Notes on transmission lines

I mainly follow Pozar's book

                Equipo de Propulsión Espacial y Plasmas, Mario Merino, 2021

Lumped circuits and transmission lines

  • When the electrical size of a circuit is negligible (L/(c1ω)1L/(c_1\omega)\ll 1, with c1c_1 the phase velocity in the wires) we can use the lumped circuit theory: circuits are made of voltage/current sources, resistors, capacitors, inductors, etc.
  • Otherwise, the finite propagation speed of signals on wires must be taken into account. Transmission line theory must be used: voltage v(z,t)v(z,t) and current i(z,t)i(z,t) on cables are not only functions of time, but also of position.

Telegrapher's model

Lines have an impedance per unit length: resistance RR and inductance LL in series, conductance GG and capacitance CC in parallel:


In a differential length, the telegrapher's equations read:

vz=RiLit;iz=GvCvt\frac{\partial v}{\partial z} = -Ri - L\frac{\partial i}{\partial t}; \quad \frac{\partial i}{\partial z} = -Gv - C\frac{\partial v}{\partial t}

In frequency domain, we define the complex phasors VV(z) and I(z)I(z) with v(z,t)=[V(z)exp(jωt)]v(z,t)=\Re[V(z)\exp(-j\omega t)]:

Caution: book uses exp(+jωt)\exp(+j\omega t) instead.

dVdz=(R+jωL)I;dIdz=(G+jωC)V\frac{d V}{d z} = (-R+j\omega L)I; \quad \frac{d I}{d z} = (-G+j\omega C)V

We eliminate II and introduce the complex propagation constant:

d2Vdz2γ2V=0;γ=(RjωL)(GjωC)=jωLCβ\frac{d^2 V}{dz^2} - \gamma^2 V = 0; \quad \gamma=\sqrt{(R-j\omega L)(G-j\omega C)} = j\omega\sqrt{LC}\beta


β=(1+jRωL)(1+jGωC)=(1RGω2LC)+j(RωL+GωC)1\beta = \sqrt{\left(1+\frac{jR}{\omega L}\right)\left(1+\frac{jG}{\omega C}\right)} = \sqrt{\left(1-\frac{RG}{\omega^2 LC}\right) + j\left(\frac{R}{\omega L}+\frac{G}{\omega C}\right)} \simeq 1

The general solution is the sum of two traveling waves, one right and one left:

V(z)=V0+exp(+γz)+V0exp(γz)=V+(z)+V(z)V(z) = V_0^+\exp(+\gamma z) + V_0^-\exp(-\gamma z) = V^+(z) + V^-(z)

  • V0+V_0^+ and V0V_0^- are the (initial) complex amplitudes of the voltage waves
  • The imaginary part (γ)=k>0\Im(\gamma)=k>0 is the wavenumber of the propagating waves, and λ=2π/k\lambda =2\pi/k is the wavelength. The phase velocity satisfies c1=ω/kc_1=\omega/k
  • The real part (γ)=α<0\Re(\gamma)=-\alpha<0 gives the attenuation constant

We can write:

v(z,t)=V0+exp(jωt+jkzαz)+V0exp(jωtjkz+αz)v(z,t) = V_0^+\exp(-j\omega t +jkz -\alpha z) + V_0^-\exp(-j\omega t -jkz +\alpha z)

It is easy to check that the current follows an analogous equation

d2Idz2γ2I=0;I(z)=I0+exp(+γz)+I0exp(γz)\frac{d^2 I}{dz^2} - \gamma^2 I = 0; \quad I(z) = I_0^+\exp(+\gamma z) + I_0^-\exp(-\gamma z)

Using the first order equations we find that voltage and current complex amplitudes

V0+=Z0I0+;V0=Z0I0;V_0^+ = Z_0 I_0^+; \quad V_0^- = -Z_0 I_0^-;

where we define the characteristic impedance of the transmission line as:

Z0=RjωLGjωC=LCρZ_0 = \sqrt{\frac{R-j\omega L}{G-j\omega C}} = \sqrt{\frac{L}{C}} \rho


ρ=1+jR/(ωL)1+jG/(ωC)1\rho = \sqrt{\frac{1+jR/(\omega L)}{1+jG/(\omega C)}} \simeq 1

Ideal and real transmission lines

In an ideal transmission line RR and GG are zero, so

γ=ik=jωLC;α=0;Z0=L/C;c1=1/LC\gamma = ik = j\omega \sqrt{LC}; \quad \alpha = 0; \quad Z_0 = \sqrt{L/C}; \quad c_1 = 1/\sqrt{LC}

In a real transmission line there is some attenuation α>0\alpha>0, Z0Z_0 is not purely real, and the phase velocity c1c_1 becomes weakly dependent on ω\omega

  • Whenever c1c_1 is a function of ω\omega, there is dispersion

The parameters L,C,R,GL,C,R,G are geometry- and material-dependent. They also depend on the constants ε0\varepsilon_0 and μ0\mu_0

Read 2.2, 'Field analysis of a transmission line'

Line termination

For an ideal line (R,G=0R,G=0):

V(z)=V0+exp(+jkz)+V0exp(jkz)V(z) = V_0^+\exp(+jkz) + V_0^-\exp(-jkz)

I(z)=V0+Z0exp(+jkz)V0Z0exp(jkz)I(z) = \frac{V_0^+}{Z_0}\exp(+jkz) - \frac{V_0^-}{Z_0}\exp(-jkz)

The ratio between V0+V_0^+ and V0V_0^- on a transmision line depend on the termination load.

The load impedance fixes the ratio between II and VV at the end of the line, where we place our origin z=0z=0 for simplicity

ZL=V(0)I(0)Z_L = \frac{V(0)}{I(0)}

z=0V(0)=V0++V0;I(0)=V0+Z0V0Z0z=0 \Rightarrow \quad V(0) = V_0^+ + V_0^-; \quad I(0) = \frac{V_0^+}{Z_0} - \frac{V_0^-}{Z_0}


ZL=V(0)I(0)=V0++V0V0+V0Z0V0=V(0)I(0)=ZLZ0ZL+Z0V0+Z_L = \frac{V(0)}{I(0)} = \frac{V_0^+ + V_0^-}{V_0^+ - V_0^-}Z_0 \quad \Rightarrow \quad V_0^- = \frac{V(0)}{I(0)} = \frac{Z_L - Z_0}{Z_L + Z_0}V_0^+

  • We define the voltage reflection coefficient Γ=V0/V0+\Gamma=V_0^-/V_0^+, which is complex in general. The power reflection coefficient is Γ2|\Gamma|^2
  • If ZL=Z0Z_L=Z_0 there is no reflected wave and Γ=0\Gamma=0. We say the load and the line are matched
  • Any other situation leads to (partially) standing waves. The ratio between the maximum and minimum voltage in the line is the standing wave ratio SWR


The input of the line + load if the line has a total length \ell is, by definition, the ratio between voltage and current at z=z=-\ell:

Zin=V()I()=1+Γexp(2jk)1Γexp(2jk)Z0Z_{in} = \frac{V(-\ell)}{I(-\ell)}=\frac{1+\Gamma\exp(-2jk\ell)}{1-\Gamma\exp(-2jk\ell)}Z_0

  • For an unmatched line (Γ0\Gamma\neq 0), it is position-dependent
  • If =nλ/2\ell= n\lambda/2, then Zin=ZLZ_{in}=Z_L, regardless of Z0Z_0
  • If =λ/4+nλ/2\ell= \lambda/4+n\lambda/2, then Zin=Z02/ZLZ_{in}=Z_0^2/Z_L (this is a so-called quarter-wave convertor)

Open line

The load impedance is infinite, and I(0)=0I(0)=0 at the end of the line. This means that Γ=1\Gamma = 1 and Zin=jZ0tan(k)Z_{in}=jZ_0 \tan(k\ell)

Shortcircuited line

The load impedance is zero, and V(0)=0V(0)=0 at the end of the line. This means that Γ=1\Gamma = -1 and Zin=jZ0cot(k)Z_{in}=-jZ_0 \cot(k\ell)

Discontinuity in characteristic impedance

Consider that we join two lines with Z0Z_0 and Z1Z_1 characteristic impedances. We assume a wave V0+V_0^+ arriving from the left and a wave V1V_1^- arriving from the right.

Equating voltages, currents at z=0z=0:

V0++V0=V1++V1;V0+Z0V0Z0=V1+Z1V1Z1V_0^+ + V_0^- = V_1^+ + V_1^-; \quad \frac{V_0^+}{Z_0} - \frac{V_0^-}{Z_0} = \frac{V_1^+}{Z_1} - \frac{V_1^-}{Z_1}

We have 4 variables and 2 equations. We can compute V0V_0^- and V1+V_1^+ as the sum of a reflected and a transmitted wave:

V0=Z1Z0Z1+Z0V0++2Z0Z1+Z0V1;V1+=Z0Z1Z1+Z0V1+2Z1Z1+Z0V0+V_0^- = \frac{Z_1-Z_0}{Z_1+Z_0}V_0^+ + \frac{2Z_0}{Z_1+Z_0}V_1^-; \quad V_1^+ = \frac{Z_0-Z_1}{Z_1+Z_0}V_1^- + \frac{2Z_1}{Z_1+Z_0}V_0^+

Looking from the left:

  • The reflection and transmission coefficients are defined as

Γ=(Z1Z0)/(Z1+Z0)\Gamma = (Z_1-Z_0)/(Z_1+Z_0)


  • The insertion loss is TT in dB

Power conservation takes the form:

1/Z0=Γ2/Z0+T2/Z11/Z_0 = \Gamma^2/Z_0 + T^2/Z_1

Interlude: decibels

Any energy or power ratio P1/P2P_1/P_2 can be expressed in dB as

10log10P1P2dB10 \log_{10} \frac{P_1}{P_2} \,\, \text{dB}

Since powers are the squares of amplitudes (e.g. voltages or currents), we can also express amplitude ratios such as V2/V1V_2/V_1 in dB with

20log10V1V2dB20 \log_{10} \frac{V_1}{V_2} \,\, \text{dB}

Electrical engineers like expressing not just ratios, but absolute quantities refered to e.g. 1 mW (for power), 1 V (for voltage), etc. Then we write:

10log10P1(in mW)dBmW10 \log_{10} P_1 (\text{in mW}) \,\, \text{dBmW}

20log10V1(in V)dBV20 \log_{10} V_1 (\text{in V}) \,\, \text{dBV}

Common coaxial connectors

BNC: bigger. Female/male. Quarter turn to mate. Up to 2 GHz roughly. Common variants exist for 50 or 75 Ohm lines. Typically, male connector is on the cable, female connector on the equipment. High voltage alternatives: MHV, SHV

SMA: smaller. Female/male. Threaded. Up to several GHz and very common in microwave systems. 50 Ohm.

N connector: common for power applications

Balanced - unbalanced transmission lines

Transmission lines discussed so far are of the "unbalanced" type: the signal wire and the ground wire in the cable are physically different and serve different roles (e.g. a coaxial cable).

If the two wires are identical, they both have opposite voltage and current at each point. This is a "balanced" line, where both leads may float with respect to ground. Optionally, there may be a "shield" around the two wires playing the role of ground. This setup can be studied with two lines + ground line.

Side comment, partially related:

Observe that our transmission line model assumes a lumped ground with constant voltage everywhere. The real ground wire may exhibit varying voltage and current with distance.

Ground loops

A ground loop occurs whenever there are more than two paths to ground in a circuit. Since the wire ground loop usually has very low resistance, often below one ohm, even weak external magnetic fields can induce significant currents.

The alternating ground current flowing through the cable can introduce AC voltage drops due to the resistance, and therefore interferences.

Ground currents through the shield of a coaxial also create noise on the signal and voltage differences among the various grounds. Ground currents can be induced by external oscillating B fields or by an equipment that leaks to ground

Single-point grounding to the building and ``breaking'' ground loops with transformers, etc, are techniques to reduce ground noise.