Pulses in Transmission Lines

Professor Jeff Filippini Physics 401 Spring 2020 Key Concepts for this Lab 1. Networks with distributed parameters What if the R/L/C are spread out? Thevenin-equivalent networks

2. Pulse propagation in transmission lines When signals move like waves Reflections from resistive and reactive loads

3. Impedance matching Getting power where you want it to go

Physics 401 2 When do Wires Carry Waves?

Thus far we’ve implicitly assumed that V(t) and I(t) are synchronized throughout our circuits – but signals travel at finite speed (� = �/�)! • Speed of light: � = 3×10 m/s = 30 cm/ns = 1 ft/ns Real signals are typically slower by a factor (n) of order unity

Over distances that are a significant fraction of the wavelength, we’re better off thinking of wave propagation

Frequency Application nλ 60 Hz AC power lines 5000 km 580 kHz WILL-AM broadcast 500 m (0.3 mile) 2.4 GHz WiFi 12.5 cm (5 inches) 430 THz Red light 0.7 μm

Physics 401 3 Transmission Lines

Transmission line: a specialized cable (or other structure) designed to conduct alternating current of radio frequency (RF). • Limit wave reflection by maintaining uniform impedance (details below!) • Reduce power loss to radiation

Wikipedia: Transmission Line Design goal: Source Maximize power from the impedance source delivered to the load Load impedance “Line impedance”

Physics 401 4 Common Types of Transmission Lines

Twisted pair Twin lead Wikipedia Wikipedia

Analog Devices Coaxial cable

Physics 401 5 Coaxial Cable

Specification: Signal voltage between central conductor and Impedance: 53 Ω braid shield Capacitance: 83 pF/m Conductor: Bare Copper Wire (1/1.02mm) Shield reduces external dipole radiation (and response to RF interference)

Physics 401 6 Experimental Setup

Wavetek 81 Tektronix 3012B

Sync output Triggering input

RG8U Signal output

Load

Physics 401 7 Modeling a Transmission Line

Simplified I(t) Ideal equivalent case Model as distributed network circuit Li-1 Li Li+1 rather than lumped components Ci-1 Ci Ci+1 V(t) Ideal line has inductance / capacitance per unit length I(t) (lossless)

Ri Li Real lines have finite Real conductance � = between situation Gi+1 Ci conductors (i.e. loss)

Physics 401 8 Pulses in Transmission Lines

reflected

rout i(x,t)

V(t) V(x,t) ZL

x V 0 forward

Physics 401 9 The Telegrapher’s Equations

rout i(x,t)

Lossless V(t) V(x,t) ZL line (G=0) i(x,t)

x dx C = capacitance per unit length L = inductance per unit length

Distributed capacitance � �� = −�� Distributed inductance �� = − = −� �� = −(� ��) �� = − = −� ��

Physics 401 10 The Wave Equation

Distributed capacitance Distributed inductance �� C = − = −� �� = −C = −� ��

Combining

�� = LC = LC ��

Physics 401 11 Voltage and Current Waves

�� Waves of velocity v = LC = LC � �� , � = � sin � � − Seek solutions � of the form… � I �, � = � sin � � − Substitute � �, � , I �, � into… � C = − = −� … to find two key consequences: 2. � �, � � 1. 1 � ≡ = � = � �, � � �� Characteristic impedance of line Speed of wave propagation

Physics 401 12 Characteristic Impedance

� C = capacitance per unit length (F/m) � = L = inductance per unit length (H/m) � � = 8.854×10 F/m � = 4�×10 H/m 2πε ε �� = � C= 0r (F/m) d D æöD µµ0 r æöD lnç÷ L = lnç÷(H/m) èød 2p èød

er – dielectric permittivity µr-magnetic permeability ≈1

138 æöD Finally for coaxial cable: ZOhmsk = log10 ç÷ ( ) e r èød

Physics 401 13 Wave Propagation Speed 2πε0r ε µµ D � = 8.854×10 F/m 1 C= 0 r æö � = L = lnç÷ æöD 2p èød � = 4�×10 H/m �� lnç÷ èød �� = �

1 1 � � � = = = ≈ d D ��

The delay time of a signal is τ = (s/m) ≈ 3.336 � ns/m Inner insulation material: Polyethylene RG-8/U For polyethylene below 1 GHz, � ≈ 2.25 RG58U Nominal impedance: 52 ohm Delay time: ~5 ns/m (speed ~2/3c)

Physics 401 14 Reflection in Transmission Lines Wikipedia: Reflection Coefficient reflected ... Any wave will be (partially) reflected when it reaches a ZL ... change in impedance x forward

Forward wave Reflected wave � � � �, � = � sin � � − � �, � = � sin � � + � � At Load � � I �, � = � sin � � − I �, � = � sin � � + �(�) = ��(�) � � � �, � = ��(�, �) � �, � = −��(�, �)

Physics 401 15 Reflection in Transmission Lines

reflected Anywhere in Transmission Line At Load ... � = �� = �� ZL � = −��

... Match at the boundary x forward � = � + � � � � = � + � = − � �

� + � � � − � = or � = � � − � � � + �

Physics 401 16 Reflection from an Open Transmission Line

� + � � � − � = or � = � � − � � � + �

Open line: �� = ∞ ⟹ � = �, and voltage at load � = � + � = 2�

Incident pulse Reflected pulse

End of line

Physics 401 17 Transmission Line Losses Attenuation (dB per 100 ft) Why is the reflected pulse smaller? MHz 30 50 100 146 150 RG-58U 2.5 4.1 5.3 6.1 6.1 Experiment: RG 58U

Vi incident

reflected Vr

æöV ATTN( db )= 20logç÷i èøVr

Cable characterized by attenuation per unit length This is slowly frequency dependent!

Physics 401 18 Reminder: Units of Ratio We can compare the relative strength of two signals by taking the logarithm (base-10) of the ratio of their powers. This (rarely used) unit was named a bel, after A.G. Bell.

We more commonly use the decibel (dB), 1/10th of a bel � �� = 10 log Power ratio � �� = 20 log Voltage ratio or current, field, … Ex: -3 dB = ½ power; -20 dB = 1/100 power (1/10 voltage) Alexander Graham Bell (1847 – 1922) Related units: dBm = dB relative to 1 mW (absolute unit) Neper (Np) = like a bel, but natural log (ln)

Physics 401 19 Transmission Line Losses

Experiment: RG 58U In our case: 4.18 �� 200�� = 20 log ≈ 1.46 �� 3.54

Where does it come from?

Ri Li

Finite Gi+1 Ci conductance

Physics 401 20 Losses in Different Cables

RG-58U RG-8U

4.18 3.932 �� 200�� = 20 log ≈ 1.46 �� 200�� = 20 log ≈ 0.335 �� 3.54 3.78 Core ø=0.81 mm Core ø=2.17 mm

Dielectric ø=2.9 mm Dielectric ø=7.2 mm Physics 401 21 Frequency-Dependent Transmission

Why is the reflected pulse distorted relative to the incident pulse?

1. Loss: Attenuation is frequency-dependent 2. Dispersion: Delay (i.e. speed) depends on RG-58U frequency

RG-58U

Physics 401 22 Frequency-Dependent Loss Example

FFT Spectrum correction

Incident pulse spectrum reflected pulse spectrum

IFFT

Physics 401 23 Reflection from a Resistive Load

reflected � + � � � − � = or � = � ... � = � � − � � � + � ZL

... 3.0 2.5 x forward Incident pulse 2.0 1.5

1.0 0.5

0.0

-0.5 V(V) Shorted line -1.0 � = 0 ⟹ V = −V -1.5 -2.0

-2.5

Reflected pulse -200 0 200 400 600 800 time (ns) Physics 401 24 Reflection from a Resistive Load Matched Impedance � + � � � − � = or � = � � = � ⟹ V = 0 � − � � � + �

1.5

1.0 Incident

0.5 0.8 Incident pulse 24.8W pulse 0.0 179.9W

V(V) -0.5 0.6 100.2W 149.6W -1.0 0.4 50 W -1.5

0 200 400

0.2 time (ns) V (V) V 1.5 End of 0.0 1.0

0.5 line -0.2 0.0

V(V) -0.5

0 100 200 300 400 500 600 700 800 -1.0

time (ns) Reflected pulse -1.5

0 200 400 Physics 401 time (ns) 25 Thevenin’s Theorem: Simplifying Networks

Any combination of voltage sources and resistive/reactive impedances with two terminals (“linear one-port network”) can be replaced by an equivalent single voltage source and series impedance. Hermann Ludwig Ferdinand von Helmholtz Léon Charles Thévenin R2 R3 (1821-1894) r (1857–1926)

R1 R4

+ N.B. replacement is exactly

+ equivalent from the load’s - v

V V - 1 2 + point of view, but e.g. internal - power dissipation in the equivalent network may differ

Physics 401 26 Thevenin’s Theorem and Transmission Lines Conditions at the Load Eliminate Vr 2 � � = � + � = � � � = � � � + � � = � + � = − Zk � �

2Vi RL

From this equivalent circuit we can find the maximum possible power delivered to the load: 2� � = � � = � � + � Matched Impedance 2 � P = Pmax if RL = Zk � = � (i.e. no reflection!) � 1 + � Physics 401 27 Thevenin’s Theorem - Experiment

RG-8U Zk Pulse at end of line 2Vi RL

Incident Reflected This experiment works best when performed with RG-8U cable due to its lower attenuation

When RL=∞ (open line) the pulse amplitude at line’s end is expected to be 2Vi, where Vi is the amplitude of the incident pulse

Physics 401 28 Reflection from a Capacitive Load

2 � � = � = � � � + � � � = ≈ 3.2 �� T1 Zk

V0 C Load / � = 1 − �

/ ( )/ � = 2 � 1 − � 1 − � T1

Physics 401 29 Reflection from an Inductive Load 2 � � = �� 2� = � � − � � + � �� / � = � 1 − �

V0 � � = ≈ 50 �� Load L �

� = �� ≈ 2.5 ��

Physics 401 30 Reflection from an Inductive Load 20 2 � Vi � = �� 2� = � � − � 10 � + � �� / � = � 1 − � 0 20 Zk VL 10

0 V0

� -10 L � = 20 � V = V - V 10 r L i

-10 -10 0 10 20 30 40 50 60 time Physics 401 31 Appendix #1: Exporting graphs from Origin

Physics 401 32 Appendix #2: Some Reminders 1. Reports should be uploaded only to the proper folder for your activity and section • For example, folder Frequency domain analysis_L1 should only be used by students from section L1 • Submit only one copy (no need to submit e.g. both Word and PDF) • I recommend the following file name style: L1_lab3_LastName

2. Origin template for this week’s lab: \\engr-file-03\phyinst\APL Courses\PHYCS401\Common\Origin templates\Transmission line\Time trace.otp

Physics 401 33 Appendix #3: Error Propagation

Suppose that I’m interested in a derived quantity � = �(�, �, … , �) I’ve made lab measurements with errors: � ± �� What is the error on y?

�� 1.15 ∆� = ∆� �, ∆� = (�) ∆� Δf ��

± ) f

( f

1.10 Intuition: If y depends strongly (weakly) on xi, then an error in xi will have a large (small) effect on our estimate of y xi±∆xi Technical aside: This assumes no correlations among errors in 1 1.5 x the various x’s. If this isn’t true, we must complicate this i formula with a covariance matrix.

Physics 401 34 Appendix #3: Error Propagation - Example

1 Derive the resonance frequency f from � �, � = 2� �� measured inductance and capacitance � = 10 ± 1 ��; � = 10 ± 2 ��

�� ∆� = ∆� �, �, ∆�, ∆� = Δ� + Δ� ��

Results: �� −1 = � � f(L,C) = 503.29212104487… Hz �� 4� �� −1 Δf = 56.26977… Hz = � � �� 4� f(L,C) = 503 ± 56 Hz

Physics 401 35 Appendix #3: Error Propagation - Practicalities

� = 10 ± 1 ��; � = 10 ± 2 �� Where are these numbers coming from?

1. Commercial resistors, capacitors, inductors, … have quoted tolerances (use if you haven’t measured!)

C=500pF ±5% F=35mH ±10%

2. Measure components with standard equipment, use equipment accuracy

SENCORE “Z” meter model LC53 Capacitance measurement accuracy ±5% Inductance measurement accuracy ±2%

Agilent E4980A Precision LCR Meter Basic accuracy ±0.05%

Physics 401 36 Appendix #4: Nonlinear Fitting

Fitting is a minimization problem: what choice of parameter values minimizes some cost function that expresses how far the fit function is from the data?

• Data: ordered pairs �, � , often in the form of an �×2 matrix • Independent variable �, e.g. frequency, time, etc. • Dependent variable �, e.g. signal magnitude • Parameterized function: � = �(�; �), which takes some set of parameters β

; Our usual cost function is the sum of squared deviations: � = ∑ is further � � = � � ; � − � normalized by the r.m.s. error, which doesn’t matter for fitting Origins uses the Levenberg-Marquardt algorithm for nonlinear fitting, which is optimized for quadratic cost functions like this one and requires a starting guess. In some cases � = (1,1, … 1) works, but a more reasonable guess is often required

Physics 401 37 Appendix #5: Unknown Load Fitting • Transmission line: unknown load simulation

Load parameters

Function generator X-axes scaling parameters Line characteristic impedance

Expected load

Location: \\engr-file-03\PHYINST\APL Courses\PHYCS401\Lab Software And Manuals\LabSoftware\Transmission lines

Physics 401 38 Appendix #5: Unknown Load Fitting • Transmission line: unknown load simulation

Location: \\engr-file-03\PHYINST\APL Courses\PHYCS401\Lab Software And Manuals\LabSoftware\Transmission lines

Physics 401 39