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Step-function response of a transmission line
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Figure 3-3 shows a parallel transmission line with characteristic impedance Zo connected to a load impedance ZL The generator at the input of the line consists of a voltage source V in series with a source impedance Zs and a switch S1 Assume for the present that all impedances are pure resistances (ie, R + j0) Also, assume that Zs = Zo When the switch is closed at time To (Fig 3-4A), the voltage at the input of the line (Vin) jumps to V/2 In Fig 3-2, you may have noticed that the LC circuit resembles a delay line circuit As might be expected, therefore, the voltage wavefront propagates along the line at a velocity v of: v= where v is the velocity, in meters per second L is the inductance, in henrys C is the capacitance, in farads 1 LC [39]
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3-3 Schematic example of transmission line
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Transmission line responses 71
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3-4 Step-function propagation along transmission line at three points
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At time T1 (Fig 3-4B), the wavefront has propagated one-half the distance L, and by Td it has propagated the entire length of the cable (Fig 3-4C) If the load is perfectly matched (ie, ZL = Zo), then the load absorbs the wave and no component is reflected But in a mismatched system (ZL is not equal to Zo), a portion of the wave is reflected back down the line toward the generator Figure 3-5 shows the rope analogy for reflected pulses in a transmission line A taut rope (Fig 3-5A) is tied to a rigid wall that does not absorb any of the energy in the pulse propagated down the rope When the free end of the rope is given a vertical displacement (Fig 3-5B), a wave is propagated down the rope at velocity v (Fig 3-5C) When the pulse hits the wall (Fig 3-5D), it is reflected (Fig 3-5E) and propagates back down the rope toward the free end (Fig 3-5F) If a second pulse is propagated down the line before the first pulse dies out, then there will be two pulses on the line at the same time (Fig 3-6A) When the two pulses interfere, the resultant will be the algebraic sum of the two In the event that a pulse train is applied to the line, the interference pattern will set up standing waves, an example of which is shown in Fig 3-6B
Reflection coefficient
The reflection coefficient of a circuit containing a transmission line and load impedance is a measure of how well the system is matched The absolute value of the re-
72 Transmission lines
3-5 Rope analogy to transmission line
3-6A Interfering opposite waves
Transmission line responses 73
3-6B Standing waves
flection coefficient varies from 1 to +1, depending upon the magnitude of reflection; = 0 indicates a perfect match with no reflection, while 1 indicates a short-circuited load, and +1 indicates an open circuit To understand the reflection coefficient, let s start with a basic definition of the resistive load impedance Z = R + j0: ZL = where ZL is the load impedance R + j0 V is the voltage across the load I is the current flowing in the load Because there are both reflected and incident waves, we find that V and I are actually the sum of incident and reflected voltages and currents, respectively Therefore: ZL = ZL = where Vinc is the incident (ie, forward) voltage Vref is the reflected voltage Iinc is the incident current Iref is the reflected current V I Vinc + Vref Iinc + Iref [311A] [311B] V I [310]
74 Transmission lines Because of Ohm s law, you can define the currents in terms of voltage, current, and the characteristic impedance of the line: Vinc Iinc = [312] Zo and Iinc = Vref Zo [313]
(The minus sign in Eq 313 indicates that a direction reversal has taken place) The two expressions for current (Eqs 312 and 313) may be substituted into Eq 311 to yield ZL = Vinc + Vref Vinc Zo Vref Zo [314]
The reflection coefficient is defined as the ratio of reflected voltage to incident voltage: = Using this ratio in Eq 314 gives = ZL Zo ZL + Zo [316] Vref Vinc [315]
Example 3-3 A 50- transmission line is connected to a 30- resistive load Calculate the reflection coefficient Solution: ZL Zo ZL + Zo (50 ) (30 ) (50 ) + (30 ) 20 = 025 80
Transmission line responses 75 Example 3-4 In Example 3-3, the incident voltage is 3 V rms Calculate the reflected voltage Solution: If Vref = Vinc then Vref = Vinc = (025) (3 V) = 075 V The phase of the reflected signal is determined by the relationship of load impedance and transmission line characteristic impedance For resistive loads (Z = R + j0): if the ratio ZL/Zo is 10, then there is no reflection; if ZL/Zo is less than 10, then the reflected signal is 180 out of phase with the incident signal; if the ratio ZL/Zo is greater than 10 then the reflected signal is in phase with the incident signal In summary: Angle of reflection No reflection 180 0
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