var params = {
    "method": "get_session_key",
    "params": {
        "username" : "admin",
        "password" : "test"
    },
    id: 1,
    jsonrpc: "2.0"
};
 
 
$.ajax({
    url: "http://LimeSurvey/index.php/admin/remotecontrol",
    type: "POST",
    contentType: "application/json",
    data: JSON.stringify(params),
    error: function(data) {
        console.log('error');
        console.log(data);
    },
    success: function(data) {
        console.log('success');
        console.log(data);
    }
});

jQuery.ajax ({
url: "http:/server/index.php/admin/remotecontrol",
type: "POST", 
dataType: 'json',
data: JSON.stringify({method:"get_session_key",params:{username:"admin", password:"test"},id:1}),
contentType: "application/json",
complete: function (xhr, status) { 
                 data = $.parseJSON(xhr.responseText);  
                 $.each(data, function(index, val) {
    console.log(index + '=>' +  val);
});
} 
});

Request:
  HTTP headers:
    content-type=application/json
    connection=Keep-Alive
    host=mylimesurveyhost.com
    content-length=65
    user-agent=Apache-HttpClient/4.2.2 (java 1.5)
 
  Post body:
    {"method":"get_session_key","params":{"username":"admin","password":"mypassword"},"id":1}

    HTTP headers:
	Accept	*/*
	Accept-Encoding	gzip, deflate
	Accept-Language	en-US,en;q=0.5
	Content-Length	83
	Content-Type	application/json; charset=UTF-8
	Host	10.0.0.2
	Referer	http://10.0.0.2/
	User-Agent	Mozilla/5.0 (Macintosh; Intel Mac OS X 10.8; rv:19.0) Gecko/20100101 Firefox/19.0
	X-Requested-With	XMLHttpRequest
 
    Post body:
	{"method":"get_session_key","params":{"username":"admin","password":"test"},"id":1}

<?php
 
require_once 'php/vendor/jsonRPCClient.php';
 
define( 'LS_BASEURL', 'http://10.0.0.2');  // adjust this one to your actual LimeSurvey URL
define( 'LS_USER', 'admin' );
define( 'LS_PASSWORD', 'test' );
 
// instanciate a new client
$myJSONRPCClient = new jsonRPCClient( LS_BASEURL.'/admin/remotecontrol' );
 
// receive session key
$sessionKey= $myJSONRPCClient->get_session_key( LS_USER, LS_PASSWORD );
echo $sessionKey;
 
// release the session key
$myJSONRPCClient->release_session_key( $sessionKey );
?>