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17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics17: Transmission Lines SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 1 / 13
Transmission Lines17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristicsPreviously assume that any change in v0 (t) appears instantly at vL (t). SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 2 / 13
Transmission Lines17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristicsPreviously assume that any change in v0 (t) appears instantly at vL (t).This is not true. SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 2 / 13
Transmission Lines17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics SummaryPreviously assume that any change in v0 (t) appears instantly at vL (t).This is not true. If fact signals travel at around half the speed of light (c 30 cm/ns).E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 2 / 13
Transmission Lines17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics SummaryPreviously assume that any change in v0 (t) appears instantly at vL (t).This is not true. If fact signals travel at around half the speed of light (c 30 cm/ns).Reason: all wires have capacitance to ground and to neighbouringconductors and also self-inductance. It takes time to change the currentthrough an inductor or voltage across a capacitor.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 2 / 13
Transmission Lines17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics SummaryPreviously assume that any change in v0 (t) appears instantly at vL (t).This is not true. If fact signals travel at around half the speed of light (c 30 cm/ns).Reason: all wires have capacitance to ground and to neighbouringconductors and also self-inductance. It takes time to change the currentthrough an inductor or voltage across a capacitor.A transmission line is a wire with a uniform goemetry along its length: thecapacitance and inductance of any segment is proportional to its length.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 2 / 13
Transmission Lines17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics SummaryPreviously assume that any change in v0 (t) appears instantly at vL (t).This is not true. If fact signals travel at around half the speed of light (c 30 cm/ns).Reason: all wires have capacitance to ground and to neighbouringconductors and also self-inductance. It takes time to change the currentthrough an inductor or voltage across a capacitor.A transmission line is a wire with a uniform goemetry along its length: thecapacitance and inductance of any segment is proportional to its length.We represent as a large number of small inductors and capacitors spacedalong the line.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 2 / 13
Transmission Lines17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics SummaryPreviously assume that any change in v0 (t) appears instantly at vL (t).This is not true. If fact signals travel at around half the speed of light (c 30 cm/ns).Reason: all wires have capacitance to ground and to neighbouringconductors and also self-inductance. It takes time to change the currentthrough an inductor or voltage across a capacitor.A transmission line is a wire with a uniform goemetry along its length: thecapacitance and inductance of any segment is proportional to its length.We represent as a large number of small inductors and capacitors spacedalong the line.The signal speed along a transmisison line is predictable.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 2 / 13
Transmission Line Equations A short section of line δx long:17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward Backwardv(x, t) and i(x, t) depend on bothposition and time.Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 3 / 13
Transmission Line EquationsA short section of line δx long:17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward Backwardv(x, t) and i(x, t) depend on bothposition and time.Small δx ignore 2nd order derivatives:Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics v(x,t) t v(x δx,t) t, v t . SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 3 / 13
Transmission Line EquationsA short section of line δx long:17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward Backward Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Line Summaryv(x, t) and i(x, t) depend on bothposition and time.Small δx ignore 2nd order derivatives:WavesCharacteristics v(x,t) t v(x δx,t) t, v t .Basic EquationsKVL: v(x, t) V2 v(x δx, t) V1KCL: i(x, t) iC i(x δx, t)E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 3 / 13
Transmission Line EquationsA short section of line δx long:17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward Backward Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Line Summaryv(x, t) and i(x, t) depend on bothposition and time.Small δx ignore 2nd order derivatives:WavesCharacteristics v(x,t) t v(x δx,t) t, v t .Basic EquationsKVL: v(x, t) V2 v(x δx, t) V1KCL: i(x, t) iC i(x δx, t) iCapacitor equation: C v t iC i(x, t) i(x δx, t) x δxE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 3 / 13
Transmission Line EquationsA short section of line δx long:17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward Backward Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Line Summaryv(x, t) and i(x, t) depend on bothposition and time.Small δx ignore 2nd order derivatives:WavesCharacteristics v(x,t) t v(x δx,t) t, v t .Basic EquationsKVL: v(x, t) V2 v(x δx, t) V1KCL: i(x, t) iC i(x δx, t) iCapacitor equation: C v t iC i(x, t) i(x δx, t) x δxInductor equation (L1 and L2 have the same current): v i V1 V2 v(x, t) v(x δx, t) xδx(L1 L2 ) tE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 3 / 13
Transmission Line EquationsA short section of line δx long:17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward Backward Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Line Summaryv(x, t) and i(x, t) depend on bothposition and time.Small δx ignore 2nd order derivatives:WavesCharacteristics v(x,t) t v(x δx,t) t, v t .Basic EquationsKVL: v(x, t) V2 v(x δx, t) V1KCL: i(x, t) iC i(x δx, t) iCapacitor equation: C v t iC i(x, t) i(x δx, t) x δxInductor equation (L1 and L2 have the same current): v i V1 V2 v(x, t) v(x δx, t) xδx(L1 L2 ) tTransmission Line Equations iC0 v t x v iL0 t xE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 3 / 13
Transmission Line EquationsA short section of line δx long:17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward Backward Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Line Summaryv(x, t) and i(x, t) depend on bothposition and time.Small δx ignore 2nd order derivatives:WavesCharacteristics v(x,t) t v(x δx,t) t, v t .Basic EquationsKVL: v(x, t) V2 v(x δx, t) V1KCL: i(x, t) iC i(x δx, t) iCapacitor equation: C v t iC i(x, t) i(x δx, t) x δxInductor equation (L1 and L2 have the same current): v i V1 V2 v(x, t) v(x δx, t) xδx(L1 L2 ) tTransmission Line Equations iC0 v t x v iL0 t xE1.1 Analysis of Circuits (2017-10213)Cis the capacitance per unit lengthwhere C0 δx L2(Farads/m) and L0 L1δxis the totalinductance per unit length (Henries/m).Transmission Lines: 17 – 3 / 13
Solution to Transmission Line Equations17: Transmission LinesTransmission Line Equations: Transmission Lines Transmission Line iC0 v t x i vL0 t xEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 4 / 13
Solution to Transmission Line Equations17: Transmission LinesTransmission Line Equations: Transmission Lines Transmission LineEquations Solution to TransmissionLine EquationsGeneral solution: Forward Wave Forward Backward Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Line i vL0 t xv(t, x) f (t ux ) g(t ux )i(t, x) WavesCharacteristics iC0 v t xwhere u xx) g(t u)f (t uZ0q1L0 C0and Z0 qL0C0. SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 4 / 13
Solution to Transmission Line Equations17: Transmission LinesTransmission Line Equations: Transmission Lines Transmission LineEquations Solution to TransmissionLine EquationsGeneral solution: Forward Wave Forward Backward Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Linewhere u i vL0 t xv(t, x) f (t ux ) g(t ux )i(t, x) WavesCharacteristics iC0 v t xxx) g(t u)f (t uZ0q1L0 C0and Z0 qL0C0.u is the propagation velocity and Z0 is the characteristic impedance. SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 4 / 13
Solution to Transmission Line Equations17: Transmission LinesTransmission Line Equations: Transmission Lines Transmission LineEquations Solution to TransmissionLine EquationsGeneral solution: Forward Wave Forward Backward Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Linewhere u i vL0 t xv(t, x) f (t ux ) g(t ux )i(t, x) WavesCharacteristics iC0 v t xxx) g(t u)f (t uZ0q1L0 C0and Z0 qL0C0.u is the propagation velocity and Z0 is the characteristic impedance. Summaryf () and g() can be any differentiable functions.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 4 / 13
Solution to Transmission Line Equations17: Transmission LinesTransmission Line Equations: Transmission Lines Transmission LineEquations Solution to TransmissionLine EquationsGeneral solution: Forward Wave Forward Backward Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Linewhere u i vL0 t xv(t, x) f (t ux ) g(t ux )i(t, x) WavesCharacteristics iC0 v t xxx) g(t u)f (t uZ0q1L0 C0and Z0 qL0C0.u is the propagation velocity and Z0 is the characteristic impedance. Summaryf () and g() can be any differentiable functions.Verify by substitution: i x E1.1 Analysis of Circuits (2017-10213) xx) g ′ (t u) f ′ (t uZ0 1u Transmission Lines: 17 – 4 / 13
Solution to Transmission Line Equations17: Transmission LinesEquations Solution to TransmissionLine EquationsGeneral solution: Forward Wave Forward Backward Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Linewhere u i vL0 t xv(t, x) f (t ux ) g(t ux )xx) g(t u)f (t uZ0i(t, x) WavesCharacteristics iC0 v t xTransmission Line Equations: Transmission Lines Transmission Lineq1L0 C0and Z0 qL0C0.u is the propagation velocity and Z0 is the characteristic impedance. Summaryf () and g() can be any differentiable functions.Verify by substitution: i x xx) g ′ (t u) f ′ (t uZ0′ C0 f (t E1.1 Analysis of Circuits (2017-10213)xu)′1u xu) g (t C0 v tTransmission Lines: 17 – 4 / 13
Forward Wave17: Transmission LinesSuppose: Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWavesu 15 cm/nsand g(t) 0 x v(x, t) f t u Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 5 / 13
Forward Wave17: Transmission LinesSuppose: Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves At x 0 cm [N],vS (t) f (t u0 ) Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristicsu 15 cm/nsand g(t) 0 x v(x, t) f t uf(t-0/u)02468Time (ns)10 SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 5 / 13
Forward Wave17: Transmission LinesSuppose: Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves At x 0 cm [N],vS (t) f (t u0 ) Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristicsu 15 cm/nsand g(t) 0 x v(x, t) f t u At x 45 cm [N],v(45, t) f (t 45u )f(t-0/u)0f(t-45/u)2468Time (ns)10 SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 5 / 13
Forward Wave17: Transmission LinesSuppose: Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves At x 0 cm [N],vS (t) f (t u0 ) Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics Summaryu 15 cm/nsand g(t) 0 x v(x, t) f t u At x 45 cm [N],v(45, t) f (t 45u )f (t E1.1 Analysis of Circuits (2017-10213)45u )f(t-0/u)0f(t-45/u)2468Time (ns)10is exactly the same as f (t) but delayed by 45u 3 ns.Transmission Lines: 17 – 5 / 13
Forward Wave17: Transmission LinesSuppose: Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves At x 0 cm [N],vS (t) f (t u0 ) Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics Summaryu 15 cm/nsand g(t) 0 x v(x, t) f t u At x 45 cm [N],v(45, t) f (t 45u )f(t-0/u)0f(t-45/u)24f(t-90/u)68Time (ns)1045f (t 45)isexactlythesameasf(t)butdelayedbyuu 3 ns. At x 90 cm [N], vR (t) f (t 90u ); now delayed by 6 ns.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 5 / 13
Forward Wave17: Transmission LinesSuppose: Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves At x 0 cm [N],vS (t) f (t u0 ) Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics Summaryu 15 cm/nsand g(t) 0 x v(x, t) f t u At x 45 cm [N],v(45, t) f (t 45u )f(t-0/u)0f(t-45/u)24f(t-90/u)68Time (ns)1045f (t 45)isexactlythesameasf(t)butdelayedbyuu 3 ns. At x 90 cm [N], vR (t) f (t 90u ); now delayed by 6 ns.Waveform at x 0 completely determines the waveform everywhere else.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 5 / 13
Forward Wave17: Transmission LinesSuppose: Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves At x 0 cm [N],vS (t) f (t u0 ) Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics Summaryu 15 cm/nsand g(t) 0 x v(x, t) f t u f(t-0/u) At x 45 cm [N],v(45, t) f (t 45u )0f(t-45/u)24f(t-90/u)68Time (ns)1045f (t 45)isexactlythesameasf(t)butdelayedbyuu 3 ns. At x 90 cm [N], vR (t) f (t 90u ); now delayed by 6 ns.Waveform at x 0 completely determines the waveform everywhere else.Snapshot at t0 4 ns:the waveform has justarrived at the pointx ut0 60 cm.E1.1 Analysis of Circuits (2017-10213)t 4 ns0f(4-x/u)20406080Position (cm)Transmission Lines: 17 – 5 / 13
Forward Wave17: Transmission LinesSuppose: Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves At x 0 cm [N],vS (t) f (t u0 ) Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics Summaryu 15 cm/nsand g(t) 0 x v(x, t) f t u f(t-0/u) At x 45 cm [N],v(45, t) f (t 45u )0f(t-45/u)24f(t-90/u)68Time (ns)1045f (t 45)isexactlythesameasf(t)butdelayedbyuu 3 ns. At x 90 cm [N], vR (t) f (t 90u ); now delayed by 6 ns.Waveform at x 0 completely determines the waveform everywhere else.Snapshot at t0 4 ns:the waveform has justarrived at the pointx ut0 60 cm.E1.1 Analysis of Circuits (2017-10213)t 4 ns0f(4-x/u)20406080Position (cm)Transmission Lines: 17 – 5 / 13
Forward Wave17: Transmission LinesSuppose: Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves At x 0 cm [N],vS (t) f (t u0 ) Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics Summaryu 15 cm/nsand g(t) 0 x v(x, t) f t u f(t-0/u) At x 45 cm [N],v(45, t) f (t 45u )0f(t-45/u)24f(t-90/u)68Time (ns)1045f (t 45)isexactlythesameasf(t)butdelayedbyuu 3 ns. At x 90 cm [N], vR (t) f (t 90u ); now delayed by 6 ns.Waveform at x 0 completely determines the waveform everywhere else.Snapshot at t0 4 ns:the waveform has justarrived at the pointx ut0 60 cm.t 4 ns0f(4-x/u)20406080Position (cm)f (t ux ) is a wave travelling forward (i.e. towards x) along the line.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 5 / 13
Forward Backward Wavesx) is a wave travelling backwards, i.e. in the x direction.Similarly g(t u17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardWaves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 6 / 13
Forward Backward Wavesx) is a wave travelling backwards, i.e. in the x direction.Similarly g(t u17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward Backwardv(x, t) f (t ux ) g(t ux )Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 6 / 13
Forward Backward Wavesx) is a wave travelling backwards, i.e. in the x direction.Similarly g(t u17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardAt x 0 cm [N],Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristicsv(x, t) f (t ux ) g(t ux )vS (t) f (t) g(t) SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 6 / 13
Forward Backward Wavesx) is a wave travelling backwards, i.e. in the x direction.Similarly g(t u17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardAt x 0 cm [N],Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics Summaryv(x, t) f (t ux ) g(t ux )vS (t) f (t) g(t) At x 90 cm [N], g starts at t 1 and f starts at t 6.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 6 / 13
Forward Backward Wavesx) is a wave travelling backwards, i.e. in the x direction.Similarly g(t u17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardAt x 0 cm [N],Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics Summaryv(x, t) f (t ux ) g(t ux )vS (t) f (t) g(t) At x 45 cm [N], g is only 1 ns behind f and they add together.At x 90 cm [N], g starts at t 1 and f starts at t 6.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 6 / 13
Forward Backward Wavesx) is a wave travelling backwards, i.e. in the x direction.Similarly g(t u17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardAt x 0 cm [N],Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics Summaryv(x, t) f (t ux ) g(t ux )vS (t) f (t) g(t) At x 45 cm [N], g is only 1 ns behind f and they add together.At x 90 cm [N], g starts at t 1 and f starts at t 6.A vertical line on the diagramgives a snapshot of the entireline at a time instant t.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 6 / 13
Forward Backward Wavesx) is a wave travelling backwards, i.e. in the x direction.Similarly g(t u17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardAt x 0 cm [N],Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics Summaryv(x, t) f (t ux ) g(t ux )vS (t) f (t) g(t) At x 45 cm [N], g is only 1 ns behind f and they add together.At x 90 cm [N], g starts at t 1 and f starts at t 6.A vertical line on the diagramgives a snapshot of the entireline at a time instant t.f and g first meet at t 3.5and x 52.5.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 6 / 13
Forward Backward Wavesx) is a wave travelling backwards, i.e. in the x direction.Similarly g(t u17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations Forward Wave Forward BackwardAt x 0 cm [N],Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics Summaryv(x, t) f (t ux ) g(t ux )vS (t) f (t) g(t) At x 45 cm [N], g is only 1 ns behind f and they add together.At x 90 cm [N], g starts at t 1 and f starts at t 6.A vertical line on the diagramgives a snapshot of the entireline at a time instant t.f and g first meet at t 3.5and x 52.5.Magically, f and g passthrough each other entirelyunaltered.E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 6 / 13
Power Flowxu17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations and gx (t) g t Define fx (t) f t backward waveforms at any point, x.xu to be the forward and Forward Wave Forward BackwardWaves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristics SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 7 / 13
Power Flowxu17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equations and gx (t) g t Define fx (t) f t backward waveforms at any point, x. to be the forward andi is always Forward Wave Forward Backwardmeasured in the ve x direction.Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission LineCharacteristicsxuThenvx (t) fx (t) gx (t)andix (t) Z0 1 (fx (t) gx (t)). SummaryE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 7 / 13
Power Flowxu17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equationsand gx (t) g t Define fx (t) f t backward waveforms at any point, x. to be the forward andmeasured in the ve x direction.Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Line Summaryxui is always Forward Wave Forward BackwardCharacteristics Then vx (t) fx (t) gx (t) and ix (t) Z0 1 (fx (t) gx (t)).Note: Knowing the waveform fx (t) or gx (t) at any position x, tells you it aty xy xand gy (t) gx t u .all other positions: fy (t) fx t uE1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 7 / 13
Power Flowxu17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equationsand gx (t) g t Define fx (t) f t backward waveforms at any point, x. to be the forward andmeasured in the ve x direction.Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Line Summaryxui is always Forward Wave Forward BackwardCharacteristics Then vx (t) fx (t) gx (t) and ix (t) Z0 1 (fx (t) gx (t)).Note: Knowing the waveform fx (t) or gx (t) at any position x, tells you it aty xy xand gy (t) gx t u .all other positions: fy (t) fx t uPower FlowThe power transferred into the shaded region across the boundary at x isPx (t) vx (t)ix (t)E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 7 / 13
Power Flowxu17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equationsand gx (t) g t Define fx (t) f t backward waveforms at any point, x. to be the forward andmeasured in the ve x direction.Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Line Summaryxui is always Forward Wave Forward BackwardCharacteristics Then vx (t) fx (t) gx (t) and ix (t) Z0 1 (fx (t) gx (t)).Note: Knowing the waveform fx (t) or gx (t) at any position x, tells you it aty xy xand gy (t) gx t u .all other positions: fy (t) fx t uPower FlowThe power transferred into the shaded region across the boundary at x isPx (t) vx (t)ix (t) Z0 1 (fx (t) gx (t)) (fx (t) gx (t))E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 7 / 13
Power Flowxu17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equationsand gx (t) g t Define fx (t) f t backward waveforms at any point, x. to be the forward andmeasured in the ve x direction.Waves Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Line Summaryxui is always Forward Wave Forward BackwardCharacteristics Then vx (t) fx (t) gx (t) and ix (t) Z0 1 (fx (t) gx (t)).Note: Knowing the waveform fx (t) or gx (t) at any position x, tells you it aty xy xand gy (t) gx t u .all other positions: fy (t) fx t uPower FlowThe power transferred into the shaded region across the boundary at x isPx (t) vx (t)ix (t) Z0 1 (fx (t) gx (t)) (fx (t) gx (t))2fx2 (t)gx(t) Z0 Z0E1.1 Analysis of Circuits (2017-10213)Transmission Lines: 17 – 7 / 13
Power Flowxu17: Transmission Lines Transmission Lines Transmission LineEquations Solution to TransmissionLine Equationsand gx (t) g t Define fx (t) f t backward waveforms at any point, x. to be the forward andmeasured in the ve x direction.Waves Power Flow Reflections Reflection Coefficients Driving a line
Power Flow Reflections Reflection Coefficients Driving a line Multiple Reflections Transmission Line Characteristics Summary E1.1 Analysis of Circuits (2017-10213) Transmission Lines: 17 - 2 / 13 Previously assume that any change in v 0(t) appears instantly at vL(t). This is not true.
