1. no service codes 2. engine starts perfect 3. smooth power at all speeds 4. when idling, in park or drive the engine has a cyclical engine surge. The surge rpm is equal to fast idle. The repeating cycle is 10 seconds normal idle, and 2 seconds surge to fast idle then back to normal idle. To confirm this cycle I watched for 10 consecutive cycles. You could set your watch by it, the cycles were so consistent. Please advise.

More information please. I am assuming this is a Toyota. What year is this vehicle? What model is this vehicle? I am also assuming this is the 1.8 litre, four cylinder engine. The most common firing order on a 4 cylinder engine is 1, 3, 4, 2.
Does this engine idle smoothly when in Neutral and in Park? When in Park/Neutral gently press and hold the gas pedal at a slightly higher engine speed. You don't need to scream the engine just push it up to its normal working speed. Does it still run smoothly? If the engine idles and runs smoothly it suggests that it isn't the firing order causing the issue.
Do you have this problem when you select Drive or when you are actually moving? Your best option here is to take the vehicle to a good mechanic that way they will be able to see, hear and feel the vibration. This will help work out what the problem is.

Ecu temperature sensor. Or idle control valve. I would start with the sensor since is only a few dollars

there is a sensor that will increase engine revs when turning at slow speed , i expect this is the issue, to check get the OBD codes read to confirm

there are a few things that could cause this , have you had the engine serviced regulary ?get the OBD codes read as may be the EGR valve or the Idle Control , clogged injectors , clogged exhaust , i'd try a fuel addative that covers the catalytic convertor as well as the engine parts , take for a longish run say 100 kilometers at a highish speed to get the engine to a good hot operating temperature

HI
first I would like to tell you, do this procedure only if the 5th digit of your VIN is C or K.

Description

The engine control module (ECM) learns the idle position of the throttle plate to ensure the correct idle. Anytime the throttle body is cleaned or replaced, the ECM must learn the new throttle position. The idle may be unstable or a DTC may set if the throttle position is not learned.

Conditions for Running the Throttle Learn Procedure

• DTCs P0101, P0102, P0103, P0107, P0108, P0111, P0112, P0113, P0506, and P0507 are not set.

• The engine speed is between 450-4,000 RPM.

• The manifold absolute pressure (MAP) is greater than 5 kPa.

• The mass air flow (MAF) is greater than 2 g/s.

• The ignition 1 voltage is greater than 10 volts.

Throttle Learn

Important: Do NOT perform this procedure if DTCs are set. Refer to Diagnostic Trouble Code

(DTC) List - Vehicle.

1. Start and idle the engine in PARK for 3 minutes.

2. With a scan tool, monitor desired and actual RPM.

3. The ECM will start to learn the new idle cells and Desired RPM should start to decrease.

4. Ignition OFF for 60 seconds.

5. Start and idle the engine in PARK for 3 minutes.

6. After the 3 minute run time the engine should be idling normal.

Important: During the drive cycle the check engine light may come on with idle speed DTCs. If idle speed codes are set, clear codes so the ECM can continue to learn.

* If the engine idle speed has not been learned the vehicle will need to be driven at

speeds above 70 km/h (44 mph) with several decelerations and extended idles.

7. After the drive cycle, the engine should be idling normally.

* If the engine idle speed has not been learned, turn OFF the ignition for 60 seconds

and repeat step 6.
8. Once the engine speed has returned to normal, clear DTCs.
Good Luck

Since you changed the sensor we will assume it is working. When you disconnect it you put the computer in limp mode which stops it from using sensor values. If code 300 is the only code we will assume the coolant temp sensor and others are ok.
I would check fuel pressure with a gauge next.

check out the idle air control valve and the passages

you can't clean a egr , been tried by many, works for temporary solution only, best place to start

put a throttle position sensor on the throttle boby this should fix the problem!

Sounds like a possible vaccume hose leak?