Transmission Lines

1. Transmission Lines Kevin BoldingElectrical EngineeringSeattle Pacific University

2. Transmission Lines • Long electrical wires are known as transmission lines • In short lines, it is safe to assume that wires have the same voltage and current at all points • In transmission lines, the voltage and current may vary along the length of the line How long is “long”? Wires longer than1/10 of the wavelength of the signal applied. (Assume travelling at the speed of light, 3.0x108 m/s)At 1MHz, this is 30m. At 3GHz, this is 1cm.

3. Transmission Line Thought Experiments 3,000,000,000 m i 5 + 5V - When the switch is closed, what current flows? What if the switch is closed for 1 second, then opened?

4. Transmission Line Model • Any wire has inductance • Inductors oppose changes in current • Any wire has resistance… but let’s pretend it doesn’t • Any pair of wires has some capacitance between them • Charging the capacitors will take current. • A balance will be reached, with some resultant current, i • This implies an resistance, R = V/i Infinitely Long i + 5V -

5. Characteristic Impedance • A transmission line appears to have a resistance that is related to the distributed capacitance and inductance • Z0 cannot be measured in traditional ways • If we put an Ohmmeter on an open transmission line, it will measure “open” or high resistance because we’re making a DC measurement • An Ohmmeter would work if we had an infinite line… • AC instruments (scopes) can be used Characteristic Impedance – Z0 Apparent resistance seen at the input of a transmission line. Determined primarily by the capacitance and inductance of the line, especially for high frequency signals. Applies to infinitely long lines, but also to lines with AC inputs if the line is “long” compared with the AC wavelength.

6. Factors affecting Characteristic Impedance • Z0 depends on C, L, and R per length • Higher C  Lower Z0 (More ability to “soak up” charge) • C↑ Z0↓ • Higher L  Higher Z0 (More opposition to current) • L ↑ Z0 ↑ • Higher R  Higher Z0 (Duh!) • R ↑ Z0 ↑ + 5V -

7. Factors affecting Characteristic Impedance • Capacitance per length depends on: • Conductor spacing – d – Larger spacing means less capacitance • d ↑ C ↓, butC ↑  Z0↓, thusd ↑  Z0 ↑ • Relative permittivity of the dielectric separating the conductors • Inductance per length depends on: • Conductor radius – r – Larger radius means less inductance • r ↑ L ↓,but L↑  Z0 ↑, thusr ↑ Z0↓ • Conductor spacing – d – When conductors are close together, the magnetic fields cancel out • d ↑  L↑,and L↑  Z0 ↑, thusd ↑  Z0 ↑ d Dielectric Insulator Conductors r

8. Determining Characteristic Impedance • For two parallel wires: • For coaxial cable: Dielectric Insulatorwith relative permittivity k d Conductors r Looking for k? Try “relative static permittivity” on wikipedia. Dielectric Insulatorwith relative permittivity k d2 d1(inside diameter of outer conductor) Signal Propagation speed (c = speed oflight in vacuum):

9. Z0 and Resistance • Characteristic Impedance looks like a resistance, but it isn’t! • Composed of C’s and L’s: No resistance (ideally) • Z0 doesn’t convert electrical energy into heat like a resistance • Z0 does not contribute to attenuation • Z0 does not vary with the length of the line • Real transmission lines have real resistance • Loss due to metal resistance - proportional to square root of frequency; goes up with line length • Loss due to dielectric resistance – proportional to frequency; goes up with line length • It is these two resistances that are the primary causes of attenuation Dielectric Insulatorwith relative permittivity k d Conductors r

10. The end of the line • After switch is closed, a wave travels down the t-line • What happens when it hits the open end? • What if the end is shorted? • Open or shorted t-lines will cause (unwanted) reflections. NOT Infinitely Long i open + 5V - shorted

11. Termination • Place a resistor of value Z0 at the end of the line • What happens when the wave hits the terminated end? • A properly terminated line will appear to be an infinite line and have no reflections • Any impedance change will cause reflections NOT Infinitely Long i R = Z0 + 5V -

12. Why we dislike Reflections • Reflections can mess up values on the line in locations between the source and end • If a periodic signal is applied to an un-terminated transmission line, the reflections will set up a standing wave pattern • The amplitude of the standing wave may be very large and will radiate power away • This is great if we’re building an antenna, but we’re not! • Un-terminated lines are a major source of Electromagnetic Interference (EMI)

13. Creating Reflections – Things to Avoid • Obviously, un-terminated t-lines • Remember, a line should be treated as a t-line if it is longer than 1/10 of the wavelength of the applied signal • Any change in impedance of the line will create reflections • Change of cable type • Change of the separation between conductors • Change in the width of conductors • Change in the dielectric between conductors • Use of standard cable with standard connectors avoids problems • Match the impedance of the cable to the equipment being used

14. Creating Reflections – Things to Avoid • Printed Circuit Boards (PCBs) present a challenge signal If the spacing between conductors changes, the impedance changes! ground signal If the width of the conductors changes, the impedance changes! ground signal Square corners not only change the spacing between conductors, but the width of the conductors too! ground Ground planes help a lot because the signal’s return path will follow directly underneath the signal. However, the impedance of the line will depend on the thickness and type of material in the PCB.