The academic way is to write differential equations and solve for i(t) , which defines both the startup transient and steady state.
Steady state is a lot easier
i think in pictures and figure out the formulas
here's an approach to get you started
examine the circuit
can we assume ideal diodes too ?
What do we know? I wrote it in brown.
Ideal diodes allow no forward voltage drop so we get at junction of the diodes perfect half wave rectified 20Vac and i assumed RMS.
But since it's now DC (lumpy but unidirectional) we should switch to average
average of a halfwave rectified sinewave is 0.318V
peak, see http://my.ece.ucsb.edu/York/Bobsclass/2B/Extras/Half-Wave%20Rectifier.pdf
and V
peak is V
RMS X √2
20 X √2 V 0.318 = 8.99 (to slide rule accuracy of 3 figures)
and that'll be the average voltage at junction of diodes and L
d.
Now an inductor cannot oppose DC beyond a few time constants so the
average voltage at other side of L
d must also be 8.99 volts
meaning average current through L
d and R
d must be by Ohm's law 8.99 /0.5 = 18.0 amps
During positive half cycles the current increases to a value above average
during negative half cycles it decreases to a value below average
with a time constant L/r = ##\frac {0.04} {0.5} ## = 0.08 sec
Between positive half cycles , while current is decreasing, , i = i
max e
t/0.08
in 30 miliseconds , just a little longer than 1/2 cycle, it'll decrease to I
min = ~68% of whatever I
max was, 32% decrease
so we can estimate that current swings between I
max and I
min , roughly I
avg ±16% , as sketched above.
Of course Ohm's Law dictates voltage across R
d must do exactly the same.
..........
Why the circuit works:
During positive half cycles the inductor absorbs energy from the supply and transformer
during negative half cycles the inductor delivers some of its absorbed energy to R
d
Of course at startup, current will start from zero and increase during every positive half cycle, after several cycles approaching this steady state sawtooth looking waveform , ±16% about average .
...........
Now you have an intuitive :"Sanity Check" method to compare against whatever Pspice (or your differential equation) tells you.
That's how i always had to flush out my algebra mistakes, and later on when QBasic and computers replaced my slide rule, my programming errors .
.......................
Your homework circuit is the basis of many kinds of regulators,
both simple ones like we just analyzed and
complex ones where they use switching transistors instead of diodes, so called "Switch Mode Power Supplies" ...
So it's important that you get accustomed to working it in your head .
Doubtless your math is fifty years more fresh than mine so i wouldn't presume to lecture you there.
I offer this "simple thinking" approach to complement your math skill. Maybe it'll help you spot an impossible result before it propagates through an entire derivation.
Hope it helps.
old jim
