# SECTION 2 Transmission Line Theory

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1 SEMICONDUCTOR DESIGN GUIDE Transmission Line Theory SECTION 2 Transmission Line Theory Introduction The ECLinPS family has pushed the world of ECL into the realm of picoseconds. When output transitions move into this picosecond region it becomes necessary to analyze system interconnects to determine if transmission line phenomena will occur. A handy rule of thumb to determine if an interconnect trace should be considered a transmission line is if the interconnect delay is greater than 1/8th of the signal transition time, it should be considered a transmission line and afforded all of the attention required by a transmission line. If this rule is applied to the ECLinPS product line a typical PCB trace will attain transmission line behaviors for any length >1/4. Thus, a brief overview of transmission line theory is presented, including a discussion of distributed and lumped capacitance effects on transmission lines. For a more thorough discussion of transmission lines the reader is referred to Motorola s MECL Systems Design Handbook. Background Exact transmission line analysis can be both tedious and time consuming; therefore, assumptions for simplifying these types of calculations must be made. A reasonable assumption is that interconnect losses caused by factors such as bandwidth limitations, attenuation, and distortion are negligible for a typical PCB trace. The basis for this assumption is that losses due to the interconnect conductor are only a fraction of the total losses in the entire interface scheme. In addition, the conductivity of insulating material is very small; as a result, the dielectric losses are minimal. A second, and more fundamental, assumption is that transmission line behavior can be described by two parameters: line characteristic impedance (ZO), and propagation delay (TPD) Characteristic Impedance An interconnect which consists of two conductors and a dielectric, characterized by distributed series resistances and inductances along with distributed parallel capacitances between them, is defined as a transmission line. These transmission lines exhibit a characteristic impedance over any length for which the distributed parameters are constant. Since the contribution of the distributed series resistance to the overall impedance is minimal, this term can be neglected when expressing the characteristic impedance of a line. The characteristic impedance is a dynamic quantity defined as the ratio of the transient voltage to the transient current passing through a point on the line. Thus, ZO can be expressed in terms of the distributed inductance and capacitance of the line as shown by Equation 1. ZO=V/I= (LO/CO) (Equation 1) where: LO = Inductance per unit length (H) CO = Capacitance per unit length (F) Propagation Delay Propagation delay (TPD) is also expressed as a function of both the inductance and capacitance per unit length of a transmission line. The propagation delay of the line is defined by the following equation: TPD= (LO*CO) (Equation 2) If LO is expressed as microhenry s per unit length and capacitance as picofarad s per unit length, the units for delay are nanoseconds per unit length. The propagation velocity is defined as the reciprocal of the propagation delay: ν=1/tpd=1/ (LO*CO) LO and CO can be determined using the easily measured parameters of line delay (TD), line length (L), and the line characteristic impedance (ZO) in conjunction with Equations 1 and 2. The propagation delay is defined as the ratio of line delay to line length: TPD=TD/L Combining equations 1 and 2 yields: CO=TPD/ZO (Equation 3) LO=TPD*ZO (Equation 4) Termination and Reflection Theory Figure 2.1 shows an ECLinPS gate driving a lossless transmission line of characteristic impedance ZO, and terminated with resistance RT. Modifying the circuit of Figure 2.1 such that the driving gate is represented by its equivalent circuit gives the configuration shown in Figure 2.2. For a positive step function, a voltage step VS, travels down the transmission line. The initial current in the transmission line is determined by the ratio VS/ZO. When the traveling wave arrives at the termination resistor RT, Ohm s RO = 6Ω ZO RE Figure 2.1. Typical Transmission Line Driving Scenario NODE a 5 1 MOTOROLA

2 Law must be maintained. If the line characteristic impedance and the termination resistance match (i.e. ZO=RT),the traveling wave does not encounter a discontinuity at the line-load interface; thus, the total voltage across the termination resistance is the incident voltage VS. However, if mismatches between the line characteristic impedance and the termination resistance occur, a reflected wave must be set up to ensure Ohm s Law is obeyed at the line-load interface. In addition, the reflected wave may also encounter a discontinuity at the interface between the transmission line and the source resistance, thereby sending a re-reflected wave back towards the load. When neither the source nor the load impedance match the line characteristic impedance multiple reflections occur with the reflected signals being attenuated with each passage over the transmission line. The output response of this configuration appears as a damped oscillation, asymptotically approaching the steady state value, a phenomenon often referred to as ringing. RS VS, IS ZO VR IR RT NODE a Figure 2.2. Thevenin Equivalent Circuit of Figure 2.1 IT IS = VS/ZO IR = VR/ZO Using substitution: /RT = VS/ZO VR/ZO (Equation 5) Since only one voltage can exist at node a at any instant in time: = VS + VR (Equation 6) Combining Equations 5 and 6, and solving for VR yields: (VS + VR)/RT = VS/ZO VR/ZO VR = ((RT ZO)/(RT+ZO))*VS VR/VS = ρ L = (RT ZO)/(RT+ZO) (Equation 7) Therefore: VR = ρ L *VS The term ρ L referred to as the load reflection coefficient, represents the fraction of the voltage wave arriving at the lineload interface that is reflected back toward the source. Similarly, a source reflection coefficient can be derived as: In performing transmission line analysis, designers may encounter one of three impedance situations: 1. RS < ZO; RT ZO 2. RS ZO; RT = ZO 3. RS = ZO; RT ZO ρ S = (RS ZO)/(Rs+ZO) (Equation 8) From equations 7 and 8 it is apparent that multiple reflections will occur when neither the source nor the load impedances match the characteristic impedance of the line. A general equation for the total line voltage as a function of time and distance is expressed by Equation 9. where: RS RT = Source Resistance = Termination Resistance V(x,t) VA(t)*[U(t TPD*x) + ρ L *U(t TPD(2L x) + ρ L *ρ S *U(t TPD(2L+x)) + (ρ L **2)*(ρ S *U(t TPD(4L x)) + (ρ L **2)*ρ S **2)*U(t TPD(4L+x)) + ] + VDC (Equation 9) Case 1: RS < ZO; RT ZO The initial current in the transmission line is determined by the ratio VS/ZO. However, the final steady state current is determined by the ratio VS/RT; assuming ohmic losses in the transmission line are negligible. For case 1, an impedance discontinuity exists at the line-load interface which causes a reflected voltage and current to be generated at the instant the initial signal arrives at this interface. To determine the fraction of the traveling wave that will be reflected from the line-load interface, Kirchoff s current law is applied to node a in Figure 2.2. This results in the following: IT = IS + IR where: IT = /RT where: VA = Voltage Entering the Transmission Line TPD = Propagation Delay of the Line L = Total Line Length x = Distance to an Arbitrary Point on the Line VDC = Initial Quiescent Voltage of the Line Finally, the output voltage, can be derived from the reflection coefficient by combining Equations 6 and 7: =(1+(RT ZO)/(RT+ZO))* VS =(2*RT/(RT+ZO))*VS The two possible configurations for the Case 1 conditions are RT > ZO and RT < ZO. The following paragraphs will describe these two conditions in detail. MOTOROLA 5 2

3 Configuration 1: RT > ZO For the case in which RT > ZO, ρ L is positive, and the initial current at node a is greater than the final quiescent current: IINITIAL > IFINAL Hence: VS/ZO)>(VS/RT) Thus, a reflected current, IR, must flow toward the source in order to attain the final steady state current as shown in Figure 2.3. An example of a line mismatched at both ends, with the termination resistance greater than the load resistance is shown ρ S = (RS Z0)/(RS+Z0) = (6 50)/(6+50) = 0.79 From Equation 9, the output voltage after one line delay is: (L,TPD) = VA(t)*[1 + ρ L ] + VDC = 0.71V Likewise, after a time equal to three times the line delay, the output voltage is (L,3TPD) = VA(t)*[ρ L *ρ S + ρ L **2*ρ S ] + (L,TPD) = 0.83V Additional iterations of Equation 9 can be performed to show that the ringing asymptotically approaches the final line voltage of 0.82V. Ringing is a characteristic response for transmission lines mismatched at both ends with RT > RO. A SPICE representation of configuration 1 is illustrated in Figure 2.5. VR IS + IR LINE VOLTAGE IR VS IS TL < t < 2TL L DISTANCE Figure 2.3. Reflected Voltage Wave for RT > ZO OUTPUT VOLTAGE- in Figure 2.4. The initial steady state output voltage is given by: I = (65/71)*( 1.75) = 1.60V The final steady state output voltage is given by: RS = RO = 6Ω Figure 2.5. SPICE Results for Circuit of Figure 2.4 Configuration 2: RT < ZO For the case in which RT<ZO, ρ L is negative, and the initial current at node a is less than the final quiescent current. RT = 65Ω IINITIAL < IFINAL Hence: Figure 2.4. Transmission Line Model for RT > ZO F = (65/71)*( 0.9) = 0.82V The input voltage is a ramp from 1.75V to 0.9V. The initial voltage traveling down the line is: VS = (50/56)*0.85 = 0.76V From Equations 7 and 8: ρ L = (RT Z0)/(RT+Z0) = (65 50)/(65+50) = 0.13 (VS/ZO) < (VS/RT) The reflected current, IR, flows in the same direction as the initial source current in order to attain the final steady state current. The unique characteristic of configuration 2 is the negative reflection coefficient at both the source and load ends of the transmission line (Figure 2.6). Thus, signals approaching either end of the line are reflected with opposite polarity. In addition, the line voltage is a function of the pulse duration yielding steps of decreasing magnitude for input pulse durations greater than the line delay, and a series of attenuated pulses for input pulse durations less than the line delay. 5 3 MOTOROLA

4 LINE VOLTAGE VS IS IR VR IS + IR volts. Stair-steps are characteristic responses for transmission lines mismatched at both ends with RT < ZO, and a pulse width greater than the line delay. Figure 2.8 shows the results of a SPICE simulation for the case described by configuration 2. TL < t < 2TL Figure 2.6. Reflected Voltage Wave for RT < ZO L DISTANCE An example of a line mismatched at both ends, with the termination resistance less than the line resistance, and the input pulse width greater than the line delay is shown in Figure 2.7. The initial steady state output voltage is defined as: I = (35/41)*( 1.75) = 1.49V The final steady state output voltage is given by RS = RO = 6Ω RT = 35Ω OUTPUT VOLTAGE-V T Figure 2.8. SPICE Results for Circuit of Figure 2.7 with Input Pulse Width > Line Delay Figure 2.9 shows the line response for the same circuit as above, but for the case in which the input pulse width is less than the line delay. As in the previous example, the initial steady state voltage across the transmission line is 1.49 volts, and the reflection coefficients are 0.18 and 0.79 for the load and source respectively. However, the intermediate voltage across the transmission line is a series of positive-going pulses of decreasing amplitude for each round Figure 2.7. Transmission Line Model for RT < ZO F = (35/41)*( 0.9) = 0.77V For an input pulse from 1.75V to 0.9V the initial voltage traveling down the line is: VS = (50/56)*0.85 = 0.76V OUTPUT VOLTAGE-V T From Equations 7 and 8, ρ L = (35 50)/(35+50) = 0.18 ρ S = (6 50)/(6+50) = 0.79 From Equation 9, the output voltage after one line delay is: (L,TPD) = VA(t) * [1 + ρ L ] + VDC = 0.87V Likewise, after a time equal to three times the line delay, the output voltage is: (L,3TPD)=VA(t)*[ρ L *ρ L +ρ L **2*ρ S ]+(L,TPD)= 0.78V Additional iterations of Equation 9 can be performed to show that the output response asymptotically approaches 0.77 Figure 2.9. SPICE Results for Circuit of Figure 2.7 with Input Pulse < Line Delay trip of the reflected voltage, until the final steady state voltage of 1.49 volts is reached. Shorted Line The shorted line is a special case of configuration 2 in which the load reflection coefficient is 1.0, and the reflections tend toward the steady state condition of zero line voltage and a current defined by the source voltage and the source resistance. MOTOROLA 5 4

5 RS = 16.7Ω Figure Transmission Line Model for Shorted Line the load and source respectively. However, the intermediate voltage across the transmission line is a series of negative pulses with the amplitude of each pulse decreasing for each round trip of the reflected voltage until the final steady state voltage of zero volts is attained. Again, for this example, the amplitude of the output response decreases by 50% for each successive reflection An example of a shorted line is shown in Figure The transmission line response for the case in which the input pulse width is greater than the line delay is shown in Figure The initial and final steady state voltages across the transmission line are zero. The source is a step function with a 0.85 volt amplitude. The initial voltage traveling down the line is: VS = (50/66.7)*0.85 = 0.64V From Equations 7 and 8, ρ L = (0 50)/(0+50) = 1 ρ S = ( )/( ) = 0.5 Upon reaching the shorted end of the line, the initial voltage waveform is inverted and reflected toward the source. At the source end, the voltage is partially reflected back toward the shorted end in accordance with the source reflection coefficient. Thus, the voltage at the shorted end of the transmission line is always zero while at the source end, the voltage is reduced for each round trip of the reflected voltage. The voltage at the source end tends toward the final steady state condition of zero volts across the transmission line. The values of the source and line characteristic impedances in this example are such that the amplitude decreases by 50% with each successive round trip across the transmission line. TRANSMISSION LINE VOLTAGE Figure SPICE Results for Shorted Line with the Input Pulse Width > Line Delay Figure 2.12 shows the line response for the same circuit as above, but for the case in which the input pulse width is less than the line delay. As in the previous example, the initial and final steady state voltages across the transmission line are zero, and the reflection coefficients are 1.0 and 0.5 for TRANSMISSION LINE VOLTAGE Figure SPICE Results for Shorted Line with the Input Pulse Width < Line Delay due to the choice of source and transmission line characteristic impedances. Case 2: RS ZO; RT = ZO As in Case 1, the initial current in the transmission line is determined by the ratio of VS/ZO. Similarly, since RT = ZO the final steady state current is also determined by the ratio VS/ZO. Because a discontinuity does not exist at the line-load interface, all the energy in the traveling step is absorbed by the termination resistance, in accordance with Ohm s Law. Therefore, no reflections occur and the output response is merely a delayed version of the input waveform. RS = RO = 6Ω Figure Transmission Line Model for Matched Termination RT = 50Ω An example of a line mismatched at the source but matched at the load is shown in Figure For an input pulse of 1.75V to 0.9V is given by: I = (50/56)*( 1.75) = 1.56V The final steady state output voltage is given by F = (50/56)*( 0.9) = 0.80V The source is a step function with an 0.85 volt amplitude. The initial voltage traveling down the line is: VS = (50/56)*0.85 = 0.76V 5 5 MOTOROLA

6 From Equations 7 and 8, ρ L = (50 50)/(65+50) = 0 ρ S = (6 50)/(6+50) = 0.79 From Equation 9, the output voltage after one line delay is: (L,TPD) = VA(t)*[1 + ρ L ] + VDC = 0.80V Likewise, after a time equal to three times the line delay, the output voltage is: (L,3TPD) = VA(t)*[ρ L *ρ S + ρ L **2*ρ S ] + (L,TPD) = 0.80V Thus, the output response attains its final steady state value (Figure 2.14) after only one line delay when the termination An example of a line mismatched at the load but matched at the source is shown in Figure For an input pulse of 1.75V to 0.9V the initial steady state output voltage is given by: I = (65/115)*( 1.75) = 0.99V The final steady state output voltage is given by F = (65/115)*( 0.9) = 0.51V The source is a step function with a 0.85 volt amplitude. The initial voltage traveling down the line is: VS = (50/100)*0.85 = 0.43V From Equations 7 and 8, ρ L = (65 50)/(65+50) = 0.13 ρ S = (50 50)/(50+50) = 0 From Equation 9, the output voltage after one line delay is: (L,TPD) = VA(t)*[1 + ρ L ] + VDC = 0.51V OUTPUT VOLTAGE-V T H=400ps/div Figure SPICE Results for Matched Termination Likewise, after a time equal to three times the line delay, the output voltage is: ( L,3TPD) = VA(t)*[ρ L *ρ S + ρ L **2*ρ S ] + (L,TPD) = 0.51V Thus, the output response attains its final steady state value after one line delay when the source resistance matches the line characteristic impedance. Again, ringing or a stair-step output does not occur since the load reflection coefficient is zero (Figure 2.16). resistance matches the line characteristic impedance. Ringing or stair-step output responses do not occur since the load reflection coefficient is zero. Case 3: RS = ZO; RT = ZO When the termination resistance does not match the line characteristic impedance reflections arising from the load will occur. Fortunately, in case 3, the source resistance and the line characteristic impedance are equal, thus, the reflection coefficient is zero and the energy in these reflections is completely absorbed at the source; thus, no further reflections occur. OUTPUT VOLTAGE-V T H=400ps/div V=100mV/div RS = 50Ω Figure SPICE Results for Circuit of Figure 2.15 RT = 65Ω Figure Transmission Line Model for VS = ZO Series Termination Series termination represents a special subcategory of Case 3 in which the load reflection coefficient is +1 and the source resistance is made equal to the line characteristic impedance by inserting a resistor, RST, between and in series with, the transmission line and the source resistance RO. The MOTOROLA 5 6

7 reflections tend toward the steady state conditions of zero current in the transmission line, and an output voltage equal to the input voltage. This type of termination is illustrated by the circuit configuration of Figure The initial voltage down the line will be only half the amplitude of the input signal due to the voltage division of the equal source and line impedances. Since the load reflection coefficient is unity, the voltage at the output attains the full ECL swing, whereas the voltage at the beginning of the transmission line does not attain this level until the reflected voltage arrives back at the source termination (Figures 2.17 and 2.18). No other reflections occur because the source impedance and line characteristic impedance match. VS = (ZO/(2*ZO))* = /2 (Equation 10) The load reflection coefficient tends to unity, thus, a voltage wave arriving at the load will double in amplitude, and a reflected wave with the same amplitude as the incident wave RO = 6Ω Capacitive Effects on Propagation Delay Lumped Capacitive Loads The effect of load capacitance on propagation delay must RST = 44Ω Figure Series Terminated Transmission Line will be reflected toward the source. Since the source resistance matches the line characteristic impedance all the energy in the reflected wave is absorbed, and no further reflections occur. This source absorption feature reduces the effects of ringing, making series terminations particularly useful for transmitting signals through a backplane or other interconnects where discontinuities exist. As stated previously, the signal in the line is only at half amplitude and the reflection restores the signal to the full amplitude. It is important to ensure that all loads are located near the end of the transmission line so that a two step input signal is not seen by any of the loads. For the series terminated circuit of Figure 2.17 with RO + RST = ZO and an input pulse rising from 1.75V to 0.9V, the initial line voltage, I is 1.325V and the final line voltage, F, is 0.9V. The source is a step function with a 0.85 volt amplitude. The initial voltage traveling down the line is: VS = (50/100)*0.85 = 0.43V From Equations 7 and 8, OUTPUT VOLTAGES H=400ps/div Figure SPICE Results for Series Terminated Line be considered when using high performance integrated circuits such as the ECLinPS family. Although capacitive loading affects both series and parallel termination schemes, it is more pronounced for the series terminated case. Figure 2.19a illustrates a series terminated line with a capacitive load CL. Under the no load condition, CL=0, the delay between the 50% point of the input waveform to the 50% point of the output waveform is defined as the line delay TD. A capacitive load VS ρ L = ( 50)/( +50) = 1 ρ S = (50 50)/(50+50) = 0 From Equation 9, the output voltage after one line delay is: RS = Z0 CL NODE a (L,TPD) = VA(t)*[1 + ρ L ] + VDC = 0.9V Likewise, after a time equal to three times the line delay, the output voltage is: (L,3TPD) = VA(t)*ρ L *ρ S + ρ L **2*ρ S ] + (L,TPD) = 0.9V Figure 2.19a. Lumped Load Transmission Line Model placed at the end of the line increases the rise time of the output signal, thereby increasing TD by an amount TD (Figure 2.19b). Figure 2.20 shows the increase in delay for load capacitances of 0, 1, 5, 10 and 20 picofarads. 5 7 MOTOROLA

8 LINE INPUT 50% ZO CL Z = ZO Z C = ZO CL CL TD THEVENIN EQUIVALENT SERIES TERMINATION TD LINE OUTPUT 50% LOADED RESPONSE UNLOADED RESPONSE ZO Z = ZO/2 /2 RT CL Z C = (ZO/2)CL THEVENIN EQUIVALENT PARALLEL TERMINATION CL Figure 2.19b. TD Introduced by Capacitive Load OUTPUT VOLTAGES CL = 0pF CL = 1pF CL = 5pF CL = 10pF CL = 20pF Figure Line Delay vs Lumped Capacitive Load The increase in propagation delay can be determined by using Thevenin s theorem to convert the transmission line into a single time constant network with a ramp input voltage. The analysis applies to both series and parallel terminations, since both configurations can be represented as a single time constant network with a time constant, τ, and a Thevenin impedance Z. Figure 2.21 shows the Thevenized versions for the series and parallel terminated configurations. The Thevenin impedance for the series configuration is approximately twice that for the parallel terminated case, thus the time constant will be two times greater for the series terminated configuration. Since τ is proportional to the risetime, the rise time will also be two times greater; thus the reason for the larger impact of capacitive loading on the series terminated configuration. NORMALIZED DELAY INCREASE ( T D / t R ) VLOW 80% 50% 20% t = 0 t = a tr = 0.6a a = 1.67tR VHIGH Figure Thevenin Equivalent Lumped Capacitance Circuits NORMALIZED LINE-LOAD CONSTANT (Z C/tR) Figure Normalized Delay Increase Due to Lumped Capacitive Load The relationship between the change in delay and the line-load time constant is shown in Figure Both the delay change ( TD) and the line-load time constant (Z C) are normalized to the 20 80% risetime of the input signal. This chart provides a convenient graphical approach for approximating delay increases due to capacitive loads as illustrated by the following example. MOTOROLA 5 8

10 The reduced line impedance is obtained by adding CD to CO in Equation 1. ZO (LO/CO) ZO (LO/(CO + CD)) = (LO/(CO * (1+CD/CO))) ZO = ZO/ (1 + CD/CO) (Equation 12) For the circuit used to obtain the traces in Figure 2.25, the distributed load capacitance is 4pF. From Equation 3, CO is calculated as CO = 750ps/93Ω = 8pF Hence: ZO 93Ω/ (1 + 4pF/8pF) = 76Ω Thus, the effective line impedance is 17Ω lower than the actual impedance while reflections are occurring on the line. Line Delay Increase The increase in line delay caused by distributed loading is calculated by adding the distributed capacitance (CD) to the intrinsic line capacitance in Equation 2. TPD = (LO * CO) TPD = (LO * (CO+CD)) TPD = TPD * (1 + CD/CO) (Equation 1 3) Once again, for the circuit used to obtain the traces in Figure 2.25, the distributed load capacitance is 4pF. From the previous example, the intrinsic line capacitance is 8pF therefore, TPD = 750ps * (1 + 4pF/8pF) = 919ps Thus, the effect of distributed load capacitance on line delay is to increase the delay by 169ps. From Equation 13 it is obvious that the larger the CO of the line the smaller will be the increase in delay due to a distributive capacitive load. Therefore, to obtain the minimum impedance change and lowest propagation delay as a function of gate loading, the lowest characteristic impedance line should be used as this results in a line with the largest intrinsic line capacitance. MOTOROLA 5 10

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