90400 Transients

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Revision Lecture 4: Transients &

Lines

Revision Lecture 4:

Transients & Lines

Transients: Basic Ideas

Steady States

Determining Time

Constant

Determining Transient

Amplitude

Transmission Lines Basics

Reflections

Sinewaves and Phasors

Standing Waves

E1.1 Analysis of Circuits (2014-4649) Revision Lecture 4 1 / 9

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Transients happen in response to asudden change

Input voltage/current abruptly changes its magnitude,

frequency or phase

A switch alters the circuit

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




frequency or phase


1st order circuits only: one capacitor/inductor

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




frequency or phase



All voltage/current waveforms are: Steady State + Transient

Steady States:find with nodal analysis or transfer function

Note:Steady Stateis not the same asDC Level

Need steady states beforeandafter the sudden change

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




frequency or phase







Transient: Always a negative exponential: Ae

t

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




frequency or phase








t

Time Constant: =RCor LR whereR is the Thveninresistance at the terminals ofCorL

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




frequency or phase








t


Find transient amplitude,A, from continuity sinceVC or

IL cannot change instantly.

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




frequency or phase








t


Find transient amplitude,A, from continuity sinceVC or

IL cannot change instantly. andA can also be found from the transfer function.

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Steady States

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Asteady-stateoutput assumes the input frequency, phase and amplitude

are constant forever. You need to determinetwoySS(t)steady stateoutputs: one forbeforethe transient (t

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Steady States

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




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Steady States

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




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Steady States

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




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Steady States

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




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Steady States

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




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Steady States

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




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Steady States

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




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Determining Time Constant

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Method 1: Thvenin

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Method 1: Thvenin

(a)Remove the capacitor/inductor

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Method 1: Thvenin


(b)Set all sources to zero (including the input

voltage source). Leave output unconnected.

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Method 1: Thvenin




(c)Calculate the Thvenin resistancebetween the capacitor/inductor terminals:

RTh = 8R||4R||(6R+ 2R) = 2R

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Method 1: Thvenin





RTh = 8R||4R||(6R+ 2R) = 2R(d)Time constant:=RThCor

LRTh

=

RTh

C= 2

RC

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Method 1: Thvenin






LRTh

=

RTh

C= 2

RC

Method 2: Transfer function

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Method 1: Thvenin






LRTh

=

RTh

C= 2

RC


(a)Calculate transfer function using nodal analysis

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Method 1: Thvenin






LRTh

=RTh

C= 2RC



KCL @ V:

VX

4R + V

8R +jCV + VY

2R = 0KCL @ Y: YV2R +

YX6R = 0

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Method 1: Thvenin






LRTh

=RTh

C= 2RC



KCL @ V:

VX

4R + V

8R +jCV + VY

2R = 0KCL @ Y: YV2R +

YX6R = 0

EliminateV to get transfer Function: YX = 8jRC+1332jRC+16

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Method 1: Thvenin






LRTh

=RTh

C= 2RC



KCL @ V:

VX

4R + V

8R +jCV + VY

2R = 0KCL @ Y: YV2R +

YX6R = 0


(b)Time Constant = 1

Denominator corner frequency

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Method 1: Thvenin






LRTh

=RTh

C= 2RC



KCL @ V:

VX

4R + V

8R +jCV + VY

2R = 0KCL @ Y: YV2R +

YX6R = 0


(b)Time Constant = 1

Denominator corner frequencyd = 1632RC =

1d

= 2RC

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Determining Transient Amplitude

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


After an input change att = 0,y(t) =ySS(t) +Ae t .

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves



y(0+) =ySS(0+) +A A=y(0+) ySS(0+)

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves



y(0+) =ySS(0+) +A A=y(0+) ySS(0+)Method:(a)calculate true outputy(0+),(b)subtractySS(0+)to getA

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




(i)Version 1: vC oriL continuity

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Revision Lecture 4:Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




(i)Version 1: vC oriL continuityx(0) = 2

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mA

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current source

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




(i)Version 1: vC oriL continuityx(0) = 2 iL(0) = 1 mAContinuity iL(0+) =iL(0)ReplaceL with a1 mA current sourcey(0+) =x(0+) iR= 6 1 = 5

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves





(i)Version 2: Transfer function

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves






H(j) = YX (j) = R+jL2R+jL

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves







Input step,x=x(0+) x(0) = +4-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves







Input step,x=x(0+) x(0) = +4y(0+) =y(0) +H(j)x

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves








= 1 + 1 4 = 5

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves








= 1 + 1 4 = 5

(ii)A =y(0+) ySS(0+) = 5 3 = 2

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

yTr

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves








= 1 + 1 4 = 5

(ii)A =y(0+) ySS(0+) = 5 3 = 2

(iii)y(t) =ySS(t) +Aet/

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

yTr

ySS

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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves








= 1 + 1 4 = 5

(ii)A =y(0+) ySS(0+) = 5 3 = 2

(iii)y(t) =ySS(t) +Aet/

= 3 + 2e

t/2

-5 0 5 10 15 20

0

2

4

6

t (s)

-5 0 5 10 15 20

0

2

4

6

t (s)

yTr

ySS

y

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Steady States

Determining Time

Constant


Amplitude

Transmission Lines Basics Reflections


Standing Waves


Transmission Line: constantL0 andC0 : inductance/capacitance permetre.

Forward wavetravels along the line: fx(t) =f0

t xu

.

Velocityu =

1L0C0

12c= 15cm/ns

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Steady States

Determining Time

Constant


Amplitude



Standing Waves




t xu

.

Velocityu =

1L0C0

12c= 15cm/ns

fx(t)equalsf0(t)but delayed by xu .

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Steady States

Determining Time

Constant


Amplitude



Standing Waves




t xu

.

Velocityu =

1L0C0

12c= 15cm/ns


0 2 4 6 8 10Time (ns)

f(t-0/u)

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Steady States

Determining Time

Constant


Amplitude



Standing Waves




t xu

.

Velocityu =

1L0C0

12c= 15cm/ns


0 2 4 6 8 10Time (ns)

f(t-0/u) f(t-45/u)

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Steady States

Determining Time

Constant


Amplitude



Standing Waves




t xu

.

Velocityu =

1L0C0

12c= 15cm/ns


Knowingfx(t)forx=x0 fixes it for allotherx.

0 2 4 6 8 10Time (ns)

f(t-0/u) f(t-45/u) f(t-90/u)

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Steady States

Determining Time

Constant


Amplitude



Standing Waves




t xu

.

Velocityu =

1L0C0

12c= 15cm/ns



0 2 4 6 8 10Time (ns)

f(t-0/u) f(t-45/u) f(t-90/u)

Backward wave: gx(t)is the same but travelling : gx(t) =g0

t+ xu

.

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Steady States

Determining Time

Constant


Amplitude



Standing Waves




t xu

.

Velocityu =

1L0C0

12c= 15cm/ns



0 2 4 6 8 10Time (ns)

f(t-0/u) f(t-45/u) f(t-90/u)


t+ xu

.

Voltage and current are: vx =fx+gx andix = fxgx

Z0whereix is

positive in the+xdirection () andZ0 =

L0C0

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Steady States

Determining Time

Constant


Amplitude



Standing Waves




t xu

.

Velocityu =

1L0C0

12c= 15cm/ns



0 2 4 6 8 10Time (ns)

f(t-0/u) f(t-45/u) f(t-90/u)


t+ xu

.

Voltage and current are: vx =fx+gx andix = fxgx

Z0whereix is

positive in the+xdirection () andZ0 =

L0C0

Waveforms offx andgx are determined by the connections at both ends.

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0

Atx =L, Ohms law vL(t)iL(t) =RL gL(t) = RLZ0RL+Z0

fL(t).

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).

Reflection coefficient:L= gL(t)fL(t)

= RLZ0RL+Z0

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).


= RLZ0RL+Z0L [1, +1]and increases withRL

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).



Knowingfx(t)forx =x0 now tells youfx, gx, vx, ix x

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).




Atx = 0:f0(t) = Z0RS+Z0

vS(t) +RSZ0RS+Z0

g0(t)

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).




Atx = 0:f0(t) = Z0RS+Z0

vS(t) +RSZ0RS+Z0

g0(t) =0vS(t) +0g0(t)

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).




Atx = 0:f0(t) = Z0RS+Z0

vS(t) +RSZ0RS+Z0

g0(t) =0vS(t) +0g0(t)

Wave bounces back and forth getting smaller with each reflection:

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).




Atx = 0:f0(t) = Z0RS+Z0

vS(t) +RSZ0RS+Z0

g0(t) =0vS(t) +0g0(t)


vS(t) 0 f0(t)

R fl i

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).




Atx = 0:f0(t) = Z0RS+Z0

vS(t) +RSZ0RS+Z0

g0(t) =0vS(t) +0g0(t)


vS(t) 0 f0(t)

L g0(t +

2Lu )

R fl i

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).




Atx = 0:f0(t) = Z0RS+Z0

vS(t) +RSZ0RS+Z0

g0(t) =0vS(t) +0g0(t)


vS(t) 0 f0(t)

L g0(t +

2Lu )

0 f0(t +

2Lu )

R fl ti

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).




Atx = 0:f0(t) = Z0RS+Z0

vS(t) +RSZ0RS+Z0

g0(t) =0vS(t) +0g0(t)


vS(t) 0 f0(t)

L g0(t +

2Lu )

0 f0(t +

2Lu )

L g0(t +

4Lu )

R fl ti

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).




Atx = 0:f0(t) = Z0RS+Z0

vS(t) +RSZ0RS+Z0

g0(t) =0vS(t) +0g0(t)


vS(t) 0 f0(t)

L g0(t +

2Lu )

0 f0(t +

2Lu )

L g0(t +

4Lu )

0

R fl ti

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).




Atx = 0:f0(t) = Z0RS+Z0

vS(t) +RSZ0RS+Z0

g0(t) =0vS(t) +0g0(t)


vS(t) 0 f0(t)

L g0(t +

2Lu )

0 f0(t +

2Lu )

L g0(t +

4Lu )

0

Infinite sum:

f0(t) =0vS(t)+0L0vS(t2Lu )+. . .

Reflections

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Reflections



Steady States

Determining Time

Constant


Amplitude



Standing Waves


vx =fx+gxix =

fxgxZ0


fL(t).




Atx = 0:f0(t) = Z0RS+Z0

vS(t) +RSZ0RS+Z0

g0(t) =0vS(t) +0g0(t)


vS(t) 0 f0(t)

L g0(t +

2Lu )

0 f0(t +

2Lu )

L g0(t +

4Lu )

0

Infinite sum:

f0(t) =0vS(t)+0L0vS(t2Lu )+. . .=

i=00iL

i0vSt

2Liu


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Steady States

Determining Time

Constant


Amplitude



Standing Waves


Sinewaves are easier because:

1. Use phasors to eliminatet:

2. Time delays are just phase shifts:


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Steady States

Determining Time

Constant


Amplitude



Standing Waves



1. Use phasors to eliminatet:f0(t) =A cos(t+) F0=Aej



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Steady States

Determining Time

Constant


Amplitude



Standing Waves





fx(t) =A cos

t xu

+ Fx =Ae

j(ux)=F0ejkx


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Steady States

Determining Time

Constant


Amplitude



Standing Waves





fx(t) =A cos

t xu

+ Fx =Ae

j(ux)=F0ejkx

kis thewavenumber: radians per metre (c.f. in rad per second)


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Steady States

Determining Time

Constant


Amplitude



Standing Waves





fx(t) =A cos

t xu

+ Fx =Ae

j(ux)=F0ejkx


As before: Vx =Fx+Gx andIx = FxGx

Z0


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Steady States

Determining Time

Constant


Amplitude



Standing Waves





fx(t) =A cos

t xu

+ Fx =Ae

j(ux)=F0ejkx



Z0

As before:

GL =LFLF0 =0VS+0G0


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Steady States

Determining Time

Constant


Amplitude



Standing Waves





fx(t) =A cos

t xu

+ Fx =Ae

j(ux)=F0ejkx



Z0

As before:

GL =LFLF0 =0VS+0G0

ButG0 =F0Le2jkL : roundtrip delay of 2Lu + reflection atx =L.


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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves





fx(t) =A cos

t xu

+ Fx =Ae

j(ux)=F0ejkx



Z0

As before:

GL =LFLF0 =0VS+0G0


Substituting forG0 in source end equation: F0 =0VS+0F0Le2jkL


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Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves





fx(t) =A cos

t xu

+ Fx =Ae

j(ux)=F0ejkx



Z0

As before:

GL =LFLF0 =0VS+0G0


Substituting forG0 in source end equation: F0 =0VS+0F0Le2jkL

F0= 0

10Lexp(2jkL)VSso no infinite sums needed

Standing Waves

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Standing Waves



Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves


Standing wavesarise whenever a wave meets its reflection:at places where the two waves arein phasetheir amplitudes add

but where they areanti-phasetheir amplitudes subtract.

Standing Waves

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Standing Waves



Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




At any pointx,

delay of xu

Fx =F0ejkx

Standing Waves

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Standing Waves



Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




At any pointx,

delay of xu

Fx =F0ejkx

Backward wave: Gx =LFxe2jk(Lx)

Standing Waves

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Standing Waves



Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




At any pointx,

delay of xu

Fx =F0ejkx

Backward wave: Gx =LFxe2jk(Lx): reflection + delay of2Lxu

Standing Waves

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Standing Waves



Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




At any pointx,

delay of xu

Fx =F0ejkx


Voltage atx: Vx =Fx+Gx

Standing Waves

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Sta d g a es

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




At any pointx,

delay of xu

Fx =F0ejkx


Voltage atx: Vx =Fx+Gx=F0ejkx

1 +Le

2jk(Lx)

Standing Waves

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g

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




At any pointx,

delay of xu

Fx =F0ejkx



1 +Le

2jk(Lx)

Voltage Magnitude : |Vx| = |F0|1 +Le2jk(Lx)

Standing Waves

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g

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




At any pointx,

delay of xu

Fx =F0ejkx



1 +Le

2jk(Lx)


: depends onx

Standing Waves

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g

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




At any pointx,

delay of xu

Fx =F0ejkx



1 +Le

2jk(Lx)


: depends onx

IfL 0,max magnitudeis(1 +L) |F0| whenevere2jk(Lx)

= +1

Standing Waves

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g

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




At any pointx,

delay of xu

Fx =F0ejkx



1 +Le

2jk(Lx)


: depends onx


= +1 x=Lorx =L k orx =L

2k or. . .

Standing Waves

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g

Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves




At any pointx,

delay of x

u Fx =F0e

jkx



1 +Le

2jk(Lx)


: depends onx



2k or. . .

Min magnitudeis(1 L) |F0| whenevere2jk(Lx) = 1

Standing Waves

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Revision Lecture 4:

Transients & Lines


Steady States

Determining Time

Constant


Amplitude


Reflections


Standing Waves



At any pointx,

delay of x

u Fx =F0e

jkx

Backward wave: Gx =LFxe2jk(Lx): reflection + delay of2LxuVoltage atx: Vx =Fx+Gx=F0e

jkx

1 +Le2jk(Lx)

Voltage Magnitude : |Vx| = |F0|

1 +Le2jk(Lx): depends onx



2k or. . .

Min magnitudeis(1 L) |F0| whenevere2jk(Lx) = 1

x=L 2k orx =L 32k orx =L

52k or. . .

90400 Transients

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