Understanding the Transmission Line Theory

This section:

Lossless Transmission Line Model

As a signal propagates down the pair of conductors, each new section acts electrically as a small lumped circuit element. In its simplest form, called the lossless model, the equivalent circuit of a transmission line has just inductance and capacitance. These elements are distributed uniformly down the length of the line, as shown in Equivalent Circuit Model of a Lossless Transmission Line.

Figure C-44: Equivalent Circuit Model of a Lossless Transmission Line

From this electrical circuit model, the two important terms that characterize a transmission line can be derived: the velocity of a signal (v) and the characteristic impedance (Z0).




LL = inductance per length

CL = capacitance per length

This is the basis for the T Element used in Star-Hspice. It accounts for a characteristic impedance (Z0) and a time delay (TD). The time delay depends on the distance (d) between the two ends of the transmission line:


Lossy Transmission Line Model

When loss is significant, the effects of the series resistance (R) and the dielectric conductance (G) should be included. Equivalent Circuit Model of a Lossy Transmission Line shows the equivalent circuit model of a lossy transmission line, with distributed "lumps" of R, L, and C Elements.

Figure C-45: Equivalent Circuit Model of a Lossy Transmission Line

The U Element used in Star-Hspice is the equivalent circuit model for the lossy transmission line. In a transient simulation, the U Element automatically accounts for frequency-dependent characteristic impedance, dispersion (frequency dependence in the velocity), and attenuation.

The most common types of transmission line cross sections are microstrip, stripline, coax, wire over ground, and twisted pair. There is no direct relationship between cross section, velocity of propagation, and characteristic impedance.

In a balanced transmission line, the two conductors have similar properties and are electrically indistinguishable. For example, each wire of a twisted pair has the same voltage drop per length down the line The circuit model for each wire has the same resistance capacitance and inductance per length.

This is not the case with a microstrip line or a coaxial cable. In those structures, the signal conductor has a larger voltage drop per length than the other conductor. The wide reference plane in a microstrip or the larger diameter shield in a coax have lower resistance per length and lower inductance per length than the signal line. The equivalent circuit model for unbalanced lines typically assumes the resistance and inductance per length of the ground path is zero and all the voltage drop per length is on the signal conductor. Even though the inductance of the reference plane is small, it can play a significant role when there are large transient currents.


The impedance of a device (Z) is defined as the instantaneous ratio of the voltage across the device (V) to the current through it:


Impedance of Simple Lumped Elements

The impedance of a device can be thought of as the quality of the device that causes it to transform a current through it into a voltage across it:


The admittance (Y) is less often used to characterize a device. It is the inverse of the impedance:


There are three ideal circuit elements used to describe passive components: a resistor, a capacitor, and an inductor. They are defined by how they interact with voltage across them and current though them:

Resistor, with resistance (R):


Capacitor, with capacitance (C):


Inductor with inductance (L):


When the voltage or current signals are time dependent, the impedance of a capacitive or inductive element is a very complicated function of time. You can simulate it with Star-Hspice, but it is difficult to build an intuitive model.

The impedance of a capacitor rotates the phase of the current 90° in the negative or direction to generate the voltage across the capacitor. The impedance of an inductor rotates the current 90° in the positive direction to generate the voltage across the inductor. For a resistor, the current and voltage have the same phase.

In the frequency domain, when all signals are sine waves in the time domain, the impedance of a capacitor and an inductor is frequency dependent, decreasing with frequency for a capacitor and increasing with frequency for an inductor. The impedance of a resistor is constant with frequency.

In the real world of finite dimensions and engineered materials, ideal circuit elements have parasitics associated with them, which cause them to behave in complex ways that are very apparent at high frequencies.

Characteristic Impedance

A controlled impedance transmission line is a pair of conductors that have a uniform cross section and uniform distribution of dielectric materials down their length. A short segment, x, of the transmission line has a small capacitance associated with it, C, which is the capacitance per length, CL, times the x:


When a voltage signal is introduced at one end, the voltage between the conductors induces an electric that propagates the length of the line at the speed of light in the dielectric. As the voltage signal moves down the line, each new section of line charges up. The new section of line, x, is charged up in a time t:


If the voltage (V) moves down the line at a constant speed and the capacitance per length is uniform throughout the line, then the constant voltage applied to the front end draws a constant charging current (I):


This constant voltage with constant current has the behavior of a constant impedance (Z):


The impedance is determined by the speed of the signal and the capacitance per length of the pair of conductors, both intrinsic properties of the line. This intrinsic impedance is termed the characteristic impedance of the line (Z0).

If a measurement is made at one end of the line in a short time compared to the round trip time delay, the line behaves like a resistor with a resistance equal to the characteristic impedance of the line. Transmission line effects are only important when rise times are comparable or shorter than the round trip time delay.

For example, if the rise time of a device is 1 ns, and it drives an interconnect trace in FR4 which is longer than three inches, the load on the device during the risetime is purely resistive. For CMOS devices, which are used to drive high resistance loads, the typical 50 ohm resistance they see initially can significantly distort the waveform from what is expected.

It is only during the initial surge of the voltage that a transmission line behaves as a constant impedance, with a value equal to its characteristic impedance. For this reason the characteristic impedance of a line is also called the surge impedance. The surge time during which the impedance is constant is the round trip time of flight, or twice the time delay. Reflections from the far end complicate the electrical behavior of the line after the surge time.

The instantaneous impedance measured at the front end of a transmission line is a complicated function of time. It depends on the nature of the terminations at the far end. When the line is shunted to ground with a resistor of value equal to the characteristic impedance of the line, there is no reflection back, and the front end of the line behaves as a resistive load. When the termination at the far end is open, the impedance at the front end starts out at the characteristic impedance and eventually, after multiple reflections, approaches an infinite impedance. During some periods the instantaneous impedance may be zero. These transient effects are fully simulated with T Elements and U Elements in Star-Hspice.


Mutual Inductance and Self Inductance

The most confusing, subtle and important parameter in high-speed packaging and interconnect design is inductance. It plays a key role in the origin of simultaneous switching noise, also called common ground inductance, and a key role in crosstalk between transmission line structures.

Operational Definition of Inductance

Consider an inductor to be any section of circuit element which carries current: an interconnect trace, a ground plane, a TAB lead frame, a lead in a DIP package, the lead of a resistor or a pin in a connector. An inductor does not have to be a closed circuit path, but can be a small section of a circuit path.

A changing current passing through an inductor generates a voltage drop. The magnitude of the voltage drop ( V) for an inductance (L) and change in current (dI/dt) is:


This definition can always be used to evaluate the inductance of a section of a circuit. For example, with two long parallel wires, each of radius (r) and a center-to-center separation (s), you can measure the voltage drop per length for one of the wires when a changing current dI/dt flows through one wire and back through the other. The induced voltage per length on one of the wires is:

[V in mV/inch, l in mA, t in ns]

From this expression, the effective inductance per length of one wire is found to be:


Mutual Inductance

A second effect also is important: the induced voltage from currents that are adjacent to, but not in, the same circuit path. This is caused by the mutual inductance between two current elements. A section of conductor in a circuit, labeled 1, may have an induced voltage generated across it because of currents not in circuit 1, but from circuits 2, 3, and 4.

The voltage generated across the section of circuit 1, V1, is given in its general form by:


The notation for mutual inductance (Lab) is related to the induced voltage on circuit element a, from the current element, b. In some texts, the symbol used is M, rather than L. The special case of the induced voltage on a circuit element from its own current (Laa) is called self inductance.

Mutual inductance relates to the magnitude of induced voltage from an adjacent current. The magnitude of this voltage depends on the flux linkages between the two circuit elements.

Self Inductance

The self inductance of an isolated single trace is a well-defined, absolute mathematical quantity, but not a measurable physical quantity. There is always a return current path somewhere, and the mutual inductance from this return current path induces a voltage on the circuit element that subtracts from the self inductance. Self inductance can never be measured or isolated, independent of a mutual inductance of a return current path.

In the example above of two long parallel wires, the measured inductance per length (LL) of one wire is neither the self inductance nor the mutual inductance of the wire. It is a combination of these two terms. If the universe contained just the two wires, the measured voltage drop per length would be:


The minus sign reflects the opposite directions of the currents I1 and I2. Operationally, when the inductance per length of one wire is measured, what is really being measured is the difference between its self inductance and the mutual inductance of the return path. Because of this effect, it is clear that the nature of the return path greatly influences the measured inductance of a circuit element.

Reference Plane Return Paths

The capacitance per length (CL) of any planar transmission line is:


The inductance per length of the signal line (LL) is:


This is the self inductance of the signal line, minus the mutual inductance of the return current in the reference plane.

For example, the inductance per length of a transmission line with characteristic impedance of 50 ohms in an FR4 printed circuit board is 9.6 nH/inch. The capacitance per length is 3.8 pF/inch. In the equivalent circuit of a lossless transmission line, the series inductance per length is 9.6 nH/inch, and the shunt capacitance to ground is 3.8 pF/inch.

In the notation of the U Element used in Star-Hspice, for an ELEV=2 (RCLK equivalent model) and a PLEV=1 (microstrip cross section), the parameters to model this lossless transmission line are

C11 = 3.8 pF/inch, L11 = 9.6 nH/inch

In the LLEV=0 parameter, Star-Hspice simplifies the inductance problem by automatically calculating the inductance of the line as the difference between the self inductance of the line and the mutual inductance of the return signal path.

In many texts, the term L11 is generically used as the self inductance. For LLEV=1, Star-Hspice assumes a circuit ground point separate from the reference plane of the transmission line. Thus the LLEV=1 option includes an approximation to the self inductance of both the signal conductor and the reference plane, while LLEV=0 assumes a reference plane return current.

Crosstalk in Transmission Lines

When there are adjacent transmission lines, for instance line 2 and line 3, the coupling capacitance and inductance between them and the quiet line, line 1, lead to crosstalk.

In the notation of Star-Hspice, the voltage per length on transmission line 1, V1, including the mutual inductance to lines 2 and 3 is:


In the LLEV=0 case, Star-Hspice simplifies the inductance analysis by automatically including the effects of the return current path. The first inductance term (L11) is the inductance per length of the transmission line (1) including the self inductance of the line and the mutual inductance of the return ground path, as discussed in the previous section.

The second term, the coupling inductance of the second transmission line (L12), includes the mutual inductance of the second signal line and the mutual inductance of the return current path of the second line. Because these two currents are in opposite directions, the mutual inductance of the pair is much less than the mutual inductance of just the second signal trace alone.

The third term (LL13) includes the mutual inductance of the signal path in the third transmission line and the mutual inductance of its return path through the reference plane.

It is important to keep in mind that the coupling inductances (L12 and L13) include the mutual inductances of the adjacent signal lines and their associated return paths. They are more than the mutual inductance of the adjacent traces. In this sense, Star-Hspice deals with operational inductances, those that could be measured with a voltmeter and a dI/dt source.

Risetime, Bandwidth, and Clock Frequency

In the time domain, a clock waveform can be described in terms of its period (Tclock), its frequency (Fclock), and a risetime ( ). Clock Waveform illustrates these features.

Figure C-46: Clock Waveform

The risetime is typically defined by the time between the 10% to 90% points.

To describe this waveform in terms of sine wave components, the highest sine wave frequency required (the bandwidth, BW) depends on the risetime. As the bandwidth increases and higher sine wave frequency components are introduced, the risetime of the reconstructed waveform decreases. The bandwidth of a waveform is determined by the fastest risetime it contains. The risetime and bandwidth are related by:


The risetime of a clock waveform and the clock period are only indirectly related. The risetime of a system is determined by the output driver response and the characteristics of the packaging and interconnect. In general, the risetime is made as long as possible without degrading the clock period.

Without specific information about a system, it is difficult to know precisely what the risetime is, given just the clock frequency or period. In a fast system such as an oscillator with only one gate, the period might be two times the risetime:


For a complex system such as a microprocessor board, the period might be 15 times as long as the risetime:


In each case, the bandwidth is always related to the risetime by the first expression in this section, and the clock frequency and clock period are always related by:


The example of the microprocessor would give the worst case of the shortest risetime for a given clock period. Combining these expressions shows the relationship between clock frequency and bandwidth:


In general, the highest sine wave frequency component contained in a clock waveform is five times the clock frequency. The important assumption is that there are about 15 risetimes in one period. If the risetime is actually faster than this assumption, the bandwidth is higher. To provide a safety margin, a package or interconnect is characterized or simulated at a bandwidth of about 10 to 20 times the clock frequency, which corresponds to roughly two to four times the bandwidth of the signal.

Definitions of Transmission Line Terms

Table C-10: Transmission Line Terms


Clock frequency in units of frequency such as Hz


Clock period in units of time such as secs or ns


Rise or fall time in units of time such as sec or ns


Bandwidth in units of frequency such as Hz or MHz


A repetitive frequency in units of Hz or MHz


A sine wave frequency in units of Hz or MHz


An angular frequency in units of radians/sec


Time or a conductor thickness, units of sec or length


Instantaneous voltage in units of volts


Instantaneous current in units of amps or mA


Instantaneous impedance in units of ohms

Z( )

Frequency domain, complex impedance in units of ohms


Capacitance in units of Farads or microFarads


Resistance in units of ohms


Inductance in units of Henrys or nanoHenries


Speed of light in the medium in units of length/time


A length in units such as inches


Time delay in units of time such as sec or nsec


Capacitance per length in units such as pF/inch


Inductance per length in units such as nH/inch


Resistance per length in units such as ohms/inch


Relative dielectric constant, , dimensionless


Reflection coefficient, dimensionless


A small change in characteristic impedance


Conductance per length units of Mhos (Siemens)/length

tan( )

Dissipation factor of a material, dimensionless


Skin depth of a conductor, units of length such as meter


Resistivity of a conductor, units of ohm-length


Permeability of free space = 4 p x 10-7 Henry/meter


Attenuation per length, units of dB/len or nepers/len


A conductor width


Number of squares in a planar conductor, dimensionless


Sheet resistance of a planar conductor, units: ohms/sq


Permittivity of free space=0.225 pF/inch=0.0885 pF/cm


Effective dielectric constant due to mixed dielectrics


A dielectric thickness in units of length such as mils


Resistance in ohms

Relationships and Rules of Thumb

Time and Frequency Relationships





Transmission Line Effects

Transmission line analysis recommended for:



On FR4 material,

[BW in MHz, d in inches]

[Fclock in MHz, d in inches]

[ in ns, d in inches]

Intrinsic Properties

[CL is in pF/inch



[LL in nH/inch, CL in pF/inch]

Typical polymers

[TD in ns, d in inches]



50 ohm lines

[pF/inch = 3.4 pF/inch (typical polymer)]

[nH/inch = 8.6 nH/inch (typical polymer)]

Transmission line of length d




Reflection coefficient from Z1 to Z2:


Reflection from a Z, of short length with time delay, TD, > TD:


Reflection from a series lumped L surrounded by a transmission line:


Reflection from a lumped C load to ground, on a transmission line:


Loss and Attenuation

Skin depth of a conductor:


[ in meters, in ohm·meter, f in Hz: in copper, at f = 1e+9 and =1.7e-8 ohm·meter, = 2.0e-6]

Low loss approximation for attenuation per length:



Attenuation per length due to just dielectric loss:

[dB/inch, f in GHz]

[dB/inch for FR4 at 1 GHz]

Attenuation per length due to metal, t < :

[dB/inch, with in ohm·meter, t in microns, w in mils]

For 1 ounce copper microstrip, 5 mils wide, 50 ohm:


Attenuation per length due to metal, 50 ohm line, skin depth limited, t > :

[dB/inch, with in ohm·meter, w in mils, f in GHz]

For 1 ounce copper microstrip, 5 mils wide, 50 ohm, at 1 GHz:


Physical Design Quantities

For a planar interconnect:

Sheet resistance:


Number of squares:



[ohm/inch, Rsq in ohm/sq]

Resistance per length:

[ohm/inch, in ohm·meter, t in microns, w in mils]

Parallel plate, no fringe fields:


Microstrip capacitance per length good to ~20%:


Microstrip capacitance per length, good to ~5%:



Stripline capacitance per length good to ~20%:


Stripline capacitance per length, good to ~5%:


Inductance per length of one wire in a pair of two parallel wires:

(r = radius, s = center to center spacing, )


Inductance per length for a circular loop:

[nH/inch of perimeter]

Inductance per length of controlled impedance line, when the return line is a reference plane:

[nH/inch with CL in pF/inch]

Conductance per length:


Attenuation in Transmission Lines

The T Element used in Star-Hspice and common to most Berkeley-compatible SPICE tools uses the lossless model for a transmission line. This model adequately simulates the dominant effects related to transmission behavior: the initial driver loading due to a resistive impedance, reflections from characteristic impedance changes, reflections introduced by stubs and branches, a time delay for the propagation of the signal from one end to the other and the reflections from a variety of linear and nonlinear termination schemes.

In systems with risetimes are on the order of 1 ns, transmission line effects dominate interconnect performance.

In some high-speed applications, the series resistance seriously effects signal strength and should be taken into account for a realistic simulation.

The first-order contribution from series resistance is an attenuation of the waveform. This attenuation decreases the amplitude and the bandwidth of the propagating signal. As a positive result, reflection noise decreases, so that a lossless simulation is a worst case. As a negative result, the effective propagation delay is longer because the risetimes are longer. A lossless simulation shows a shorter interconnect related delay than a lossy simulation.

The second-order effects introduced by series resistance are a frequency dependence to the characteristic impedance and a frequency dependence to the speed of propagation, often called dispersion. Both the first order and second order effects of series resistance generic to lossy transmission lines are simulated using the Star-Hspice U model.

The Physical Basis of Loss

The origin of loss is the series resistance of the conductors and the dielectric loss of the insulation. Conductor resistance is considered in two parts, the DC resistance and the resistance when skin depth plays a role. The dielectric loss of the insulation, at low frequency, is described by the material conductivity ( ) (SIG in Star-Hspice), and at high frequency, by the dissipation factor, tan( ). These material effects contribute a shunt conductance to ground (G).

In a planar interconnect such as a microstrip or stripline, the resistance per length of the conductor RL is


= bulk resistivity of the conductor
w = the line width of the conductor
t = thickness of the conductor

Example 1: FR4, 5 mil wide line, half ounce copper:

RL = 0.24 ohm/inch

Example 2: Cofired ceramic, 4 mil wide, 0.75 mils thick Tungsten:

RL = 2.7 ohm/inch

Example 3: Thin film copper, 1 mil wide, 5 microns thick:

RL = 3.6 ohm/inch

Skin Depth

At high frequency, the component of the electric field along the conductor, which drives the current flow, does not penetrate fully into the depth of the conductor. Rather, its amplitude falls off exponentially. This exponential decay length is called the skin depth ( ). When the signal is a sine wave, the skin depth depends on the conductor's resistivity ( ), and the sine wave frequency of the current (f):

[ in meters, r in ohm·meter, f in Hz]

A real signal has most of its energy at a frequency of 1/t period , where t period is the average period. Because most of the loss occurs at this frequency, as a first approximation >> should be used to compute the skin depth. In practice, as mentioned previously, approximating tperiod by 15( ) works well. With a 1 ns rise time, the skin depth of copper is 8 microns, assuming 15( ) is used for 1/f skin . For half ounce copper, where the physical thickness is 15 microns, the skin depth thickness should be used in place of the conductor thickness to estimate the high frequency effects.

For thin film substrates with a physical thickness of the order of 5 microns or less, the effects of skin depth can, to the first order, be ignored. In cofired ceramic substrates, the skin depth for 1 ns edges with tungsten paste conductors is 27.6 microns. This is also comparable to the 19 micron physical thickness, and to the first order, the effects of skin depth can be ignored. At shorter rise times than 1 ns, skin depth plays an increasingly significant role.

Dielectric Loss

Two separate physical mechanisms contribute to conductivity in dielectrics, which results in loss: DC conduction and high frequency dipole relaxation. As illustrated in the following section, the effects from dielectric loss are in general negligible. For most practical applications, the dielectric loss from the DC conductivity and the high frequency dissipation factor can be ignored.

To be cautious, estimate the magnitude of the conductance of the dielectric and verify that, for a particular situation, it is not a significant issue. Exercise care in using these material effects in general application.

The bulk conductivity of insulators used in interconnects ( ), typically specified as between 10-12 and 10-16 siemens/cm, is often an upper limit, rather than a true value. It is also very temperature and humidity sensitive. The shunt conductance per length (GL) depends on the geometrical features of the conductors in the same way as the capacitance per length (CL). It can be written as:


At high frequencies, typically over 1 MHz, dipole relaxations begin to dominate the conduction current and cause it to be frequency dependent. This effect is described by the dissipation factor of a material, which ranges from 0.03 for epoxies down to 0.003 for polyimides and less than 0.0005 for ceramics and Teflon. The effective conductivity of a dielectric material at high frequency is:


The shunt conductance per length of an interconnect, when dipole relaxation dominates, is:


As a worst case, the frequency corresponding to the bandwidth of the signal can be used to estimate the high frequency conductivity of the material.

The Lossy Transmission Line Model

In the lossless transmission line model, only the distributed capacitance (C) and inductance (L) of the interconnect is considered:

Figure C-47: Lossless Transmission Line Model

In the lossy transmission line model, the series resistance and dielectric conductance are introduced into the equivalent circuit model:

Figure C-48: Lossy Transmission Line Model

These four circuit elements, normalized per unit length, can be used to describe all the high frequency properties of a transmission line. When the equivalent circuit equation is solved in the frequency domain, the characteristic impedance is modified to:


and the propagation phase term, , is:


In the propagation phase term, is related to the phase velocity by:


To first order, when RL << LL and GL << CL, the characteristic impedance and phase velocity (v) are unchanged from their lossless values. However, a new term, the attenuation per length ( ) is introduced.

The attenuation per length is approximately:



The total attenuation ( d) determines the fraction of the signal amplitude that remains after propagating the distance (d). When has the units of dB/length, the fraction of signal remaining is:


It is useful to keep in mind that a 2 dB attenuation in a signal corresponds to a final amplitude of 80% of the original and 6 dB attenuation corresponds to a final amplitude of 50% of the original. Attenuation on the order of 6 dB significantly changes the signal integrity.

Attenuation Due to Conductor Resistance

In the typical case of a 50 ohm transmission line, the attenuation per length due to just the series resistance is


When the resistance per length is of the order of 0.2 ohm/inch or less, as is the case in typical printed circuit boards, the attenuation per length is about 0.02 dB/inch. Typical interconnect lengths of 10 inches yields only 0.2 dB, which would leave about 98% of the signal remaining. Using the lossless T Element to approximate most applications provides a good approximation.

However, in fine line substrates, as the examples in the previous section illustrated, the resistance per length can be on the order of 2 ohms/inch. In such a case, the attenuation is on the order of 0.2 dB/inch. So a 10 inch interconnect line then has an attenuation on the order of 2 dB, which would leave only about 80% of the signal. This is large enough that its effects should be included in a simulation.

Attenuation Due to the Dielectric

When the dielectric completely surrounds the conductors, the attenuation due to just the conductance per length of the dielectric is:

[dB/inch, f in GHz]

The worst case and highest attenuation per length is exhibited by FR4 boards, with tan( ) of the order of 0.02 and a dielectric constant of 5. The attenuation at 1 GHz is about 0.1 dB/inch. For an interconnect 10 inches long, this is 1 dB of attenuation, which would leave about 90% of the signal remaining, comparable to the attenuation offered by a conductor with 1 ohm/inch resistance.

When the resistance per length is larger than 1 ohm/inch-- for example in cofired ceramic and thin film substrates, and the dissipation factor is less than 0.005, the attenuation from the conductor losses can be on the order of 10 times greater than dielectric loss. In these applications, the dielectric losses can be ignored.

Integrating Attenuation Effects in Star-Hspice

All of the first-order effects of attenuation are automatically simulated with the U Element in Star-Hspice.

With ELEV=1, the inputs can be the cross sectional geometry and the material properties of the conductor, bulk resistivity (RHO), the relative dielectric constant of the insulation (KD), and the conductivity of the dielectric (SIG). From these features, the equivalent capacitance per length, inductance per length, series resistance per length, and conductance per length are calculated by Star-Hspice.

With ELEV=2, the equivalent capacitance per length, inductance per length, series resistance per length, and conductance per length are input directly using estimates, measurements or third-party modeling tools.

Star-Hspice automatically generates a model for the specified net composed of a series of lumped elements that resembles the model for a lossy transmission line. The parameter WLUMPS controls the number of lumped elements included per wavelength, based on the estimated rise time of signals in the simulation.

The attenuation effects previously described are a natural consequence of this model. The U Element allows realistic simulations of lossy transmission lines in both the AC and the transient domain.

Star-Hspice Manual - Release 2001.2 - June 2001