Electromagnetic Waves, Antennas, and RF: Transmission Lines and the Telegrapher's Equations

Electromagnetic Waves, Antennas, and RF: Transmission Lines and the Telegrapher’s Equations

EM21 showed that a traveling electromagnetic wave reflects whenever it encounters an impedance discontinuity. Transmission-line theory is the one-dimensional distributed version of the same physics. Instead of describing the fields everywhere in the cross-section, we describe the line by a voltage v(z,t) and current i(z,t) that vary with both position and time. The result is not a lumped circuit in the ordinary low-frequency sense: it is a wave system governed by partial differential equations [1, 2, 3, 4].

The central chain developed here is

(1)

The algebra will strongly resemble EM21. That is not an accident. A material boundary and a transmission-line load are both wave-impedance discontinuities.

1 Why a wire becomes a wave problem

At sufficiently low frequency, a short connection can often be approximated as an ideal wire: voltage is taken to be the same everywhere along it, and current changes everywhere essentially at once. That approximation fails when the physical length becomes a significant fraction of a wavelength or when the signal rise time becomes comparable to the propagation delay.

A transmission line is therefore modeled as a distributed electromagnetic system. Over a small segment dz, four per-unit-length parameters summarize the local field storage and loss:

R	[Ω∕m]		series conductor resistance,	(2)
L	[H∕m]		series magnetic-energy storage,	(3)
G	[S∕m]		shunt dielectric leakage,	(4)
C	[F∕m]		shunt electric-energy storage.	(5)

The parameters L and C are especially important physically. A propagating line wave continuously exchanges energy between electric-field storage and magnetic-field storage, just as the plane waves of EM16–EM20 do. The parameters R and G dissipate part of that energy into Conductor and dielectric loss.

2 Deriving the first telegrapher equation

Consider the small line segment from z to z + dz. Apply Kirchhoff’s voltage law around the series path. The voltage drop across the resistance is

(6)

and the inductor drop is

(7)

Therefore

(8)

Expand the voltage at z + dz to first order:

(9)

Ignoring terms of order dz² and higher gives

(10)

Divide by dz:

(11)

This is the first telegrapher equation. It says that spatial change of voltage is produced by both resistive drop and time-varying magnetic energy storage.

3 Deriving the second telegrapher equation

Now apply current conservation to the same segment. The current entering at z differs from the current leaving at z + dz because some current flows through the shunt conductance and capacitance:

(12)

Using

(13)

gives

(14)

Together,

(15)

These are the time-domain telegrapher’s equations. They are coupled first-order partial differential equations in space and time.

4 The full lossy wave equations

The first-order system can be combined into second-order equations. Differentiate the voltage equation with respect to z:

(16)

Substitute

(17)

Then

	= RGv + RC + LG + LC	(18)
	= RGv + (RC + LG) + LC.	(19)

Therefore

(20)

The current obeys the identical form:

(21)

The LC term produces wave propagation, the terms proportional to RC + LG produce first-order damping, and RG is a loss-coupling term. This is the distributed-circuit counterpart of the damped electromagnetic wave equation developed in EM20.

5 Lossless line: the wave equation appears immediately

Before moving to the full sinusoidal solution, consider the ideal lossless case

(22)

Then

(23)

Differentiate the first equation with respect to z:

(24)

Substitute the second equation:

(25)

Hence

(26)

Similarly,

(27)

Comparing with the standard one-dimensional wave equation,

(28)

we identify

(29)

Thus the finite propagation speed of a transmission-line signal follows directly from the distributed electric and magnetic energy storage.

6 Sinusoidal steady state and phasor form

Assume sinusoidal time dependence with the convention

(30)

Then time differentiation becomes multiplication by iω, and the telegrapher equations reduce to ordinary differential equations:

(31)

(32)

These equations show that the natural series impedance per unit length is

(33)

and the natural shunt admittance per unit length is

(34)

7 Deriving the propagation constant

Differentiate the voltage equation once more:

(35)

Using the current equation,

(36)

Define the propagation constant

(37)

Then

(38)

The same derivation gives

(39)

Exactly as in EM20,

(40)

is the attenuation constant and

(41)

is the phase constant.

8 Forward and backward traveling waves

The general voltage solution is

(42)

The first term propagates in the +z direction; the second propagates in the −z direction.

To obtain the current, substitute the voltage solution into

(43)

Differentiating gives

(44)

Therefore

(45)

The ratio multiplying the traveling-wave voltage naturally defines the characteristic impedance.

9 Deriving the characteristic impedance

For the forward wave,

(46)

From the preceding equation,

(47)

Hence

(48)

Using

(49)

we obtain

(50)

The complete current solution is therefore

(51)

The minus sign in the reflected current is crucial. The reflected wave transports energy toward decreasing z, so its current direction is opposite to the reference direction chosen for the forward wave.

10 Lossless-line results

For

(52)

the propagation constant becomes

γ	=	(53)
	= iω.	(54)

Thus

(55)

The characteristic impedance is purely real:

(56)

The phase velocity is

(57)

and the wavelength is

(58)

These relations reveal the physics of a line very cleanly: L and C together set the propagation speed, while their ratio sets the wave impedance.

11 Low-loss approximation

Many practical RF lines satisfy

(59)

Expanding the exact propagation constant to first order in the small loss terms gives

(60)

where the Z₀ on the right is approximately the lossless value , while

(61)

The two contributions to α separate conductor loss and dielectric loss. The exact formulas should be used when the low-loss inequalities are not satisfied.

12 The load creates the reflected wave

Let the load be located at z = 0 and let the line extend toward negative z. The load impedance is

(62)

At the load,

(63)

while

(64)

Thus

(65)

Define the voltage reflection coefficient at the load by

(66)

Then

(67)

Solving for Γ_L gives

(68)

This is the transmission-line counterpart of the normal-incidence electromagnetic boundary result from EM21,

(69)

The symbols are different, but the wave physics is the same.

13 Special load cases

Three cases should be recognized immediately.

13.1 Matched load

If

(70)

then

(71)

There is no reflected wave.

13.2 Open circuit

For

(72)

we obtain

(73)

The reflected voltage has the same phase as the incident voltage at the load, while the forward and reflected currents cancel there.

13.3 Short circuit

For

(74)

we obtain

(75)

The reflected voltage reverses phase so that the total load voltage vanishes.

14 Reflection coefficient away from the load

At an arbitrary position z, the ratio of reflected to incident voltage is

Γ(z)	=	(76)
	= ΓLe2γz.	(77)

Therefore

(78)

For a lossless line,

(79)

so |Γ| is constant along the line while its phase rotates with position.

For a lossy line extending toward negative z, the magnitude of the reflection coefficient decreases as one moves away from the load because the reflected wave has traversed additional lossy distance.

15 Standing waves

For a lossless line, write the load reflection coefficient as

(80)

The voltage is

(81)

Its magnitude satisfies

(82)

Maximum voltage occurs when the cosine equals +1:

(83)

Minimum voltage occurs when the cosine equals −1:

(84)

The distance between adjacent maxima is

(85)

and the distance from a maximum to the neighboring minimum is

(86)

16 Standing-wave ratio

The voltage standing-wave ratio, usually abbreviated VSWR or simply SWR, is

(87)

Substituting the extrema gives

(88)

Solving for mismatch magnitude,

(89)

Important limiting cases are

|Γ| = 0	⇒ SWR = 1,	(90)
|Γ|→ 1	⇒ SWR →∞.	(91)

SWR measures the magnitude of mismatch but does not determine the phase of Γ.

17 Power carried by the forward and reflected waves

For a lossless line with real Z₀, the time-average forward power is

(92)

and the reflected power magnitude is

(93)

Therefore

(94)

The net power flowing toward the load is

(95)

This is exactly analogous to EM21, where the reflected power fraction at a lossless material boundary is R = |Γ|².

18 Input impedance at an arbitrary distance from the load

A transmission line transforms impedance with distance. For a point a distance ℓ from the load, located at z = −ℓ, the input impedance is

(96)

Substituting the forward and reflected solutions and eliminating Γ_L gives the general result

(97)

For a lossless line, γ = iβ and

(98)

so

(99)

This equation explains why a reactive or mismatched load can appear as a very different impedance when viewed through a finite length of line.

19 Worked example: 50 Ω line terminated in 100 Ω

Consider a lossless line with

(100)

The load reflection coefficient is

ΓL	=	(101)
	= .	(102)

Thus

(103)

The SWR is

SWR	=	(104)
	= 2.	(105)

Hence

(106)

The reflected power fraction is

(107)

so about 11.1% of the incident power is reflected and 88.9% is delivered to the load in this ideal lossless system.

Now suppose

(108)

so

(109)

Take a line length

(110)

Then

(111)

for which

(112)

The input impedance is therefore

Zin	= 50	(113)
	= 40 + i30 Ω.	(114)

Thus

(115)

A purely resistive 100 Ω load has become a complex impedance when viewed through the line.

20 Quarter-wave and half-wave transformations

Two lossless-line lengths are especially important.

For

(116)

we have tan(βℓ) = 0, so

(117)

A half-wave line repeats the load impedance.

For

(118)

| tan(βℓ)|→∞, yielding

(119)

This is the quarter-wave impedance inversion. It is the basis of the quarter-wave transformer developed later in impedance-matching theory.

21 Where L and C come from: a coaxial example

Transmission-line parameters are compact summaries of electromagnetic fields. For an ideal coaxial line with inner-conductor radius a, outer-conductor inner radius b, permeability μ, and permittivity 𝜖, electrostatic and magnetostatic field calculations give [3, 5]

(120)

and

(121)

Therefore

(122)

so

(123)

This is exactly the bulk electromagnetic-wave speed from EM20.

Likewise,

Z0	=	(124)
	= ln .	(125)

Thus

(126)

where

(127)

is the intrinsic impedance of the dielectric filling the coax.

For example, let

(128)

Then

(129)

(130)

so

(131)

and

(132)

The reduced speed is simply the electromagnetic-wave speed in a dielectric with refractive factor = 1.5.

22 Characteristic impedance is not ordinary resistance

The symbol Z₀ has units of ohms, but it should not be interpreted as a resistor smeared along the line. In a lossless line,

(133)

yet

(134)

remains finite.

The quantity Z₀ is the voltage-to-current ratio of a single traveling wave. A matched load equal to Z₀ absorbs the wave energy without reflection; the line itself need not dissipate that energy.

23 Connection to the electromagnetic boundary problem

The conceptual parallel with EM21 can now be made explicit:

(135)

At a material interface,

(136)

At a transmission-line load,

(137)

In both cases, reflection is created because the forward wave reaches a boundary that demands a voltage/current or electric/magnetic-field ratio different from that carried by the incident wave alone. A reflected wave appears so that the total fields can satisfy the boundary condition.

24 A practical calculation workflow

For a uniform transmission line driven at one frequency:

1. identify R,L,G,C and frequency f;
2. compute

   (138)

3. compute

   (139)

4. apply the load boundary condition to obtain

   (140)

5. use |Γ| for reflected-power fraction and SWR;
6. use the phase of Γ(z) to locate standing-wave maxima and minima;
7. if necessary, transform the load through a line length with the input-impedance equation.

This workflow separates propagation, attenuation, mismatch, and impedance transformation instead of treating them as unrelated formulas.

25 Common mistakes

• Treating Z₀ as a physical resistor. A lossless line can have Z₀ = 50 Ω even though its series resistance is zero.
• Dropping the minus sign in reflected current. The backward wave reverses the current direction associated with positive power flow.
• Using |Γ| as the reflected power fraction. For a lossless line, the reflected power fraction is |Γ|².
• Confusing SWR with reflection phase. SWR determines |Γ|, not the complex phase of Γ.
• Assuming the load impedance appears unchanged at the source. A finite line transforms impedance according to its electrical length.
• Using lumped-circuit intuition when the interconnect is electrically long. Voltage and current become waves with finite propagation speed.

26 What EM22 adds to the series

EM20 established propagation constants and wave impedance in bulk media. EM21 showed how impedance discontinuities create reflection and standing-wave behavior at material boundaries. EM22 now translates that physics into the distributed voltage-current language used throughout RF hardware.

The key results are

(141)

(142)

(143)

(144)

and

(145)

These equations provide the bridge from electromagnetic wave theory into coaxial cables, PCB traces, impedance matching, Smith charts, S-parameters, microwave networks, and ultimately the RF front ends used in communication and navigation receivers.

References

References