Linear Power Supply Transformer
A typical transformer datasheet will specify a full-load voltage, which is the RMS secondary voltage measured when the secondary is delivering its rated RMS current. Is it correct to assume that this VA rating only applies when the secondary voltage and current are both sinusoidal?


Yes only sinewaves.

> In other words, how does the shape of the current waveform affect the transformer V-I ratings?

For pulse loads derate VA rating at least 30% due to increased RMS current (heating) per average output current which gives more rise in temperature.  Often 10% duty cycle pulses for 10% Voltage ripple uses RC=T=16x/f  (f=  pulse rate , line rate is doubled after full bridge)

main Reason:  the copper losses will be Pcu=Ipk^2*DCR, for a copper DC resistance DCR.  Thus a large cap loaded DC bridge will increase the losses and rise in temp. more than a  sine current load for same RMS output power.


for the advanced reader


2nd reason THe output caps charge just before peak voltage and stop at peak voltage. Thus with  10% ripple that  means the current is an amplified sine pulse that only starts near 80 deg and stops at 90 deg at a peak level of 10x the decay current into the load, so it has a 10% pulse width with very fast rise time and 10% sine peak shape,  with more than 10 harmonics of 2x the line frequency (even and odd) which for laminated steel with high mu, can result in eddy current losses. Load balance affects VAR utilization on tapped transformers.

3rd reason* if the output is center-tapped and 1 diodes used on each leg, AND only loaded on one leg, say V+ and not V- then this results in offset DC current in the shared core and reduces margin to saturation with DC current flowing thru the core secondary (  from half-wave loading) THus primary saturation can occur sooner if one assumed incorrectly they could get full VA on half wave rectified V+. The VA output must be shared with load balance on each winding to utilize the 
power.  

There are some excitation losses from the primary inductance creating a reactive current about 10% of rated current in order to activate the mutual  coupling.  


  How would one go about determining the proper current rating for a linear power supply transformer?


Use VA rating VA/V*70%=Imax dc max and derate further for lower than 50'C winding temp rise even if sine wave.


  My immediate thought would be to approximate the current through the filter capacitor as a periodic sawtooth pulse and find the RMS value of the pulse waveform, then add maybe 50% to account for the fact that the transformer supplies current to the load while the diode is conducting and also to provide a decent safety factor. Is there a better way of doing this?


I once used 10% ripple = 16/f  and derate VA by 40%. Now I use SMPS.
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  If you draw, say, 5 A of peak current on the transformer secondary, then for a 10:1 transformer, you'd be drawing 50 A of peak current on the primary. How is standard grid power able to handle such high repetitive peak current spikes without blowing a breaker?


Breakers near trip Current rating respond in a minute or few.  RC low pass filter T=  line f/8 thus ~1/10 second so the pulse is too fast to thermally affect breaker or fuse.

Normally Temp rise is the limiting factor with several causes that causes. We expect Voltage output to rise 40% with no load due to sine pk/avg and another 10% due to secondary DCR losses so V_no load will be ~ 50% more than avg.DC at max permissable load. THis makes it very inefficient compared to HF SMPS due to the much lower duty cycle charge periods and long discharge periods, so much greater storage energy and size of XFMR, Cap are necessary. 


  THere are books by Dr./Prof Keith Billings with easy nomographs that make linear supply design and SMPS design  as easy as a recipe for making  Won Ton soup.  I as a client at Burroughs Inc worked, with him when he was Eng Mgr of Hammond Power Supplies in Canada ca. 80's. He is now in Colorado I believe.