How to generate a correct JWT without using Firebase - php

Question

I am trying to generate a JWT for the DocuSign Embed API and I can't seem to figure out the private/public key part.
When I generate the JWT, the signature part is very short. Pasting it in jwt.io, it turns out both the header and payload are correct, but the signature part is wrong (and way too short when I compare it to what I get when I copy paste my public and rsa private keys.
Here is the relevant code:
$args = [
'envelope_id' => $_GET['envelope_id'],
'ds_return_url' => 'https://www.returnurl.be',
'starting_view' => 'envelope',
'base_path' => 'https://baseurl.be'
];
$args['integration_key'] = 'abc123';
if(isset($_GET['user_id'])) {
$args['user_id'] = $_GET['user_id'];
}
//generate jwt:
$headers = array('alg'=>'RS256','typ'=>'JWT');
$payload = array(
'iss'=>$args['integration_key'],
'sub'=>$args['user_id'],
'aud'=>'account-d.docusign.com', //prod: account.docusign.com
'iat'=>time(),
'exp'=>(time() + 6000),
'scope'=>'signature impersonation'
);
$secret = '-----BEGIN RSA PRIVATE KEY-----
MIIEowIBAAKCAQEAj...
-----END RSA PRIVATE KEY-----';
$jwt = generate_jwt($headers, $payload, $secret);
echo $jwt;
function generate_jwt($headers, $payload, $secret = 'secret') {
$headers_encoded = base64url_encode(json_encode($headers));
$payload_encoded = base64url_encode(json_encode($payload));
$signature = hash_hmac('SHA256', "$headers_encoded.$payload_encoded", $secret, true);
$signature_encoded = base64url_encode($signature);
$jwt = "$headers_encoded.$payload_encoded.$signature_encoded";
return $jwt;
}
function base64url_encode($str) {
return rtrim(strtr(base64_encode($str), '+/', '-_'), '=');
}