329lect26 - 26 Introduction to distributed circuits Last...

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26 Introduction to distributed circuits Last lecture we learned that voltage and current variations on TL’s are governed by telegrapher’s equations and their d’Alembert solutions — the latter can be expressed explicitly as + - Wire 2 Wire 1 + - 0 f ( t ) R g R L I ( z, t ) V ( z, t ) z l Source ckt Transmission line Load Z o I ( z, t ) V ( z, t ) = V + ( t - z v ) + V - ( t + z v ) and I ( z, t ) = V + ( t - z v ) Z o - V - ( t + z v ) Z o in terms of v = 1 LC and Z o = L C and V + ( t ) and V - ( t ) labeling signal waveforms propagated in + z and - z directions, respectively. We will next solve a sequence of distributed circuit problems containing TL segments and two terminal elements such as resistors and voltage (or current) sources. In solving the problems, we will apply the usual rules of lumped circuit analysis at element terminals and treat the TL’s in terms of d’Alembert solutions above. Consider a TL with a characteristic impedance Z o extending from z = 0 to z = l , where a two-terminal source circuit (e.g., a receiving antenna) modeled by a Thevenin equivalent with voltage f ( t ) and resistance 1
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R g is connected between the TL terminals at z = 0 and a load (e.g., a receiver circuit) modeled by a resistance R L terminates the line at z = l (see margin). + - Wire 2 Wire 1 + - 0 f ( t ) R g R L I ( z, t ) V ( z, t ) z l Source ckt Transmission line Load Z o I ( z, t ) We want to determine voltage and current signals V ( t ) and I ( t ) on the TL and the load in terms of source signal f ( t ) . Let us simplify the problem posed above by having R g = 0 f ( t ) = u ( t ) R L = Z o in which case the circuit takes the simplified form in the margin. Circuit 1: + - + - 0 u ( t ) Z o I ( z, t ) V ( z, t ) z l Z o I ( z, t ) Now applying KVL at z = 0 we have u ( t ) = V + ( t ) + V - ( t ) . Next applying KVL and KCL at z = l we have V ( l, t ) = I ( l, t ) Z o (since current I ( l, t ) flows though the load Z o to induce V ( l, t ) ), which implies that V + ( t - l v )+ V - ( t + l v ) = ( V + ( t - l v ) Z o - V - ( t + l v ) Z o ) Z o V - ( t + l v ) = - V - ( t + l v ) , 2
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