How does TACAN work?
How is TACAN different from the VHF Omnidirectional Range (VOR) system?

A very short question, but the answer calls for describing several techniques which are by themselves difficult to summarize without taking liberties with reality, so the post is rather long and should be read by sections of interest rather than totally at once. And for those not interested in the design techniques, fortunately there is a...
Short answer
Design principle:

A TACAN uses UHF to increase bearing precision. It consists on a single integrated system performing bearing and distance determination at once. In this system, the ground station is a transponder and the interrogator is on-board the aircraft (contrary to the SSR transponder). Frequency is like graduations on a scale, when frequency increases the graduations on the scale are more dense, and readings are more accurate.

A VOR works on VHF for bearing determination. No aircraft action is required to trigger the ground station signal which is permanent. To determine distance, another independent system, the DME, is used. The DME has been borrowed to military and is actually a TACAN without its bearing components (so it is a transponder interrogated by the aircraft).

When selecting a VOR frequency in a civilian aircraft, the avionics actually sets the VOR receiver on this frequency, and the standalone DME interrogator at some &quot;paired&quot; UHF frequency obtained from the ICAO standard pairing table (page 6). VOR and DME share nothing on board beyond the frequency pairing table.


Station antennas:

The original TACAN antenna is made of two small rotating drums with parasitic antenna elements (see details below). A TACAN can be installed on ships or mobile stations. The TACAN antenna is externally similar to a conventional VOR. A TACAN in Alaska during an exercise:



(Source: Wikipedia)

The cylinder contains the rotating antenna system. In more modern TACAN, the mechanical rotation has been replaced by electronically scanned arrays, reducing the size:



Transportable TACAN, source

Doppler VOR (DVOR) are more common than conventional VOR (CVOR), because they can be located on airfields (CVOR: See below for details). The DVOR antenna is a large circular array with a central reference antenna and a large counterpoise under the array.  VOR are sometimes colocated with a DME station, in that case the DME vertical antenna is above and coaxial to the VOR system.



Lambourne VOR/DME, DME antenna on top of the central VOR reference antenna. (source: Wikipedia)

Because the DME part is common to VOR/DME and TACAN, it is technically possible to associate a VOR to a TACAN to obtain a VORTAC station. Military use the TACAN, civilian use the VOR and the DME information of the TACAN:



A full TACAN in place of the previous DME antenna. Source

In addition VOR (CVOR/DVOR) uses Alford loop antennas which are horizontally polarized and radiate low on the horizon. They are sensitive to reflection on obstacles. An electrical counterpoise is required to hide the ground and raise the radiation angle. This artificial ground plane can be very large:



PFN Vortac (decommissioned), source


Signals:

A VOR transmits bearing information continuously.

A TACAN only sends reply pulse pairs when interrogated (see the explanation below). These pairs encode both the bearing and DME information.

A TACAN is usually more powerful than a VOR and has a larger range of use.


I'll focus on explaining bearing determination systems, and explain the DME as an integral component of the TACAN. In addition there are two types of VOR, conventional and Doppler, which work very differently even if they deliver compatible signals to the (unsuspecting) common receiver.

Bearing determination principle
The common principle of bearing determination is to send two signals from the ground station:

A reference signal telling any receiver around the current orientation of the active signal.

A variable signal allowing a particular receiver to determine when the active signal is &quot;pointing&quot; to the receiver (pointing is not the accurate word as the DVOR signals are omnidirectional, see more in this answer).


The receiver determines its relative bearing by comparing these two signals. Both signals are sine functions, the orientation value is represented by the current phase of this function. Both VOR and TACAN use this basic principle, albeit they realize it differently.
The signal phase has the major role in this story, so let's be sure we agree the meaning:

Any periodical (repetitive) signal can be seen as the result of a vector turning at some rate $\small \omega$. The sine function for angle $\small x$ is $\small y = sin(x)$. Applied to a sinusoidal wave of frequency $\small f$ and peak amplitude $\small A$, this becomes $\small y=A.sin(\omega t+\varphi)$ where $\small \omega = 2 \pi f$. The angle $\small \omega t+\varphi$, has been split between the phase $\small \omega t$ and the phase at origin $\small \varphi$. $\small \varphi$ is null if we start a cycle at time 0, this is usually the case. More simply, the quantity $\small \omega t$ represents how much the vector has turned at time $t$. It is an angle, reset after a full turn, therefore it indicates in the end which portion of a full cycle has already been spent (which phase we are in the cycle). Visually:


Phase angle of a sine wave
From that, it is clear that comparing phases of two signals with the same frequency (simple to do with electronics) is equivalent to comparing how much time one lags behind the other (time is actually difficult to measure).
VOR types
Conventional VOR (CVOR) and Doppler VOR (DVOR) stations are perceived identically by the receiver, though they transmit very different signals. The DVOR use tricks to mimic a CVOR and deceive the CVOR receiver. CVOR have nearly disappeared from sight because, due to their sensitivity to reflections, they cannot be located on airfields or close to roads. However en-route/high altitude CVOR can be found at isolated places, the reason is they are more compact and have a smaller cone of silence than DVOR and reflections can be minimized, e.g. when the VOR is located at the top of a hill.
Understanding the DVOR tricks without knowing how a CVOR works is difficult, and doesn't give clues on how the bearing is really determined. So I'm afraid we need to understand the CVOR before the DVOR.
Conventional VOR: CVOR
Early CVOR antenna used to be an array of four Alford loops at the corners of an imaginary square, known by their conventional names: NW, NE, SW and SE. NW+SE loops form the first pair, NE+SW loops form the second pair.

CVOR with four Alford loops
Alford loops are horizontally polarized and very sensitive to reflections on surrounding obstacles (multipath).
Recent generations of CVOR use a slotted antenna, which is a fixed cylinder with vertical slots (generally four slots):

CVOR with slotted antenna and the DME antenna on top. Source: AviaTecho.
A counterpoise is placed under the array to hide the VOR shelter and the ground and raise the radiation direction, it has the dual effect of minimizing undesirable reflection on the shelter and the ground and reducing the cone of silence above the VOR.
CVOR creates and uses reference and variable signals this way:

A low frequency generator creates three 30 Hz signals, identical except their phases. Two audio signals are derived from a reference signal: The sin signal has a phase at origin of -90° and the cos signal has a phase at origin of +90° (the point is sin and cos signals are in phase opposition).

The phase of the reference conceptually represents a direction and is often called the goniometer. As this signal frequency is 30 Hz, the imaginary direction it represents sweeps 360° 30 times per second (1,800 rpm, this is pure abstraction, there is no rotating parts in a CVOR).

A low frequency generator creates a 9960 Hz signal which is FM modulated by the reference. This signal is known as the reference subcarrier.



Conventional VOR block diagram


A HF generator creates a carrier of frequency f (f being the VOR frequency), this carrier is split in three parts:


one part is AM modulated by the reference subcarrier.
another is AM modulated by sin.
the last part is AM modulated by cos.



The HF signal with the reference subcarrier is sent to all antennas. This way the reference can be received identically regardless of the position of the receiver around the CVOR.

The two other HF signals have first their carrier removed, so that only the sidebands subsist. This is to prevent carriers to interfere in space, interferences must occur only between sidebands.

Then one signal is sent to the NW+SE pair of antennas, the other signal is sent to the other pair (remember the two pairs are perpendicular).

The space modulation magic does the rest. The sin and cos sidebands are added as field vector values, sometimes the individual amplitudes are added, sometimes they are subtracted, in variable proportion. This results in an unbalanced cardioid pattern (more precisely a Limaçon de Pascal) which rotates around the VOR antennas at 1,800 rpm, the direction being linked to the phase of the reference (or sin or cos, as they are all linked by fixed values).

The signal resulting from space modulation appears like being a carrier AM modulated according to the direction of the virtual &quot;rotating antenna&quot;. The AM modulation is also a 30 Hz signal, and is known as the variable signal.

The resulting signal also contains the unmodified (and constant amplitude) carrier with its reference subcarrier.



CVOR spectrum


To determine the bearing (the radial) of the receiver relative to the CVOR, one only needs to compare the phase of the variable signal with the phase of the reference signal. Both are contained in the resulting signal. The phase of the reference signal and the phase of the variable signal are equal when the reference &quot;points&quot; to north (per principle at this time both phases have a value of 135°, the sum of 45° and 90°, but the actual value has no influence, only the phase difference is meaningful):



VOR: Determination of the bearing by comparing phases

Now we know the principle of the CVOR, it more easy to understand the principle of the DVOR. The DVOR was created to make up for some weaknesses in the CVOR: The CVOR is not terribly accurate unless the installation site is very carefully selected (no obstacle). That means isolated points, not airfields. That's not the preferred option for maintenance and this often precludes having the CVOR aligned with the runway for a VOR approach.


From CVOR to Doppler VOR, ensuring retro-compatibility
The lack of precision of the VOR is rooted in two design choices:

The antennas are close to each other, any default in their placement has great consequences in the accuracy.

The variable signal is AM modulated, AM modulation is deadly subject to errors created by electromagnetic noise and multipath.


In the Doppler VOR (once again... there are two types of DVOR, the single sideband and the double sideband, I'll describe the DSB here):

The two active antennas are at a large distance from each other (diametrically opposed).

The variable signal is FM modulated.


In order to be compatible with the CVOR receiver, other changes had to be made:

As the receiver still compares two signals, one being AM, the other FM, the reference signal must be AM modulated.

Because the result of the phase comparison is now inverted (variable minus reference becomes reference minus variable), the direction of rotation of the pattern must also be inverted (anti-clockwise instead of clockwise).

Because the pair of antennas used for the variable signal creates deliberately a Doppler effect, the reference must be sent on a specific central antenna preserved from the Doppler effect.


Doppler VOR: DVOR
The principle of a Doppler VOR is to create the frequency modulation by Doppler effect rather than by electronic modulation. The Doppler effect occurs with a moving wave source: In spite the source has a constant frequency, when it comes closer to the receiver the apparent frequency is higher than the actual frequency. How much higher depends only on the closure rate.

Doppler effect on train noise: Sound is higher-pitched at the front than at the rear
In the DVOR, pairs of opposed antennas (still Alford loops) are constantly switched on/off, scanning the full array anti-clockwise, the full scan being done 30 times per second. Actually there are two groups of antennas rather than two antennas involved to allow for blending (smooth transition from one pair to the next), but let's simplify for a second. From a receiver standpoint, the signal seems to come from a moving source, and therefore a Doppler shift will occur in a proportion which depends on the apparent direction of the move.

DVOR Doppler effect
To allow for compatibility with the CVOR receiver, this shift must be at most 480 Hz, 480 Hz being the FM swing of the subcarrier in the CVOR. A simple calculation shows the array diameter must be about 14 m (46 feet).
To generate the FM modulated signal, the unmodulated 9960 Hz subcarrier is sent on the &quot;rotating&quot; pair of antennas. The Doppler shift is maximum when the receiver direction is tangential to the pair trajectory and minimal when the pair is perpendicular to the direction of the receiver. This shift is exactly representative of the aircraft bearing and is the variable signal modulation we need.
From a radio signal perspective, only the sideband frequencies are used to transmit the 9960 Hz subcarrier (VOR frequency f +/- 9960 Hz). The carrier is itself sent on the central antenna, AM modulated by the reference signal. This way the carrier is not subject to the Doppler shift.
Bottom line... Like in the CVOR, the receiver sees the composite signal: A carrier AM modulated with a 30 Hz (which is the reference instead of the variable signal), with a subcarrier FM &quot;modulated&quot; as the result of the Doppler effect, at 30 Hz (the frequency of the scan, it now represents the variable signal instead of the reference) and with a swing not far from the expected 480 Hz.
Blending: If a pair of antennas were used one at a time, the number of measurable bearings would be equal to the number of antennas in the array (about 50). To create a more continuous scan (and therefore a larger number of measurable bearings), the antennas which precede and follow the main antenna are also fed by the subcarrier signal, but with a smaller power. This &quot;blends&quot; the transition from a scan position to the next one.
See also What causes the phase to change in a VOR? for a better explanation of the DVOR.
TACAN bearing
A TACAN is based on a stationary antenna plus a rotating parasitic system. The base antenna is vertical and common to the distance and bearing measurement instruments.
Parasitic elements in the aerial field refer to passive antenna elements added to the actual active radiator. A reflector decreases the gain on its side, a director increases the gain on its side (more). The well-known Yagi directional antenna (here in an horizontal polarization) has the two types of parasitic elements:

(Source, modified)
These elements are used in the TACAN, but they are rotating around the active element:

(Source, modified)

The central element, which is the one also used for the DME portion, transmits a constant amplitude signal.

A rotating drum with a reflector electrically adjusts the radiation pattern, adding a signal dip (low gain) that rotates at 900 RPM, which is equivalent to a 15 Hz amplitude modulation. The radiation pattern in the horizontal plan takes the shape of a cardioid:


(Source: Advances in Electronics and Electron Physics, Volume 68, modified)

Another drum with a set of 9 directors, mechanically linked to the first one, creates a 135 Hz (9x15) additional amplitude ripple over the 15 Hz modulation:


(Source: Advances in Electronics and Electron Physics, Volume 68, modified)


Now we need to start again the reasoning taking into account that the TACAN signal isn't transmitted permanently, but only keyed (switched on/off) by bursts of information. Burst are of two kinds:

Reference bursts
DME responses.

Reference bursts are generated according to the orientation of the modulation pattern:

When the 15 Hz peak faces North a main reference burst is sent. The burst consists of 24 pulses

with an asymmetric duty cycle.

When any of the 135 Hz peaks faces East, an auxiliary reference burst is sent. The burst consists of 24 pulses with a symmetric duty cycle.



(Source: Advances in Electronics and Electron Physics, Volume 68. Modified)
The duration of these bursts is only a portion of the 15 Hz cycle, meaning that if there is few aircraft DME interrogations, most of the time the TACAN signal is not keyed, therefore not transmitted. This lack of transmission would create a difficulty for the aircraft receiver:

To adjust its receiver gain (AGC) to counter fading.
To identify the 15 Hz and 135 Hz modulations.

To maintain the capability of reception, the TACAN signal is instead keyed at a constant rate of 2,700 pairs of pulses per second, adding squitter pulses if necessary to fill the blanks. The more the DME interrogations are received by the TACAN, the more DME reply bursts are sent, the less squitter pulses are necessary (more in MIL-STD-291).


The 135 Hz signal has been removed for simplification (Source)
The 135 Hz modulation is used for the bearing determination. By comparing the time between an auxiliary burst and the subsequent reception of one of the 9 signal peaks, it is possible to determine the aircraft bearing relative to the ground station. The main burst (15 Hz) is used to disambiguate which of the 9 lobes was used, and therefore which of the 40° (360/9) sector is actually relevant for the bearing.
In theory the use of the top end of the UHF band and the 135 Hz ripple increase the bearing accuracy by one order of magnitude compared to the VOR. In practical this is less, but still better than the VOR.
The DME principle is to measure the time a radio signal takes for a roundtrip to the ground station. As radio waves travel at light speed, knowing the time is knowing the distance. &quot;The aircraft interrogates the ground transponder with a series of pulse-pairs (interrogations) and, after a precise time delay (typically 50 microseconds), the ground station replies with an identical sequence of pulse-pairs.&quot; (Wikipedia).