17: Transmission Lines

1 17: 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+ 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 LineCharacteristics+ Analysis of Circuits (2017-10213) Transmission Lines : 17 2 / 13 Previously assume that any change inv0(t)appears instantly atvL(t). 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 2 / 13 Previously assume that any change inv0(t)appears instantly atvL(t).

2 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 2 / 13 Previously assume that any change inv0(t)appears instantly atvL(t). fact signals travel at around half the speed of light (c= 30cm/ns). 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 2 / 13 Previously assume that any change inv0(t)appears instantly atvL(t). fact signals travel at around half the speed of light (c= 30cm/ns).Reason:all wires have capacitance to ground and to neighbouringconductors and also self-inductance.

3 It takes time to change the currentthrough an inductor or voltage across a 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 2 / 13 Previously assume that any change inv0(t)appears instantly atvL(t). fact signals travel at around half the speed of light (c= 30cm/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 lineis a wire with a uniform goemetry along its length : thecapacitance and inductance of any segment is proportional to its 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 2 / 13 Previously assume that any change inv0(t)appears instantly atvL(t).

4 Fact signals travel at around half the speed of light (c= 30cm/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 lineis a wire with a uniform goemetry along its length : thecapacitance and inductance of any segment is proportional to its represent as a large number of small inductors and capacitors spacedalong the 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 2 / 13 Previously assume that any change inv0(t)appears instantly atvL(t). fact signals travel at around half the speed of light (c= 30cm/ns).Reason:all wires have capacitance to ground and to neighbouringconductors and also self-inductance.

5 It takes time to change the currentthrough an inductor or voltage across a lineis a wire with a uniform goemetry along its length : thecapacitance and inductance of any segment is proportional to its represent as a large number of small inductors and capacitors spacedalong the signal speed along a transmisison line is Line Equations+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 LineCharacteristics+ Analysis of Circuits (2017-10213) Transmission Lines : 17 3 / 13A short section of line xlong:v(x, t)andi(x, t) depend on bothposition and Line Equations+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 LineCharacteristics+ Analysis of Circuits (2017-10213) Transmission Lines : 17 3 / 13A short section of line xlong:v(x, t)andi(x, t) depend on bothposition and x ignore 2nd order derivatives: v(x,t) t= v(x+ x,t) t, v Line Equations+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 LineCharacteristics+ Analysis of Circuits (2017-10213) Transmission Lines : 17 3 / 13A short section of line xlong:v(x, t)andi(x, t) depend on bothposition and x ignore 2nd order derivatives: v(x,t) t= v(x+ x,t) t, v EquationsKVL.

6 V(x, t) =V2+v(x+ x, t) +V1 KCL:i(x, t) =iC+i(x+ x, t) Transmission Line Equations+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 LineCharacteristics+ Analysis of Circuits (2017-10213) Transmission Lines : 17 3 / 13A short section of line xlong:v(x, t)andi(x, t) depend on bothposition and x ignore 2nd order derivatives: v(x,t) t= v(x+ x,t) t, v EquationsKVL:v(x, t) =V2+v(x+ x, t) +V1 KCL:i(x, t) =iC+i(x+ x, t)Capacitor equation:C v t=iC=i(x, t) i(x+ x, t) = i x xTransmission Line Equations+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 LineCharacteristics+ Analysis of Circuits (2017-10213) Transmission Lines : 17 3 / 13A short section of line xlong:v(x, t)andi(x, t) depend on bothposition and x ignore 2nd order derivatives: v(x,t) t= v(x+ x,t) t, v EquationsKVL:v(x, t) =V2+v(x+ x, t) +V1 KCL:i(x, t) =iC+i(x+ x, t)Capacitor equation:C v t=iC=i(x, t) i(x+ x, t) = i x xInductor equation (L1andL2have the same current):(L1+L2) i t=V1+V2=v(x, t) v(x+ x, t) = v x xTransmission Line Equations+17.

7 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 3 / 13A short section of line xlong:v(x, t)andi(x, t) depend on bothposition and x ignore 2nd order derivatives: v(x,t) t= v(x+ x,t) t, v EquationsKVL:v(x, t) =V2+v(x+ x, t) +V1 KCL:i(x, t) =iC+i(x+ x, t)Capacitor equation:C v t=iC=i(x, t) i(x+ x, t) = i x xInductor equation (L1andL2have the same current):(L1+L2) i t=V1+V2=v(x, t) v(x+ x, t) = v x xTransmission Line EquationsC0 v t= i xL0 i t= v xTransmission Line Equations+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 LineCharacteristics+ Analysis of Circuits (2017-10213) Transmission Lines : 17 3 / 13A short section of line xlong:v(x, t)andi(x, t) depend on bothposition and x ignore 2nd order derivatives: v(x,t) t= v(x+ x,t) t, v EquationsKVL:v(x, t) =V2+v(x+ x, t) +V1 KCL:i(x, t) =iC+i(x+ x, t)Capacitor equation:C v t=iC=i(x, t) i(x+ x, t) = i x xInductor equation (L1andL2have the same current):(L1+L2) i t=V1+V2=v(x, t) v(x+ x, t) = v x xTransmission Line EquationsC0 v t= i xL0 i t= v xwhereC0=C xis the capacitance per unit length (Farads/m) andL0=L1+L2 xis the totalinductance per unit length (Henries/m).

8 Solution to Transmission Line Equations17: 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 4 / 13 Transmission Line Equations:C0 v t= i xL0 i t= v xSolution to Transmission Line Equations17: 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 4 / 13 Transmission Line Equations:C0 v t= i xL0 i t= v xGeneral solution:v(t, x) =f(t xu) +g(t+xu)i(t, x) =f(t xu) g(t+xu)Z0whereu= 1L0C0andZ0= to Transmission Line Equations17: 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 4 / 13 Transmission Line Equations:C0 v t= i xL0 i t= v xGeneral solution:v(t, x) =f(t xu) +g(t+xu)i(t, x) =f(t xu) g(t+xu)Z0whereu= 1L0C0andZ0= thepropagation velocityandZ0is thecharacteristic to Transmission Line Equations17: 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 4 / 13 Transmission Line Equations:C0 v t= i xL0 i t= v xGeneral solution.

9 V(t, x) =f(t xu) +g(t+xu)i(t, x) =f(t xu) g(t+xu)Z0whereu= 1L0C0andZ0= thepropagation velocityandZ0is thecharacteristic ()andg()can beanydifferentiable to Transmission Line Equations17: 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 4 / 13 Transmission Line Equations:C0 v t= i xL0 i t= v xGeneral solution:v(t, x) =f(t xu) +g(t+xu)i(t, x) =f(t xu) g(t+xu)Z0whereu= 1L0C0andZ0= thepropagation velocityandZ0is thecharacteristic ()andg()can beanydifferentiable by substitution: i x= ( f (t xu) g (t+xu)Z0 1u)Solution to Transmission Line Equations17: 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 4 / 13 Transmission Line Equations:C0 v t= i xL0 i t= v xGeneral solution:v(t, x) =f(t xu) +g(t+xu)i(t, x) =f(t xu) g(t+xu)Z0whereu= 1L0C0andZ0= thepropagation velocityandZ0is thecharacteristic ()andg()can beanydifferentiable by substitution: i x= ( f (t xu) g (t+xu)Z0 1u)=C0(f (t xu) +g (t+xu))=C0 v tForward Wave17: 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 5 / 13 Suppose.

10 U= 15 cm/nsandg(t) 0 v(x, t) =f(t xu)Forward Wave17: 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 5 / 13 Suppose:u= 15 cm/nsandg(t) 0 v(x, t) =f(t xu) Atx= 0cm [N],vS(t) =f(t 0u)0246810 Time (ns)f(t-0/u)Forward Wave17: 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 5 / 13 Suppose:u= 15 cm/nsandg(t) 0 v(x, t) =f(t xu) Atx= 0cm [N],vS(t) =f(t 0u) Atx= 45cm [N],v(45, t) =f(t 45u)0246810 Time (ns)f(t-0/u)f(t-45/u)Forward Wave17: 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+ Analysis of Circuits (2017-10213) Transmission Lines : 17 5 / 13 Suppose:u= 15 cm/nsandg(t) 0 v(x, t) =f(t xu) Atx= 0cm [N],vS(t) =f(t 0u) Atx= 45cm [N],v(45, t) =f(t 45u)0246810 Time (ns)f(t-0/u)f(t-45/u)f(t 45u)is exactly the same asf(t)but delayed by45u= Wave17: 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 LineCharact