Date: 202109
Local time of day: 0601-1200
Location: ZZZ.Airport, US
Flight conditions: VMC
Light: Daylight
Aircraft: Challenger 350
Operator: Corporate
Operating under: Part 91
Flight plan: IFR
Mission: Passenger
Flight phase: Climb; Descent
Airspace: Class A ZZZ3; Class B ZZZ3; Class E ZZZ3
ATC / advisory: Center ZZZ3; TRACON ZZZ2
Aircraft component: Transponder
Component problem: Malfunctioning
Reporter function: Captain; Pilot Flying
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine; Flight Crew Instrument
Human factors: Situational Awareness; Confusion; Troubleshooting
Second reporter function: First Officer; Pilot Not Flying
Anomaly: Aircraft Equipment Problem Critical; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT
Detector: Automation Air Traffic Control; Person Air Traffic Control; Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Provided Assistance; Flight Crew Executed Go Around / Missed Approach; Flight Crew Overcame Equipment Problem; Flight Crew Requested ATC Assistance / Clarification; Flight Crew Took Evasive Action; General Flight Cancelled / Delayed; General Maintenance Action
Contributing factors: Aircraft; Human Factors; Procedure
Primary problem: Aircraft

Narrative: On Date on the flight departing ZZZ at XA:57z and landing ZZZ1 at XC54z the transponder started to send out incorrect altitude information. We were departing on the ZZZZZ 6 and just past ZZZZZ1; ATC asked us our altitude. We were level at 1;500 feet and ATC was reading 1;900 feet. At this time our radar altimeter was reading 1;500 feet also.  During our climb we checked as we were passing 10;000 feet. ATC was reading 10;600. We were assigned to level off at 16;000 feet and ATC was reading 16;700 feet.  We tried the second transponder and ATC said all they were receiving was XXX go back to number one. We did and continued to climb to 28;000 feet.  We elected to stay at 28;000 feet out of RVSM airspace because of the unreliable transponders. Level at 28;000 feet we checked with ATC and the transponder was reading correctly. We contacted Company maintenance; discussed the problem and we told to continue the flight to ZZZ1. Upon decent into ZZZ1 we were instructed to level at 3;600 feet which we did.  At this time ATC said we were at 3;000 feet and issued an alert and said climb immediately to 5;000 feet. We did and were vectored for another approach. We finished the flight with an ILS XX approach.


Second narrative: Departed ZZZ on ZZZZZ6 climbed to 1;500 feet MSL no problems. ATC reported check our altitude they were showing 1;900 feet. I quickly took a picture because this was incorrect. They asked we change transponders. We did and they reported they were only getting 'XXXX'. We changed back. We checked in multiple times to verify altitudes. Level at 10;000 feet; they said we were 10;600 feet. Level at 16;000 feet; they reported 16;700 feet. After reaching our adjusted cruise altitude of 28;000 feet (respecting RVSM) and cycling transponders off then on; ATC reported they showed level 28;000. No further issues until the ILS approach into ZZZ1. Weather was marginal VFR. When cleared for the approach starting at 3;600; ATC issued an altitude warning; we asked to go around and try again. On the 2nd approach we had no issues; but did check in to verify each altitude.


Synopsis: Flight Crew reported inaccurate transponder contributed to altimeter errors and a descent below minimum altitude at destination airport. This caused a go around and a precautionary landing.