Transmission over Coaxial Cable Notes for EE456 University of - PDF document

Transmission over Coaxial Cable Notes for EE456 University of Saskatchewan Created Nov. 7, 2016 by Eric Salt revised Nov 10, 2016: Implemented revisisons and suggestions made by Prof. Ha Nguyen revised Nov 11, 2016: Changed the title revised Nov 24, 2016: Added frequency depency to equations as suggested by Quang Nguyen and Brian Bersheid. Also reworded several paragraphs to make them more understandable. 1

TRANSMISSION OVER COAXIAL CABLE 2 1 Introduction Both analog and digital communications systems transmit bandpass analog radio frequency signals over a medium of some sort to a receiver. The difference between the two systems is that one embeds an analog message signal in the analog radio frequency signal while the other embeds digital information. For example commercial AM radio embeds an analog message signal into its radio frequency signal by amplitude modulating a carrier. Therefore, commercial AM is an analog communication system. On the other hand the radio frequency of a wireless router is modulated with digital data. Therefore, a computer linked to a router via WiFi is a digital communication system. Communications systems, whether analog or digital, use coaxial cable, or some other conduit like a micro strip transmission line, to get the radio frequency signal from one point to another. For example, a coaxial cable could be used in a wireless system to get the radio frequency (RF) signal from the high power amplifier to an antenna. Coaxial cable is also commonly used to connect the RF signal to test equipment while debugging or testing a communications system. Perhaps the best example is the role of coaxial cable in a cable TV distribution system, where it carries the radio signal from the transmitter to the receiver. In the notes to follow the properties and behavior of coaxial cables will be explored with a view to give the students a very good understanding at a macroscopic level. The detailed transmission line model that used to derive the transmission line equations is clearly presented, but the model is not analysed with calculus to obtain the equations. The resulting equations of this somewhat tedious analysis are just stated. These notes attempt to interpret and give meaning to the resulting equation so they can be applied appropriately with confidence. The notes go on to explain how devices like directional couplers work and how they can be used to divert some of the power flowing inside a coaxial cable to a test instrument or a device such as a cable modem. 2 Modelling a Coaxial Cable 2.1 The Basic Model A coaxial cable is modelled as a transmission line. Any pair of wires used to carry electrical signals can be modelled as a transmission line. For example very long power lines are modelled as transmission lines as are very short mircrostrip lines, which is simply a track on a printed circuit board above a ground plane on the other side of the board. A coaxial cable is two concentric conductors separated by a solid insulator. The outer conductor is called the sheath and the inner conductor is called the center wire. The material used for the solid insulator has a low dielectric constant and very low dielectric losses. An illustration of a coaxial cable is given at the top of Figure 1. The transmission line model for a coaxial cable is given in Figure 1. The cable is viewed as the cascade of an infinite number of cable segments of length dx with each segment being modelled with lumped circuit elements. Each of the conductors is modelled as an inductor with inductance ℓ in series with a resistor with resistance r . As the conductors are in proximity to each other they are linked by a capacitor with capacitance c . The insulator separating the conductors is certain to

2 MODELLING A COAXIAL CABLE 3 distance in meters x I(t,0) v(t,0) I(t,0) x x+dx dx center conductor ℓ ℓ r r I(t,x+dx) I(t,x) v(t,x) c g v(t,x+dx) c g I(t,x) r I(t,x+dx) r ℓ ℓ sheath Figure 1: A transmission line model of a coaxial cable suffer dielectric losses so the link between the two conductors must include a resistor. In this case the resistor is specified in term of conductance g , which has units of siemens, which is 1/ohm. Since the length of the segment represented by the lumped elements is infinitesimal, as indicated in Figure 1, the values of the elements must also be infinitesimal. The infinitesimal values are ℓ = ( L/ 2) dx , r = ( R/ 2) dx , c = Cdx and g = Gdx , where L , R , C and G are the inductance per meter, resistance per meter, capacitance per meter and conductance per meter, respectively. I.e. L and R are the inductance and resistance of 1 meter of cable measured from the source end with the load end shorted and C and G are the capacitance and conductance of 1 meter of cable measured from the source end with the load end open. The conductance, G , models the losses in the dielectric. The loss per meter in the model is the square of the voltage across the dielectric insulator times G . Many if not all of the molecules in a dielectric with a relative dielectric constant greater than 1 are dipoles (or become dipoles in the presence of an electric field). An electric field in a dielectric places a torque on the molecules (the dipoles) causing them to rotate in a direction that reduces the electric field. This rotation, no mater how slight, bumps/rubs other molecule causing them to vibrate and increase their kinetic energy. This molecular kinetic energy is also referred to as heat. Since energy is conserved, the kinetic energy (i.e. heat) comes from the energy in the electric field. Each time the polarity of the electric field changes, the molecules are rotated and energy is converted from the electric field to heat. Therefore, the power converted from the electric field to heat is proportional to the frequency of the alternating electric field. This means the conductance G is the function of frequency given by 2 πF G ′ where G ′ is a constant. Surprisingly, the resistance per unit length, i.e. R , is also a function of frequency. The reason for this has to do with a phenomenon called “skin effect”. At high frequencies most of the current in a conductor flows near the outside of the conductor reducing it effective cross sectional area. In coaxial cables this phenomenon starts at a frequency of about 0.1 MHz in coaxial cables. Above

2 MODELLING A COAXIAL CABLE 4 √ FR ′ , where R ′ is a constant. this frequency the resistance per unit length is modeled as R = 2.2 Analysis for a Cable of Infinite Length In this subsection the results of the analysis for a cable of infinite length are discussed. The results of the analysis for a finite length cable that is terminated with a load impedance Z L are presented in the next subsection. The cable is driven at one end by a time dependent voltage source denoted v ( t, 0) in Figure 1. This source voltage propagates down the cable creating a distribution in voltage as a function of time and position on the cable. For purposes of analysis the voltage of the center conductor w.r.t the sheath a distance x from the voltage source is denoted v ( t, x ) as illustrated in Figure 1. The current in both the center conductor and sheath at a distance of x from the source is similarly denoted I ( t, x ) . Perhaps unnecessary, but it is pointed out the current I ( t, x ) differs from that of I ( t, x + dx ) due to the leakage paths through the capacitor with capacitance Cdx and the resistor with conductance Gdx . Using calculus as well as Kirchoff’s voltage and current laws produces the transmission line equations: − ∂v ( t, x ) = R × I ( t, x ) + L∂I ( t, x ) ∂x ∂x − ∂I ( t, x ) = G × v ( t, x ) + C ∂v ( t, x ) ∂x ∂x To analyze these transmission line equations: 1. The input voltage must be a sinusoid of the form v ( t, 0) = A o cos(2 πFt + φ o ) , where A o is a constant with units volts, F is a constant with units Hz and φ o is a constant with units radians. However, without loss of generality φ o can be taken to be zero and the input can be, and will be, v ( t, 0) = A o cos(2 πFt ) . 2. The system must be in steady state. The analysis is done under the assumptions all the transients that result from turning on the source have died out. The analysis assumes the source is turned on at time t = −∞ and the system has reached steady state by time t = 0 . Under the conditions given above G and R , while frequency dependent (recall G = G ′ F and R = R ′ √ F ), can be treated as constants since the variable in the differential equation is x . In steady state analysis it is common practise to represent a sinusoidal function of time by a phasor and characterize the sinusoid by the complex amplitude of the phasor at t = 0 . A phasor is a complex function of time of the form Ae j (2 πFt + φ ) = Ae jφ e j 2 πFt . It is a vector in the complex plane that has a constant magnitude, which is A , and spins with a constant angular velocity, which is 2 πF in units of radians/second. Its complex amplitude at time t = 0 , which is Ae jφ , is usually referred to as its complex amplitude without stating that it is the amplitude at t = 0 . The real part of the phasor is the cosine A cos(2 πFt + φ ) . The magnitude of the complex amplitude, which is A , is the amplitude of the cosine and the angle of the complex amplitude, which is φ , is the phase of the cosine at time t = 0 . The frequency of the phasor, which is its angular velocity, is not a parameter in the complex amplitude. If the input to the cable is a sinusoid, then in steady state, the voltage of the center conductor w.r.t. to sheath at any point on the cable will also be a sinusoid. The sinusoidal voltage at a distance