior of an ideal transmission line. The bandwidth of this model is about up to the fourth resonance peak, at 2 GHz.

Figure 7-41

Figure 7-41 Simulated impedance of an ideal transmission line (circles) and simulated impedance of a 16-section LC lumped-circuit model (line).

We can estimate the bandwidth of an n-section lumped-circuit model based on the time delay of this ideal transmission line. These last examples illustrate that the more segments in the LC model, the higher the bandwidth. One section has a bandwidth up to one-quarter of the first resonant peak; 2 sections up to half the first resonance; and 16 sections up to the second resonant frequency. We can generalize that the highest frequency at which we have good agreement—the bandwidth of the model—is:



BWmodel = bandwidth of the n-section lumped-circuit model

n = number of LC sections in the model

TD = time delay of the transmission line

f0 = resonant frequency for one complete wavelength = 1/TD

We have approximated the relationship to be a little more conservative and a little easier to remember by using n = 10 × BWmodel × TD rather than n = 8 × BWmodel × TD.

For example, if the interconnect has a TD = 1 nsec, and we would like an n-section LC model with a bandwidth of 5 GHz, then we need at least n = 10 × 5 GHz × 1 nsec = 50 sections. At this highest frequency, there will be 5 GHz × 1 nsec = 5 wavelengths on the transmission line. For each wave, we need 10sections; therefore, we need 5 × 10 = 50 LC sections for a good approximation.

If the TD of a line is 0.5 nsec, and we need a bandwidth of 2 GHz, the number of sections required is n = 10 × 2 GHz × 0.5 nsec = 10 sections.

We can also evaluate the frequency to which we can use a single LC section to model a transmission line. In other words, up to what frequency does a transmission line look like a simple LC circuit? The bandwidth of one section is:

For the case of a transmission line with TD = 1 nsec, the bandwidth of a single-section LC model for this line is 0.1 × 1/1 nsec = 100 MHz. If TD = 0.16 nsec (roughly 1 inch long), the bandwidth of a simple LC model for this line is 0.1 × 1/0.16 nsec = 600 MHz. The longer the time delay of a transmission line, the lower the frequency at which we can approximate it as a simple LC model.

In evaluating the number of sections we need to describe a transmission line for a required bandwidth, we have found that we needed about 10 LC sections per wavelength of the highest-frequency component of the signal and a total number of LC segments depending on the number of wavelengths of the highest frequency component of the signal that can fit in the transmission line.

If we have a signal with a rise time, RT, the bandwidth associated with the signal (the highest-sine-wave frequency component that is significant) is BWsig = 0.35/RT. If we have a transmission line that has a time delay of TD, and we wish to approximate it with an n-section lumped-circuit model, we need to make sure the bandwidth of the model, BWmodel, is at least > BWsig:


BWsig = bandwidth of the signal

BWmodel = bandwidth of the model

RT = rise time of the signal

TD = time delay of the transmission line

n = minimum number of LC sections needed for an accurate model

For example, with a rise time of 0.5 nsec and time delay of 1 nsec, we would need n > 3.5 × 1/0.5 = 7 sections for an accurate model.

When the rise time is equal to the TD of the line we want to model, we need at least 3.5 sections for an accurate model. In this case, the spatial extent of the rise time is the length of the transmission line. This suggests a very important rule of thumb, given in the following tip.

This rule of thumb is illustrated in Figure 7-42. In FR4, if the rise time is 1 nsec, the spatial extent of the leading edge is 6 inches. We need 3.5 LC sections for every 6 inches of length, or about 1.7 inches per section. We can generalize this: If the rise time is RT, and the speed of the signal is v, then the length for each LC section is (RT × v)/3.5. In FR4, where the speed is about 6 inches/nsec, the length of each LC section required for a rise time, RT, is 1.7 × RT, with rise time in nsec.

Figure 7-42

Figure 7-42 As a general rule of thumb, there should be at least 3.5 LC sections per spatial extent of the rise time for an accurate model of the interconnect at the bandwidth of the signal.

If the rise time is 1 nsec, the length of each single LC should be less than 1.7 inches. If the rise time is 0.5 nsec, the length of each LC section should be no longer than 0.5 × 1.7 = 0.85 inches.

The analysis in this section evaluated the minimum number of many LC sections required to model a real transmission with adequate accuracy up to the bandwidth of the signal. But this is still an approximation with limited bandwidth. This is why the first choice when selecting a model for a real transmission line should always be an ideal transmission line, defined by a characteristic impedance and time delay. Only under very rare conditions, when the question is phrased in terms of the L or C values, should an n-section lumped model ever be used to model a real transmission line. Always start with an ideal transmission-line model.

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.


Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.


If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information

Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.


This site is not directed to children under the age of 13.


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure

Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact

Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice

We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020