Democrats suffered a second beating yesterday on public financing of Senate elections as the Senate failed by 13 votes to cut off the Republican filibuster.

With 60 votes needed for debate-limiting cloture, advocates mustered only a 47-to-46 majority - worse than their showing on Friday when their majority was 79 to 45.

However, Senate Majority Leader Robert C. Byrd (D-W. Va.) said he expects cloture forces to make a sizable gain today, when a third vote is taken.

Minority Leader Howard H. Baker Jr. (R-Tenn.) believes government financing will help Democrats win more elections than Republicans, and has invoked the strongest bonds of party unity to hold republicans together in the fillibuster. Only two voted for cloture yesterday - charles (Mac) Mathias Jr. (Md.) and Edward W. Brooke (Mass). Once again 10 Southern Democrats voted against cloture. Many fear they will face well-funded GOP opponents for the first time if public financing wins.

Republican John Gardner, founding chairman of Common Cause, said Baker's "claims that the campaigns reform bills discriminates against Republicans are ridiculous." Before the cloture vote, a Dick Clark (D-Iowa) amendment to make sure Republicans get major-party financing in the South was adopted.