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The HART protocol is acronym for Highway Addressable Remote Transducer. Although it is a new concept it uses the conventional modeof 4-20mA for its communication from field to PLC or DCS.
This protocol was developed by rosemount Inc in early nineties. The HART communication protocol is based on the Bell 202 telephone communication standard and operates using the frequency shift keying (FSK) principle. Since I dont know much about FSK, we can understand it from here. 
The digital signal is made up of two frequencies— 1,200 Hz and 2,200 Hz representing bits 1 and 0, respectively. Sine waves of these two frequencies are superimposed on the direct current (dc) analog signal cables to provide simultaneous analog and digital communications. The average value of the FSK signal is always zero so the 4–20 mA analog signal is not affected. The digital communication signal has a response time of approximately few data updates per second without interrupting the analog signal. A minimum loop impedance of 230 W is required for communication.
HART communication can happen in two ways. HART is a master-slave communication protocol, which means that during normal operation, each slave (field device) communication is initiated by a
master communication device. Two masters can connect to each HART loop. The primary master is generally a distributed control system (DCS), programmable logic controller (PLC), or a personal computer (PC). The secondary master can be a handheld terminal like a Handheld Communicator or another PC. Slave devices
include transmitters, actuators, and controllers that respond to commands from the primary or secondary master.
Some HART devices support the optional burst communication mode. Burst mode enables faster communication (3–4 data updates per second). In burst mode, the master instructs the slave device to continuously broadcast a standard HART reply message (e.g., the value of the process variable). The master receives the message at the higher rate until it instructs the slave to stop bursting.

Leaving all the technicalities behind, comparing with conventional 4-20mA the HART provides a whole bunch of diagnostic and configuration features to the control Engineer to setup and monitor the process effectively. One of the example that i found online is:
suppose we are monitoring a small process based on a temperature transmitter having thermocouple sensor. Then HART trnamsitter can tell me at any point of time about the
sensor: Has it failed or cable break from the sensor into the transmitter,
transmitter: is the value given by sensor in range or is it out of service,
cabling: is the cable from the transmitter to the DCS proper.

A nice article in controlglobal which dispells the myth of HART is seen here.

MYTH #4: HART offers little value for Plant Control Systems and Enterprise Level Networks.

FACT: Accessiong the data in HART devices on a "full-time" routine basis increases performance, integrity, and reliability of plant control and enterprise level systems. Retrieving digital data directly from the device can avoid potential filtering that may occur in acquiring this information through intermediate systems.

Traditionally, a HART enabled instrument’s analog signal transmitted the primary variable and additional variables and diagnostics were confined to the device.  HART technology was used only for setup, calibration or as a maintenance tool.

The practice of intermittent communications with a device for startup and troubleshooting has provided exceptional value for users of the technology.  However, using the technology in this manner limits the capabilities of a device within the real scope of a process plants operation.  The continuous monitoring of the HART digital channel can open the door to improved performance.

Continuously monitoring a HART enabled instrument’s process variable via the digital channel can lessen the impact on the precision (not the accuracy) of the measurement by reducing the number of conversions the data must go through.  Although slower than the analog channel, the digital channel provides high-resolution measurement values for up to four measurement variables including information on engineering units, data quality and the status of the device.

With a fully enabled HART system, devices can now provide the ability to take the intermittent diagnostic methodology to a new level.  Whether a complete control system or a single loop interface, the integration of a continuously monitored HART digital channel will reduce downtime by using the onboard diagnostics and data provided in every HART- enabled device.

Today, there are many ways to integrate the valuable information provided by a HART-enabled device. The big question is “How do you get started with your existing system?”

To “See What You Can Do” with HART there is no reason to start big, it’s just important to start.  First, find a control loop that needs to be monitored, add a single loop interface (an example is the HIM by Moore Industries), and begin monitoring the information provided by your HART-enabled device.
The single loop interface allows a user to access all of the additional variables provided by a HART-enabled device and transmit the values via one of the interfaces analog outputs.  In addition to the analog outputs, single loop interfaces have the ability to provide contact closure alarm outputs.  The contact closure alarms can be programmed to change status on either a process alarm or a change in the device’s status. 

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