Shields seem to be combination of graviton "membrane", EM and subspace field, with graviton "membrane" taking care of any particle weapons, while EM and subspace fields block sensors and any energy weapons.
Due to shield physics, hard to quantify, but GCS shields as of middle 24th century could not block energy of photon torpedo spread (5 torpedoes) at point-blank range, suggesting 27.68 - 78.5 gigatons versus photon torpedoes and other similar weapons (say, turbolasers). However, shield strength seems to vary, possibly due to various shield effectiveness versus different kinds of weapons.
On starships, the shield contained six sections, forward, starboard, port, aft, dorsal, and ventral, which allowed greater rendunancy. Shields could be projected from navigational deflector dish, but are more commonly projected by grid of emitters on ship's hull.
We often hear that shields are at "80% and holding", "50%", "40%" etc., which seems to indicate shield effectiveness - shields at 20% will block 80% of energy directed at them. It is possible that effectiveness is only in regards to graviton field, or entire shield system. It is specially supported by TNG "Yesterday's Enterprise", where we see E-D taking increasing amounts of damage per hit as shield percentage falls.
Enterprise was pulled inside a Dyson Sphere. We have following dialogues:
After Enterprise was pulled inside:
"Auxilliary power failling."
"The resonante frequency of tractor beams is incompatible with our power systems. Warp and impulse engine relays have been overloaded. I am attempting to compensate."
(few moments later)
"The tractor beams have released us, sir. "
"Holed position here until we can get our bearing."
"Full sensor sweep, Mr. Data. Where are we?"
"Approximately 90 million kilometers from the star's photosphere. I am reading a great deal of surface instability. It may be..."
"Sir! The inertial motion from the tractor beams is still carrying us forward. Impulse engines are off-line and I can't stop our momentum. We're falling directly into the star.
(Cut to Jenolan.)
(Cut back to Enterprise.).
Data: "We will enter the sun's photosphere in 3 minutes."
Later, they use manouvering thrusters to change course and attain orbit, implying that primary power is offline, which is confirmed in dialogue.
Helmswoman: "We're in orbit, captain. Our altitude is 150 000 kilometers.
Riker: "I'll see about getting main power back on line."
Later:
Data: "Sphere appears to be abandoned. Sensors show that star is extremely unstable. It is experiencing severe bursts of radiation and matter expulsions."
(...)
Worf: "Sir. Sensors show a large magnetic disturbance on the star's surface."
Data: "It is a solar flare, captain. Magitude 12, class B."
Picard: "Shields?"
Worf: "Shields are up, but only at 23%."
Data: "The star has entered a period of increased activity. Sensors indicate that solar flares will continue to grow. In three hours, our shields will no longer be sufficient to protect us, sir."
So:
a) Enterprise is 150 000 km from star's surface
b) shields are up to 23% of full power
c) star has entered period of increased activity
d) it will take 3 hours for shields to be overwhelmed
e) with GCS being 641 meters long and 463 meters wide, with shields being approx. 700 meters long and 500 meters wide; ellipse representing area of shields struck is piab, or pi * 700m * 500m; or 1 099 577 m^2. Shields are probably around 250 m high. However, surface area of ellipsoid is:
For this:
a is 350 m, b is 250 m and c is 125 m. p ≈ 1.6075. With that we get surface area of 720 103 square meters.
I will asume that star is same as our Sun (which is disproved by screenshot above, but still...). In next to last screenshot a (presumed) coronal mass ejection is seen.
Being 150 000 km from surface, Enterprise was inside corona, meaning that we have take into account both direct radiation and energy carried by corona.
RADIATION
The energy flux at the surface of the Sun is approximately 63 x 10e6 W/m^2 (1).
diameter of sun = 1 391 000 kilometers
So, surface area of sun is 4 x pi x r^2, r = 695 500 000 m; so SAs = 6.0786 x 10e18 m^2.
On the other hand, surface area of imaginary sphere which contains Enterprise's orbit within its shell is 7.46028 x 10e18 m^2. Meaning that energy density at that distance is 0.8148 of that on surface, or 51 332 400 W / m^2, or 51.3224 MW / m^2.
CORONAL ENERGY
In Astrophysical Journal, we have this:
"... the coronal cloud has a thermal energy of 10e51 ergs at 20 kpc..."
1 erg = 10e(-7) J
1 kpc = 3.08568025 × 10e19 meters
So, at 6.1713605 x 10e20 m ( 6.17 x 10e17 km) coronal cloud has energy of 10e44 J. However, diameter of Sun is 1 391 000 km (for radius of 695 500 km). Meaning that surface area of imaginary sphere whose shell is at this altitude is 4.783879 x 10e36 m^2, which gives energy of 20.9 MW per m^2. Still, Enterprise was at 150 000 000 m from surface which is sphere with surface area of 8.98 x 10e12 m^2; smaller than first one by factor of 5.327 x 10e23, giving energy of 1.113 x 10e31 W per m^2.
In end, combined energy absorbed by shields is 1.2 x 10e35 J at 23 %, giving total energy capacity of 5.22787 x 10e35 J... or 125 yottatons. Which is a tad too high for some other examples. For example, even high-end estimate gives TDiC yield of 20 billion megatons per torpedo. Or 20 petatons.
CORONAL EJECTIONS
Coronal mass ejections reach velocities between 20 km/s to 3200 km/s with an average speed of 489 km/s, based on SOHO/LASCO measurements between 1996 and 2003. The average mass is 1.6 x 10e12 kg. The values are only lower limits.
We see Enterprise entering one of these; with minimum values, it imparted 1 x 10e20 J, or 23.9 Gt of energy.
Basically, only way to get reasonable numbers is to assume that writers simply forgot to do the math (which they certainly did) and take estimate at beginning of page.
CONCLUSION: Star Trek shields are 27.68 - 78.5 gigatons versus photon torpedoes and other similar weapons
There are many questions and theories about whole "frequency" thing from Star Trek. To me, it seems most likely that shields are not working continuously even while they're "up", but rather we have shields turning on and off many times in second. Here we have screencap from "Generations" showing E-D shield status display and some diagrams I made to illustrate my theory:
Frequencies are matched. Shields stop weapons fire completely.
Frequencies are matched partially. Weapons partially blocked. "Bleedthrought" present.
Total frequency mismatch. Weapons pass throught shields unhindered.
It also explains whole "regenerative shielding" idea - premise is that computer analyzes weapon that struck shields and matches frequency to stop bleedthrought completely. One more thing it explains is fact that in almost all Star Trek battles ships seem to take damage even before their shields are brought down (infamous "exploding consoles" etc.).
However, damage is often too small for what it should be. One possibility is that there are different shield layers, but we have no proof either way.
