PHP JWT Invalid signature

Given this code:

function base64url_encode($data) {
    return rtrim(strtr(base64_encode($data), '+/', '-_'), '=');
}

$key = 'secret';

//setting the header: 'alg' => 'HS256' indicates that this token is signed using HMAC-SHA256
$header = array(
    'alg' => 'HS256',
    'typ' => 'JWT'
);

// Returns the JSON representation of the header
$header = json_encode($header); 

//encodes the $header with base64.  
$header = base64url_encode($header);

$payload = array("a" => "b");

$payload = json_encode($payload);       
$payload = base64url_encode($payload);

$signature = hash_hmac('SHA256','$header.$payload', $key, true);
$signature = base64url_encode($signature); 

echo "$header.$payload.$signature";

Returns the following JWT:

eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJhIjoiYiJ9.rhCKIvkwiuNcchxDZnGak8XT1q8lmLhnm8aIxzUioWg

But the signature is not verified at https://jwt.io/ The payload is decrypted well though... What may be the problem ?