pulses and transmission line theory - infineon technologies

Application Note Please read the Important Notice and Warnings at the end of this document Revision 1.0

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AN-2022-02

Pulses and transmission line theory

Driving gate driver under highspeed conditions and with long

transmission lines

About this document

Infineon strives to enhance electrical systems with comprehensive semiconductor competence. This expertise is revealed in the products themselves and their behavior under relevant use conditions, and also in the sharing of knowledge on the latest semiconductor technologies. For new technologies such as the silicon carbide (SiC) MOSFET, this is of particular importance, since a SiC MOSFET under certain operating conditions shows

different characteristics compared to silicon (Si) switches.

One important aspect to be considered for applications which use SiC MOSFET is the connection from microcontroller to the EiceDRIVER™ gate driver ICs. Both devices, the microcontroller and the gate driver itself are typically based on Complementary Metal-Oxide Semiconductor (CMOS)- technology. An exception are the optically isolated gate drivers. Due higher switching frequencies and shorter rise- and fall- times from the

pulses as well, the distance between microcontroller and the gate driver stage should be very small. However,

from an overall system design perspective, if long cables (>20…30 cm) are necessary to connect the microcontroller with the gate driver, fundamental theorems of the signal and system theory come into play and

need to be considered

Scope and purpose

• Switching frequency vs. fall- and rise time

• Review of application level impact using long cables

• Provide design guidelines on how to match longer cables used between microcontroller outputs and gate-driver inputs in the context of high switching frequencies and fast rise- and fall- times

• Amplify the main differences of optoelectronic- based gate drivers versus gate drivers with inputs based

on CMOS technology

Intended audience

• Engineers who want to learn how to use the Infineon EiceDRIVER™ gate driver ICs

• Experienced design engineers designing circuits with Infineon EiceDRIVER™ gate driver ICs, IGBTs, CoolSiC™ MOSFETs and MOSFETs

• Design engineers designing power electronic devices, like inverters, drives

Application Note 2 of 45 Revision 1.0

2022-03-07

Pulses and transmission line theory Driving gate driver under highspeed conditions and with long transmission lines

Systems with CoolSiC™ MOSFETs

Table of contents

About this document ....................................................................................................................... 1

Table of contents ............................................................................................................................ 2

1 Systems with CoolSiC™ MOSFETs ............................................................................................. 3

2 Switching frequencies, rise and fall times ................................................................................. 5 2.1 Pulse versus sinus ................................................................................................................................... 5 2.2 Observations on the 100 kHz trapezoidal signal .................................................................................... 8

2.3 Observations on the 100 MHz trapezoidal signal ................................................................................. 11

2.4 Observations with variable rising and falling switching edge ............................................................. 12 2.5 Summary of Fourier considerations ..................................................................................................... 16

3 Transmission line theory ........................................................................................................ 17 3.1 Introduction ........................................................................................................................................... 17

3.2 Line equations ....................................................................................................................................... 18 3.3 Wave propagations on transmission line ............................................................................................. 19

3.4 Special case, low loss lines ................................................................................................................... 21 3.5 Transmission line .................................................................................................................................. 22 3.6 Material properties of lines ................................................................................................................... 23

3.7 Determination of the waveform for line reflections ............................................................................ 23 3.7.1 LATTICE diagram – Example 1 ......................................................................................................... 25 3.7.2 LATTICE diagram – Example 2 ......................................................................................................... 27

3.8 Critical length of lines............................................................................................................................ 29

3.8.1 Critical length of lines for sinusoidal signals ................................................................................... 29

3.8.2 Critical length of lines for trapezoidal (or rectangular) signals ...................................................... 29

4 Circuit basics for impedance matching .................................................................................... 31 4.1 Small signal behavior of transistors ..................................................................................................... 31 4.1.1 Small-signal behavior of bipolar transistors ................................................................................... 31

4.1.2 Small-signal behavior of MOSFET transistors ................................................................................. 32 4.1.3 Push-pull output stage .................................................................................................................... 33

5 Line matching (termination) ................................................................................................... 35

5.1 Driving CMOS circuits via transmission line ......................................................................................... 35

5.1.1 Line termination via series damping ............................................................................................... 35 5.1.2 Line termination via pull-down ....................................................................................................... 36 5.1.3 Line termination via pull-up and pull-down ................................................................................... 37 5.1.4 AC-based line termination ............................................................................................................... 37

5.1.5 Chip interior ...................................................................................................................................... 38

5.2 Transmission line termination for gate drivers .................................................................................... 39

5.2.1 Line termination of single-channel opto drivers via series damping ............................................. 39 5.2.2 Line termination of multi-channel opto drivers via series damping .............................................. 39 5.2.3 Line termination of single-channel CT gate drivers via series damping ........................................ 40

5.2.4 Line termination of multi-channel CT gate drivers via series damping ......................................... 41

6 Summary ............................................................................................................................. 42

7 References and appendices .................................................................................................... 43 7.1 References ............................................................................................................................................. 43

Revision history............................................................................................................................. 44


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1 Systems with CoolSiC™ MOSFETs

In numerous applications such as solar inverters, telecom & network power and HEV/EV isolated gate drivers

are used for driving MOSFETs and IGBTs. In addition to switching the MOSFETs or IGBTs on and off, these drivers also provide galvanic isolation. The device’s switching rate depends on the application and type of switch being used. Switching frequencies of 10 to 20 kHz are common in IGBTs, however, silicon carbide (SiC)

and gallium-nitride or GaN-based systems can operate at much higher switching frequencies without significant power loss during transition. Silicon carbide (SiC) as a compound semiconductor material is formed

by silicon (Si) and carbon (C).

A short summary shows several fundamental advantages of SiC- MOSFETs over Si- IGBTs or Si- MOSFETs:

- Higher voltage operation (at the same layer thickness)

- At the same voltage and current rating, SiC- MOSFETs have a much smaller die area and smaller die thickness, leading to much lower conduction losses

- Higher switching frequeny (lower switching losses due to faster rise and fall times)

- Higher operating temperature

If all physical effects are considered in the gate driver design, the total costs of the whole system can be

reduced.

The topic of “higher switching frequency” should now be considered in more detail. One question often comes up regarding how to drive a SiC MOSFET in the right way. Infineon Technologies AG is well known for its broad

range of integrated gate drivers. Dedicated parts of this portfolio are designed to drive SiC MOSFETs, especially

the EiceDRIVER™ family. Using the EiceDRIVER™ gate driver family for developing a SiC MOSFET based system solution helps to reduce the design complexity, and development time, and reduces the bill of material in

comparison to a discrete implemented gate drive solution. Furthermore, the board space will be reduced and the reliability of the gate-drive solution will be increased.

The most important aspect of this driver stage (shown in Figure 1) is to switch the load according to its

requirements, and to keep the switching losses as small as possible. The EiceDRIVER™ gate driver IC takes the input-signal “IN+” and amplifies this signal to a level at which the switch (CoolSiC™) is fully conducting.

A second aspect becomes more and more important as soon as the application gets more complex, and the switching frequencies are becoming higher. This is valid for the interface between controller and the gate-

driver and for the interface between driver and power-switch as well. Let’s consider the switching speed of the CoolSiC™ power switch IMW120R045M1. The typical rise- time is given with 24 ns, the typical fall- time with 12

ns. These relatively sharp switching edges are the guarantee for the low switching losses of the power switch. On the other hand, these fast-switching edges lead to increased technical requirements for all the components.

All components along the signal paths, the microcontroller, the interfaces (electrical bus) from microcontroller

to gate driver, the gate driver itself and at least the interface between gate-driver and power-switch should be able to handle these sharp switching edges (see Figure 1).


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Figure 1 System overview including microcontroller – gate-driver – power switches

This document deals with the interface between the microcontroller and the gate driver. Ideally, the pulse-based control signals precisely arrive at the same time at the gate-driver inputs, which requires the microcontroller outputs to switch simultaneously in a system with well-balanced routing parasitics. However,

due to different delay times in the microcontroller outputs and different cable parasitics, the signals have

varying run times and typically do not exactly match in timing at the gate driver inputs, for example phase 2, blue lines. If the microcontroller is fast enough, it can compensate the propagation delays in most cases. A so-

called dead time is introduced into the switching process. This is an artificially inserted delay which ensures that the switches are never conducting at the same time. This, of course, delays the entire switching process. Furthermore, the low-pass characteristic of the interface should be small enough to not weaken the necessary

slope.

Another important aspect is the cable length l1 (or length of transmission lines) between the microntroller and the gate driver. First, the length of this line determines the propagation time of the signals from the

microcontroller to the gate driver. These propagation times must also be considered for very high frequencies.

In the further course of this document, it will be shown that with long cables reflections occur on these

transmission lines due to the steep rise and fall times of the high-frequency switching edges. These reflections

can significantly disrupt switching operations and lead to application failure.


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Switching frequencies, rise and fall times

2 Switching frequencies, rise and fall times

2.1 Pulse versus sinus

A brief excursion into systems theory should once again recall the relationships between switching impulses

and sinusoidal frequencies.

In the previous section, the topic of impulses was mentioned. An pulse is a process with any time course, which

belong to the periodic processes, and are characterized by the fact that the same function value reappears after

a certain time T, the period duration. The periodic-function can change the sign within a period.

One of the basic periodic- functions is the sine- function as shown in Figure 2.

t [s]

2π

0

π 32π

π 12

T

π 52

3π

= Period

1

Amplitude

-1

π 14

a(t) = A sin(ω t +

a(t) = A sin(ω t + 0)

a(t) = A sin(ω t - )π2

= A cos(ω t )

φ

)π4

π4

=

φ π2= -

Figure 2 Periodic sine and cosine function

The sine- function is an elementary function known from power electronics. This function is characterized by three important parameters:

- Amplitude

- Frequency

- Phase angle at t = 0

The sinusoidal voltage can be generated, and explained, by rotating a conductor loop at a constant angular velocity “ω” in a homogeneous magnetic field. A full revolution corresponds to an angle of 360 ° or 2 ∙ π.


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The instantaneous value can be calculated as follows:

Equation 1

The sine function plays a special role among the periodic time functions. Their differential coefficients and their

time integrals again result in functions of the same type. The sine function is an element of any periodic

function (Fourier theorem). This means that, according to Fourier, all periodic time functions can be

represented as the sum of a constant component and an infinite number “n” of harmonic oscillations, the

frequencies of which are integer multiples of the pulse repetition frequency f0 with f0=1/T. The Fourier series

can be calculated as follows:

𝑓(𝑡) = 𝑎0 + ∑ [(𝑎𝑛∙ cos(𝑛∙ω0∙t) + 𝑏𝑛∙ sin(n∙ω0∙t)]∞𝑛=0 Equation 2

with Equation 3

The Fourier coefficients can be calculated according to Equations 4.

(Represents the DC share)

Equation 4

Depending on how the signal is specified, the areas can be integrated from 0 till T or from -T/2 till+T/2,

importantly, over a full period. These equations quickly suggest that harmonics are generated with every non-

pure sine- or cosine-signal. An example shows the relations of these harmonic components and their

dependencies.

a(t) = A ∙ sin (ω ∙ t + φ)

a(t) Instantaneous value A Amplitude φ Phase angle at t = 0

ω Angular frequency -> ω = 2 ∙ π ∙ f unit [s-1

] T Period duration unit [s]

f = 1/T Frequency unit [s-1

] = 1 [Hz]; 1 Hz = 1 rotation per second

ω0= 2 ∙ π ∙ f0= 2 ∙ π

T

න𝑓(𝑡)𝑑𝑡

𝑇

0

a0= 2 T

න 𝑓(𝑡)𝑑𝑡

+𝑇/2

−𝑇/2

a0= 2 T

or


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For this we consider here a signal that is common in digital technology, the so-called trapezoidal signal. A

purely square-wave signal is theoretically conceivable, but is practically impossible to achieve. Every practical

digital signal has a rising edge and a falling edge (see Figure 3).

Figure 3 Example of trapezoid signal

It does not matter whether the amplitude of the signal ranges from 1 to -1 or for example from 0 to 2, this

can be compensated by the DC share. For further simplification, the slope of the edges tr and tf is equal. Since

the mathematical integration of this function is very complex, existing Fourier series tables can be used

(reference for example Prof. Norbert Geng, Prof. Werner Neundorf). The following Equation 5 can be displayed

as a result:

Equation 5

Equation 5 now represents the trapezoidal function and the suggestion is, that the function itself and the slope

of the edges influence the harmonics. The following graphic evaluations show the results. The basic frequency

of this signal is given with 100 kHz, T = 1∙10-5 s.

Figure 8 shows a selection of harmonics based on calculations from Equation 5. For this first calculation, the

factor d was chosen with d = π / 10000, which means that d is very small and represents very steep edges. This

would be ia rise of 0.5 ns in this case, which is of course very fast.

𝑓(𝑡) = [sin((2∙𝑛 + 1)∙𝑑)

(2∙n + 1)2(sin((2∙𝑛 + 1)∙ω0∙𝑡)]

∞

𝑛=0

4

π∙d


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2.2 Observations on the 100 kHz trapezoidal signal

In the following section, a trapezoidal signal with a frequency of 100 kHz is considered. The rise and fall of the switching edges (factor d) are kept stable. Figure 4 shows the sinusoidal signals generated according to Equation 5 with their different frequency components and amplitudes. The factor d was chosen with the value

of π /10000, which corresponds to a rise of the edge of about 0.5 ns, based on the frequency of 100 kHz.

Figure 4 Single sinusoidal shares of trapezoidal signal at d = π / 10000 and 100 kHz

As expected the basic wave sinω1 (100 kHz) has the amplitude with the biggest value. The amplitude itself looks

very small with approximate 3∙10-4, but the missing factor (4/ (π∙ d)) is calculated with the value of 4057 and

has to be multiplied. With approximately the third of the amplitude of the basic harmonic, the sinω3 (300 kHz)

still has a major influence of the result. It is clear that with every harmonic the frequency increases and the

amplitude of this corresponding harmonic decreases.

Figure 5 depicts the summary of harmonic ω1 up to harmonic ω

13 to show the reproduced signal from this

summary; Figure 6 shows the summary of harmonic ω1 up to harmonic ω

25 and Figure 7 shows the summary of

harmonic ω1 up to harmonic ω

53.

If Figures 5, 6 and 7 are compared with each other, a steady improvement in the signals can be seen. The more

harmonics included in the calculation, the closer the result comes to the ideal signal. The oscillations become

softer and the slope of the signal edge increases.

A slightly different representation of the sinusoidal signals shown in Figure 5 is made in Figure 8. Here the

individual signals are shown in a frequency spectrum with their respective maximum amplitudes at the angle of

n* π /2.


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Figure 5 Summary of sinus shares ω1 up to harmonic ω13 at d = π /10000 and 100 kHz

Figure 6 Summary of harmonic ω1 up to harmonic ω25 at d = π /10000 and 100 kHz


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Figure 7 summary of harmonic ω1 up to harmonic ω53 at d = π /10000 and 100 kHz

Figure 8 Normalized frequency spectrum at d = π / 10000 and 100 kHz


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2.3 Observations on the 100 MHz trapezoidal signal

Figure 9 Summary of harmonic ω1 up to harmonic ω53 at d = π /10000 and 100 MHz

Figure 10 Normalized frequency spectrum at d = π / 10000 and 100 MHz


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The first results can be summarized:

1.) When using signals that are not purely sinusoidal, harmonics are generated.

2.) The number of harmonics is theoretically infinite

3.) The harmonics and their amplitudes do not depend on the frequencies, compare Figure 8 and 10. Only

the frequency axis is shifted

2.4 Observations with variable rising and falling switching edge

In the following section, the rise times of the rising and falling switching edges are reduced. The base frequency

is 100 kHz, the value of the factor d was given with π /8, which corresponds to a rise of the edge of about of

about 625 ns.

Figure 11 Single sinus harmonics of trapezoidal signal at d = π /8 and 100 kHz

The function of the trapezoidal function (Figure 12) can be seen very well and the oscillations are very small. In

Figure 11 it can be seen that amplitude of the first harmonic is around 1000 times higher compared to the harmonic in Figure 5, but the factor (4/ (π∙ d)) is with a value of 3.2455 much lower as opposed to 4057, which belongs to Figure 5.

Another interesting point of view is shown in Figure 13. The magnitudes of the amplitudes of the higher

frequency harmonics in relation to the first harmonics have become significantly smaller, as can be seen in comparison to Figure 10.


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A last observation will be done with the same function (Equation 5), but with a very flat slope in relation to the period based on the 100 kHz signal. The value of d = π / 2 represents a rising edge of about 2.5 µs based on the

periodic duration from 1∙10-5 s. The factor (4/ (π∙ d)) has a value of 0.8113 and the results are shown in Figures 14, Figure 15 and Figure 16.

Figure 14 Single sinus harmonics of trapezoidal signal at d = π /2 and 100 kHz

The slope of the two lines is so great that they meet in the middle (half of 5.0E-6) and form a triangle as shown

in Figure 15. If the slope is even smaller, a trapezoid is created again, but with considerably reduced amplitude. And the smaller the amplitude, the smaller the increase. This means that the signal is now absolutely distorted.


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2.5 Summary of Fourier considerations

The following final results can be summarized:

1.) When using signals that are not purely sinusoidal, harmonics are generated.

2.) The harmonics depends strongly on the shape of the signal (i.e., triangle, rectangle) corresponding to

the Fourier analysis

3.) At least in the case of signals similar to rectangles, the weighting of the amplitudes of the harmonics is strongly dependent on the angle or the rise or fall- times of the switching edges.

4.) The steeper the switching edges, the more harmonic components can be detected, and their amplitudes increase significantly

5.) Even with pulses with relatively low frequencies, high-frequency harmonics must be expected

6.) The part of the spectrum in the low-frequency range is a function of the fundamental frequency, the higher-frequency part is a function of the edge steepness


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Transmission line theory

3 Transmission line theory

3.1 Introduction

In section one it was shown, for example, that pulses are sent from a sender (microcontroller) over electrical cables (transmission lines) to send information to a receiver (gate-driver).

An electrical transmission line, hereinafter simply referred to as a line, is to be understood to mean a device for

transmitting electrical energy or electrical signals from a transmitter to a receiver, consisting of a conductor system for forwarding and returning the electrical current. The conductors must be designed and arranged in such a way that the electromagnetic fields generated during transmission can be considered as transverse

fields (TEM waves). The last condition requires, that the ohmic resistance of the conductors is low, so that the

longitudinal components of the electric field strength can be neglected and that the distance between the

conductors is less than half the wavelength in order to prevent the formation of hollow waves.

A transmitter or receiver is to be understood here as any source or sink of electrical energy or electrical signals, for example: generator -> motor; TV station -> transmission antenna, receiving antenna -> receiver, etc.

The basic form of all cables is the symmetrical two-wire cable. It consists of two wires of the same cross- section that are routed in parallel. If you differentiate the lines according to the shape of the cross-section,

which we would like to assume does not change along a line, the basic options shown in Figure 17 result. Figure

17 shows the basic types of lines (cables). All others can be derived from it. The individual types should not be discussed in more detail.

Figure 17 Cross- sections of the most important types of transmission lines

+ +

- -

+

- -

+ + +-

Push-pull Common-mode

StriplineStripline

Stripline Stripline

Symmetrical

two-wire line

single-wire line

To GND

Coaxial cable

Shielded

two-wire cable

Asymmetrical

two-wire line


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3.2 Line equations

Some important results of the transmission line theory are summarized below. A deeper introduction would go beyond the scope of this application note. The audience should get a feel for what actions have to be taken

when high-frequency signals are transported over lines. For this, a piece of line and the associated physical properties were simply analyzed. The corresponding "equivalent circuit diagram of a short line section" can be seen in Figure 18. In the case of longer homogeneous cables (l>ʎ/4), the location dependence of current and voltage and thus the resistance is of interest. Homogeneous means that the line section of the same length dx

also has the same resistance R’dx, the same inductance L’dx, the same capacitance C’dx and the same

conductivity G’dx

Equation 6

This is valid for length l of the line pieces with dx < ʎ/4 (ʎ - Wavelength). The following Figure helps to understand how the line equations were created.

Figure 18 Line and equivalent circuit diagram of a short line section

i(x,t) iA(x,t)iE(x,t)

dx

uE(t) uA(t)

x

L dxR dx

G dx

IGC

C dx

dx

U(x,t)U(x+dx,t)

I(x,t)

URL

I(x+dx,t)

R’ = R

dx Resistance per unit length

Conductive per unit length

Capacitive per unit length

Inductive per unit length

(ohmic losses)

(dielectric losses)

(capacitive field energy)

(magnetic field energy)

G’ = G

dx

C’ = C

dx

L’ = L

dx


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With help of the coefficients R’, L’, C’ and G’ the differential equations, given by the equivalent circuit diagram in Figure 18, can be derived and we get the so-called telegraph equation.

From the voltage mesh we get:

Equation 7 and 8

The current node delivers:

Equation 9 and 10

After some reshaping and differentiation, we get:

Equation 11

With the definition of the propagation constant ɣ follows from Equation 7:

with Equation 12 and 13

and further definition of following constants:

Attenuation constant

Phase constant

3.3 Wave propagations on transmission line

The above Equation 12 represents a wave equation with two solutions. These two solutions describe two

waves, one which runs from source to the load, and the second which returns from the load.

U(x) = Uh∙γ ∙ + Ur ∙

γ ∙

Uh∙γ ∙ - Ur ∙

γ ∙

I(x) = ZW

I2

U1U2

x

I1

ZE

ZW ZL

l

U(x)

I(x)

X=0

E

Zi

Figure 19 Equivalent circuit diagram

ɣ α jβ = + = ඥ(R’ + jωL‘)∙(G’+ jωC‘)

α = Re γ

β = Im γ

d2U(x)

dx2

= (R’ + jωL‘)∙(G’+ jωC‘)∙U

d2U(x)

dx2

- ∙U γ 2 =0

u(x+x,t)-u(x,t)

dx = - R’I(x) - jωL‘I(x) - (R’ + jωL‘)I(x)

du

dx =

i(x+x,t)-i(x,t)

dx = - G’U(x) - jωC‘U(x) - (G’ + jωC‘)U(x)

di

dx =


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The general solution is an overlay of opposite, exponential damped voltage waves:

Equation 14

Equation 15

with

Running wave from source to Load Equation 16

Returning wave from load to source Equation 17

The characteristic impedance of the line can be expressed as:

Equation 18

The input impedance ZE of a line enclosed with a load- resistance:

Equation 19

Important notice: In the case of a matched enclosed line (ZL = ZW), the following statement applies for any length of line: ZE = ZW = ZL

The speed of propagation at which the wave travels across the line is defined as:

Equation 20

c0 = speed of light in vacuum

Important: Speed of a wave in vacuum is faster as that in material (example PCB)

Another important coefficient is the reflection coefficient ϱ. It is a parameter that describes how much of an electromagnetic wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected wave to the incident wave, with each expressed as phasors. The

coefficient can be calculated with the Equation 21 and 22:

ZW = ට𝑅′+𝑗ω𝐿′

𝐺′+𝑗ω𝐶′

ZE = ZL ∙

cosh(ɣ∙l) + ZW

ZL ∙ sinh(ɣ∙l)

cosh(ɣ∙l) + ZL

ZW ∙ sinh(ɣ∙l)

U(x) = Uh(x) + U

r(x)

U1 ∙ Uh = 𝑒−ɣ∙𝑋

U1 ∙ Ur = 𝑒ɣ∙𝑋

I(x) = Ih(x) + I

r(x) =

Uh(x)

ZW

-

Ur(x)

ZW


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Reflection coefficient at the load side Equation 21

Reflection coefficient at the source side Equation 22

Physical interpretation of the reflection factor, for example with special cases at load side:

ZL = ZW: ϱL = 0 line matching; no reflected power. The signal energy arriving at the line output is fully

converted.

ZL = 0: ϱL = -1 short circuit; this implies the reflected wave having a 180° phase shift (phase reversal)

with the voltages of the two waves being opposite at that point and adding to zero (as a short

circuit demands).

ZL -> ∞ ϱL = 1 no load; the line is open at the output; the incoming energy is not converted. At no time

does electricity flow at the end of the line. The energy cannot be lost and must be fully reflected.

The line output acts as an oscillator and sends the signal on the line back to the input.

Standing wave (ZL = 0 [short circuit] or ZL -> ∞ [no load at output])

If the line output is loaded with any resistance that does not correspond to the line impedance, the signal is

completely or partially reflected from the output to the input. The voltage and current waves running back and forth overlap on the line. There they form stationary wave patterns, which are referred to as standing waves.

The wave equation of the superposition is calculated by adding the wave equations for the outgoing and return signal.

3.4 Special case, low loss lines

A particularly simple, but important special case can be derived from the solutions of the line equations [3.3]. For all lines, with which you want to transmit energy or messages, you try to keep the losses as small as possible. This means that the line is constructed in such a way that the smallest possible voltage drop occurs

along this line and the lowest possible current flows between the conductors. This is expressed in the line

constants in such a way that R’ and G’ become very small or can be neglected.

In practice, the lines can often be considered as loss- less (good approximation), especially at high frequencies R’<< ωL’ and G’ << ωC’.

Condition: R’ = 0 and G’ = 0

This simplifies important Equations and the equivalent circuit diagram.

ZL+ ZW

Ur

Uh =

ZL - ZW ϱL =

Zi+ ZW

Ur

Uh =

Zi - ZW ϱS =


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C

Zi

E

L 2

L 2

U2 ZLU1 ZW

Transmission line

l

U1

L

C

Zi

E U2 ZLZW

Transmission line

l

or

Figure 20 Equivalent circuit diagram of loss less lines with source and load

Equation 22

Equation 24

The signal runtime is independent of the frequency, it is only dependent on the material properties. This

ensures that all components of the signal are transmitted regardless of their frequency.

At very high frequencies, the skin effect must be considered, since this increases the internal resistance of the

line.

3.5 Transmission line

Figure 21 shows a typical transmission line.

Signal line

Signal return line

Transmitter Receiver

ZWZA ZE

l

Figure 21 Typical transmission line

ZW = ට

𝐿′

𝐶′ Characteristic impedance

τ = v

1 = ඥ𝐿′∙𝐶′ Signal runtime


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A transmission line consists of:

- The signal line that carries the signal current

- The signal return line (in most cases GND), which has an equal return current

- ZA – output impedance of the transmitter (driver)

- ZE – input impedance of the receiver

Any random direct current connection between the ground connections of the two stations (e.g. protective conductor) does not represent a defined signal return.

The area between the signal conductor and the return conductor determines the ability of the arrangement for

radiation and its immunity to the radiation of high-frequency energy (antenna).

3.6 Material properties of lines

Typical values of different lines are summarized in Table 1.

L’(nH/cm) C’(pF/cm) ZW(Ω) τ (ns/m)

Single wire 20 0.06 600 3.4

Vacuum μ0 ε0 370 3.3

Twisted two-wire cable 5 - 10 0.5 - 1 80 - 120 5

Ribbon cable 5 - 10 0.5 - 1 80 - 120 5

PCB track 5 - 10 0.5 - 1.5 70 - 100 ~5

Coaxial cable 2.5 1.0 50 5

Bus line 5 - 10 10 - 30 20 - 40 10 - 20

Table 1 Typical values of material properties for different types of lines

These are just typical values. For example, the values for PCB tracks should be calculated using special

calculation programs

3.7 Determination of the waveform for line reflections

There are different methods to determine the waveforms of line reflections. One of these methods, the LATTICE

diagram, will be briefly discussed in this section. This method is particularly suitable for creating waveforms for lines that have been terminated with resistors. Special computer programs or graphic methods are used for complex loads or non-linear components.

First, a line is considered again in general (see Figure 22).


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Figure 22 Transmission line

As soon as switch "S" is closed, a pulse or the incident wave U1 is generated at point P1. The voltage on this

pulse is:

Equation 25

The wave is moving towards P2. After it has arrived there, U2 gets the value of the wave U1. This is reflected accordingly, as long as ϱ

L<> 0! If ϱ

L= 0, no reflection will happen. The reflected wave will be added to U2 at P2,

and Ur1 will be reflected to P1. The reflected wave will be:

Equation 26

The reflected wave Ur1 returns to point P1, and the value will be added to the prevalue of U1. Once there, it is reflected again, but with the factor ϱ

S. The reflected wave will be added to the value of P1, at P1.


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Equation 27

This process is repeated now until the energy of the wave has been fully converted. Now the steady state sets in:

Equation 28

In order to better illustrate the entire process, two examples should illustrate the wave movement and its

reflections.

3.7.1 LATTICE diagram – Example 1

Example 1 shows a line whose terminating resistance is infinitely large. The length "l" is not defined, but τ is

plotted as a variable on the x-axis, since τ depends on the line properties and the length “l” of the line.

The Figure of the transmission line is according to Figure 22; for the specification see Figure 23.

Figure 23 Values of transmission line for Example 1 according to Figure 22

Based on the given values of Ri, RL and ZW, the values of ϱL and ϱ

S are calculated. With this value the incident

wave and the reflected waves can be calculated (see Figure 24). With RL = 1 GΩ the indication is RL -> ∞.

The incident wave starts according to Equation 25 with a value of 4.412 V. After τ1 this wave reaches point P1.

Since ϱL is 1, the wave is completely reflected back.

E(V)= 5 V

R i= 10 Ω ϱS= -0,765

R L= 1.000.000.000 Ω ϱL= 1,000

ZW= 75 Ω

Ri + R

L

RL

U2(t->∞)

= E ∙


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Figure 24 Wave schedule according to example 1

Voltage U2 (τ1) now has the value of 8.823 V according to Equation 14 and U2(τ0) = 0 V. The reflected wave ur1

runs back to point P1 and is reflected with ϱS. U1 (τ2) has now the value of 5.4498 V. This ping-pong game

continues until the entire energy of the wave has been converted and the static state is set with approximately 5 V (Equation 28).

The graphic in Figure 25 shows the waveform of U2 and U1 over τ.

U1(2*tn) U2(tn+1)

τ0 4,412 V U1 = 4,412 0

8,823529 V τ1

ur1= 4,412

τ2 5,449827 V ur2= -3,373702

2,076125 V τ3

ur3= -3,373701

τ4 4,656015 V ur4= 2,579889

7,235904 V τ5

ur5= 2,579889

τ6 5,263047 V ur6= -1,972856

3,290191 V τ7

ur7= -1,972856

τ8 4,798846 V ur8= 1,508655

6,307500 V τ9


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Figure 25 Graphical waveform diagram according to Example 1

3.7.2 LATTICE diagram – Example 2

Example 2 uses the same configuration as Example 1, but the value of RL has now the value of ZW, i.e., line matching. For the complete specification, please see Figure 26.

Figure 26 Values of transmission line for Example 2 according to Figure 22

Related to the fact that RL = ZW the reflection coefficient ϱL is 0, which means that no wave should be reflected.

Figure 27 shows the corresponding wave schedule.

E(V)= 5 V

R i= 10 Ω ϱS= -0,765

R L= 75 Ω ϱL= 0,000

ZW= 75 Ω


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Figure 27 Wave schedule according to example 2

The wave schedule diagram shows clearly that only one wave, the incident wave exists. The incident wave runs to point P2, where ϱ

L is 0. The reflecting wave is deleted. After τ1 the steady state is reached.

Figure 28 Graphical waveform diagram according to example 2

U1(2*tn) U2(tn+1)

τ0 4,412 V U1 = 4,412 0

4,411765 V τ1

ur1= 0,000

τ2 4,411765 V ur2= 0,000000

4,411765 V τ3

ur3= 0,000000

τ4 4,411765 V ur4= 0,000000

4,411765 V τ5


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3.8 Critical length of lines

The question is often asked, from which line length the line theory is to be applied. Some information can be found in the literature, but it is only rough estimates.

A clear distinction must first be made between sinusoidal signals and pulse-shaped signals.

3.8.1 Critical length of lines for sinusoidal signals

For sinusoidal signals the critical line length can be roughly determined as follows:

Length of line should be << ʎ / 4 of signal

In this case there is a relatively constant potential over the line length, no zeros, no minimum and maximum

values.

3.8.2 Critical length of lines for trapezoidal (or rectangular) signals

The line theory is to be used if the rise time (or fall time) of the signal is shorter than twice the signal transit

(run) time τ.

Equation 29

This means, if the used cable is longer than the calculated value “l” in the Equation 29, the transmission line

theory should be applied.

Example: twisted two-wire cable; tr = 2 ns; τ = 5 ns/m

Answer: If the length of the used line is longer than 20 cm, the line theory should be applied.

Important notice: All critical lines on which high-frequency signals are transmitted should be treated according to the line theory.

This means that for all critical lines on which high-frequency signals or pulses are transmitted from a

transmitter via a line to a receiver, the impedances should be as equal as possible. This is because cables that are not matched lines can contribute to EMI problems due to their reflections.

In the best-case scenario: ZA = ZW = ZE

Sufficient conditions: ZA = ZW or ZE = ZW (for point-to-point connections)

l < 2 ns

2 ∙ 5 ns/m = 0.2 m = 20 cm


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Since in practice the corresponding impedances of the different circuit parts do not always have the same values, the so-called impedance converter circuits must ensure that the above-mentioned conditions are met.

These impedance converters then provide impedance matching. There are various circuit arrangements for this, which will be described in the next chapter.


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Circuit basics for impedance matching

4 Circuit basics for impedance matching

4.1 Small signal behavior of transistors

Small signal behavior is the behavior of the transistor in the immediate vicinity of the static operating point. One then approximates the (in reality curved) characteristic curves simply by straight lines (specifically by the tangent at the working point). The circuit then appears linear as a whole and the computational effort is drastically reduced. These conditions also apply at the control of CMOS, TTL or optical components by means

of the components previously mentioned via lines. The following tables show the summaries known in the literature.

4.1.1 Small-signal behavior of bipolar transistors

Table 2 shows the most important summarized operating parameters of the corresponding basic circuits for bipolar transistors.

I2

Z2

E

Z1

I2

U1

U2

I1

E

Z1

Z2

ZS

U1

U2

I1

E

Z1

Z2

I2

U2

rbe +(1+β) Z1

β Z2

Z2

Z1

1

(1+β) Z2

rbe1+

1-rbe +(1+β) ZS

β Z2

Z2

ZS

-U2

U1

Vu =

Z E rbe +(1+β) ZS

(1+β)

rbe+Z1

Emitter-circuit

rbe +(1+β ) Z2 (1+β)

rbe+Z1

Basis-circuitEmitter follower

(collector circuit)

small≈ Z2Z A ≈ Z2

I1

. .

Table 2 Operating parameters of bipolar transistor basic circuits

The values Vu denotes the voltage gain, ZE the input impedance and ZA the output impedance, the factor β

denotes the current amplification factor. The values ZE and ZA are of course relevant for the impedance matching.


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4.1.2 Small-signal behavior of MOSFET transistors

Table 3 shows the most important summarized operating parameters of the corresponding basic circuits for MOSFET transistors.

I2

Z2

E

Z1

I2

U1

U2

I1

E

Z1

Z2

ZS

U2

U2

U1

Vu =

Z E

Z Asmall

I2

U1

I1

E

Z1

Z2 U2

-

1+S∙ZS

S∙Z2≈

Z2

ZS

-1+S∙Z2

S∙Z2≈ 1 S∙Z2

∞ ∞1S

Gate-circuitSource follower

(drain circuit)Source-circuit

≈ Z21S

Z2≈ Z2

Table 3 Operating parameters of MOSFET transistor basic circuits

The factor S denotes the so-called slope, physical unit in Siemens. The Equations given here correspond to the

simplified transistor models and for low frequencies, but some general statements can be made here in terms of input and output impedances. For comparison, some typical numerical values are summarized in Table 4.

The following values should be assumed for these calculations:

Z2 = 50 kΩ; ZS = 1 kΩ; Z1 = 120 Ω; S = 0.004 Siemens; β = 100


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Bipolar MOSFET

E C B S D G

Vu -25 1 417 -25 1 200

ZE 200 kΩ 5 MΩ 120 Ω ∞ ∞ 500 Ω

ZA 50 kΩ 1.2 Ω 50 kΩ 50 kΩ 249 Ω 50 kΩ

Table 4 Some properties of the basic circuits based on values above

The values shown here can vary depending on the application-specific wiring, but some basic statements can

be made.

1.) Due to the low output impedances ZA, the drain circuit and the collector circuit are very well suited for driving lines. Please compare again with the values of ZW in Table 1.

2.) The basic circuit and the gate circuit would be very suitable as an input stage for receiving the signals from the line because of the low input impedance ZE. If the input stages of the receivers are not too complex, in many cases a simple resistance is sufficient to match the desired impedance of the line to

that of the input stage.

4.1.3 Push-pull output stage

The collector-circuit has a small disadvantage in order to drive lines. It has only one active element to push the current into the load; the discharge has to be done via the resistor. To circumvent this, and possibly achieve

higher performance, the resistance was simply replaced by a further collector circuit. The result is the so-called push-pull stage. This stage is now included in many diverse assemblies, like operational amplifiers, buffers, and

analog audio amplifiers, but is also present in TTL and CMOS digital logic circuits and microcontroller as well. In most cases these are realized as a complementary pair of transistors, one dissipating or sinking electrical

current from the load to ground or a negative power, and the other supplying current to the load from a positive power.

ICIC

Figure 29 Push-pull stage in bipolar and CMOS technology


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As shown, this type of output stage can be found in many applications. Due to the relatively low output impedance, line matching can be achieved with relatively simple circuits, as will be shown in the next section.


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Line matching (termination)

5 Line matching (termination)

As shown in Chapter 3, non-matched lines cause reflections, which can be seen as the overshoot of the voltages and rise or fall in steps. The size of the overshoot depends on the mismatch (reflection factor) and the duration on the cable length.

In this section, only the simple connecting lines are considered, i.e. the transmitter at the beginning of the line

and the receiver at the end of the line. Bus systems are somewhat more complex and must be considered separately.

We also assume that the input impedances of the receivers are very high and the output impedances of the transmitters are very low. Otherwise, the impedance values must be considered. CMOS devices usually have a high input impedance (typically 10 MΩ) and a low output impedance (10 to 1000 Ω). Only the unbalanced lines

are considered for the time being.

5.1 Driving CMOS circuits via transmission line

There are three basic types of line termination, from which the most diverse types of termination are derived.

1.) Series damping

2.) Pull-up / pull-down networks

3.) AC termination

5.1.1 Line termination via series damping

The series attenuation is achieved by inserting a small resistor RP as close as possible to the signal source in the

transmission line. The idea behind this is to make the reflection factor at the signal source zero by inserting a series resistor to make the (apparent) output impedance of the signal source (transmitter + RP) equal to the line

impedance:

Signal line

Signal return line

TransmitterReceiver

l

ZA RP ZE

Example: CMOS

ZW

ZE 1 MΩ

RP = ZW ZA-

P1 P2UA

U1 U2

with ZA 0

RP = ZW

ZA 0 Ω

Figure 30 Line termination by series damping


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With the start of the pulse (t = 0) U1 has the half of the value from UA. After the time τ the wave reaches point P2. Since ZE is close to ∞, the reflection factor ϱ

L = 1. Now the wave will be reflected with the value ϱ

L ∙ U1 back to

P1, U2 has the value of UA. After the time 2 ∙ τ the return wave reaches P1 again. Since the reflection factor here is ϱ

S = 0, no further reflection occurs, and the wave is extinguished.

5.1.2 Line termination via pull-down

With this special termination, reflections are completely avoided by using the right line termination. Although pull-up / pull-down networks are very popular, they should only be used in an emergency as they consume quite a lot of energy. The basic idea behind these circuits is the absorption of excess signal energy by cross

currents at the end of a line.

Signal line

Signal return line

TransmitterReceiver

l

ZA

RT

ZE

Example: CMOS

ZW

ZE 1 MΩ

ZW = RT ZE

P1 P2UA

U1 U2

I

RT = ZW

with ZE 1 MΩ

ZA 0 Ω

Figure 31 Line termination via pull-down resistor

With the start of the pulse (t = 0), U1 has the value from UA. After the time τ1, the wave reaches point P2. Since ZE = RT, the reflection factor ϱ

L = 0. The return wave is extinguished, U2 has the value of UA. The entire signal-

transmission is done in τ1.

The same is valid for the termination with pull-up and pull-down resistor, shown in Figure 32. This variant is

often used when switching levels have to be adjusted additionally.

In both pictures, the additional current and the associated additional power loss are marked with the red lines.


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5.1.3 Line termination via pull-up and pull-down

Signal line

Signal return line

TransmitterReceiver

l

ZA

RT

ZE

Example: CMOS

ZW

ZE 1 MΩ

ZW = RT ZE

P1 P2UA

U1 U2

I

ZW = RT

with ZE 1 MΩ

R1

ZA 0 Ω

VCC

R1

R1

Figure 32 Line termination via pull-up and-down resistor

5.1.4 AC-based line termination

In order to avoid the problem with the additional electricity and the associated additional power loss, the pull-

down termination can be slightly modified (see Figure 33).

Signal line

Signal return line

TransmitterReceiver

l

ZA

RT

ZE

Example: CMOS

ZW

ZE 1 MΩ

ZW = RT ZE

P1 P2UA

U1 U2

RT = ZW

with ZE 1 MΩ

ZA 0 Ω

CT

RT CT τ. .4>

R1

maybe R1 between 1..10 kΩ

Figure 33 AC-based line termination


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5.1.5 Chip interior

Before starting the termination of lines, the interior of the chips involved should be carefully examined.

VCC1

VCC1

1

2

3

Chip inside

4

5

R1

R1_int

R2_int

R2

R3

R4

R4

Figure 34 Example of possible inside view of a chip

For example, in some chips there are already internal pull-up or pull-down resistors for the inputs integrated. In

the example in Figure 35 input pin 2 is connected to an internal pull-down resistor R1_int. To avoid additional power losses due to static currents the connection of R1 at pin 2 should not be done; Resistor R2 would be

correct as it is at the same potential as R1. Also, different kind of outputs can be configured in most microcontrollerfamilies. Output 4 is a push-pull output, output 5 for example an open-drain. If loads or lines are

connected here, the instructions in the Sections 5.1.1 up to 5.1.4 should be observed.


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5.2 Transmission line termination for gate drivers

The transmission line termination is very important point, especially in harsh environments. In addition to the noise and interference signals, the line reflections are subsequently added, which prevents a clean and

accurate switching behavior in total. The idea is to make the reflection factor at the signal source zero by inserting a series resistor to make the (apparent) output impedance of the signal source (transmitter + RP) equal to the line impedance.

Again, all electrical lines that have a higher length than the critical length of l > tr / (2* τ), and on which high-frequency signals are transmitted, should be terminated!

Finally, the line termination on gate drivers with different technologies will be compared. On one side is the CT

(coreless transformer) gate driver with a high impedance CMOS input resistor. On the other side is the optical

gate driver, which comes very close to the characteristic impedance of the transmission line due to its

integrated LED and the low impedance input resistor RLED.

5.2.1 Line termination of single-channel opto drivers via series damping

The series attenuation is the most economical option and is achieved by choosing the value of resistor RP as close as possible to the value of the characteristic impedance of the transmission line. Assuming the

microcontroller can drive the LED current, such a circuit could look like the one in Figure 35.

Figure 35 Transmission line termination with optocoupled device

Due to the higher input current and the associated small input resistance RLED (about 150 Ω in a 3.3 V system), a

certain degree of line matching is already present. But an exact fit is not easy. It depends on the LED current IF,

and it will not be so easy to adapt the RP to the ZW.

5.2.2 Line termination of multi-channel opto drivers via series damping

In applications such as half-bridge or full-bridge, in which several drivers have to be operated, it can be assumed that a microcontroller cannot drive several opto devices at the same time without being overloaded.

With a much higher effort, e.g. an additional buffer, such an application can then work well, see Figure 36.

microcontroller Opto deviceAnode

Cathode

RLED /2

RLED /2

Signal line

Signal return line

l

ZWU1 U2

RP

2

VF

Shield

3

4 5

1 8

7

6

VCC2

OUT

VEE2

GND2

IFD

rive

r

Channel 1

ZE = RLED + RDiode

ZW RP =

ZE

, τ

ZA Ω

P1 P2


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Figure 36 Multi-channel transmission line termination with optocoupled devices

5.2.3 Line termination of single-channel CT gate drivers via series damping

It also applies here that series attenuation is the most economical option. Due to the fact that CT gate drivers have a high-resistance input, the wiring is very simple (see Figure 37).

Figure 37 Single-channel transmission line termination with CT gate-driver

Due to the high input impedance of the CT gate driver, line matching is done with only one resistor!

microcontroller Opto deviceAnode

Cathode

RLED /2

Buffer

(high impedance

input)

RLED /2

Signal line

Signal return line

l

ZWU1 U2

RP

2

VF

Shield

3

4 5

1 8

7

6

VCC2

OUT

VEE2

GND2

IF

Dri

ve

r

Channel 1

ZW RP =

Opto deviceAnode

Cathode

RLED /2

Buffer

(high impedance input)

RLED /2

Signal line

Signal return line

l

ZWU1 U2

RP

2

VF

Shield

3

4 5

1 8

7

6

VCC2

OUT

VEE2

GND2

IF

Dri

ve

r

Channel N

ZW RP =

ZA Ω

ZA Ω

microcontroller

RP Signal line

Signal return line

l

ZWU1 U2

ZW RP =

Channel 1

VEE2

OUT+

VCC2

OUT-

7

6

5

8IGATE

GND1

VCC1

2

3

4

1

IN+

IN-

15 V3.3 V

CT-driver

ZA Ω


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5.2.4 Line termination of multi-channel CT gate drivers via series damping

Figure 38 shows a multi-channel application with CT gate driver.

Figure 38 Multi-channel transmission line termination with CT gate-driver

The transmission line termination with CT gate drivers is in both cases, in single- or multi-channel applications, very easy.

Due to the high impedance resistances of the CT gate driver, the CT gate drivers are very interesting for multi-channel applications, such as drives, EV charging, etc.

microcontroller

RP

RP

Signal line

Signal return line

l

ZWU1 U2

ZW RP =

Channel N

VEE2

OUT+

VCC2

OUT-

7

6

5

8IGATE

GND1

VCC1

2

3

4

1

IN+

IN-

15 V3.3 V

CT-driver

Signal line

Signal return line

l

ZWU1 U2

ZW RP =

Channel 1

VEE2

OUT+

VCC2

OUT-

7

6

5

8IGATE

GND1

VCC1

2

3

4

1

IN+

IN-

15 V3.3 V

CT-driver

ZA Ω

ZA Ω


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Summary

6 Summary

- Transmission lines between integrated circuits in fast systems have to be treated as lines in terms of line theory.

- The application of the line theory can only be neglected below a so-called critical length of the line.

- The critical line length begins when twice the transit time of the signal becomes greater than the rise (fall) time of the signal, i.e. when the line reflections no longer fall into the rise time.

- This is, for example the case with a point-to-point connection from approximately 20 cm, for bus systems from approximately 5 cm.

- A transmission line is to be regarded as a real load. This means that delays occurring there must be

added to those of the integrated circuits.

- Lines that are not properly terminated lead to reflections when operating with high-frequency

signals.

- Line reflections lead to signal distortions that can only be accepted to the extent that their

amplitude does not exceed the signal-to-noise ratio of the circuit used. This can lead to significant

malfunctions.

- By terminating the lines, e.g. with resistors, the reflections can be reduced or prevented.


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References and appendices

7 References and appendices

7.1 References

[1] Zinke/Brunswig: Lehrbuch der Hochfrequenztechnik 1; Springer-Verlag, ISBN 3-540-51421-X, 4. Auflage, 1990

[2] R. Geißler, W. Kammerloher, H. W. Schneider: Berechnungs- und Entwurfsverfahren der

Hochfrequenztechnik 2; Vieweg Verlag, ISBN 3-528-04943-X, 1994

[3] Vielhauer: Lineare Netzwerke; VEB Verlag Technik, Bestellnummer: 5530718, 1982

[4] Schunk, Hermann: Grundlagen der Impulstechnik; Dr. Alfred Hüthig Verlag Heidelberg, 2. Auflage, ISBN 3-7785-0921-7, 1983

[5] U. Tietze, Ch. Schenk: Halbleiter Schaltungstechnik; Springer Verlag Berlin Heidelberg New York, ISBN 3-540-42849-6, 12. Auflage, 2002

[6] H. Pfeifer: Leitungen und Antennen, Wissenschaftliche Taschenbücher; Akademie-Verlag 7041, 2. Auflage, EDV760 0345

[7] Kühn: Handbuch TTL- und CMOS-Schaltkreise; VEB Verlag Technik, Bestellnummer 5534591, 1985

[8] D. Eckhardt, W. Groß: Grundlagen der digitalen Schaltungstechnik; Militärverlag der Deutschen

Demokratischen Republik (VEB), Bestellnummer: 7464064, 1981

[9] Texas Instruments: Digital Design Workshop; EH9448/04/902/00/D

[10] Avago TECHNOLOGIES: Optocouplers – Designer´s Guide; AV02-4387EN


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