LEDs are quickly replacing other lighting technologies due to their relative optoelectronic efficiency $($i$.$e$.$ the lumens of visible light per watt of electrical power$).$ Consider that a traditional $60$W incandescent bulb has a typical light output of $840lm$ for $60$ W $($average AC power$).$

A high power LED with comparable output is the Luxeon M$.$ Consider the LXR$8-$SW$30$ white LED with a typical light output of $850$ lm $.$ It has a rated forward voltage of $11\mathrm{.}2$ V and a rated forward current of $700$ mA $.$

a$.$ Calculate the lumens per watt metric for the high power LED and contrast it with that of a $60$ W incandescent bulb.

b$.$ Consider the circuit shown below for powering a set of LXR$8-$SW$30$ white LEDs from a $12$ V battery. Assume the battery is ideal. Assume all diode rated forward voltages are $11\mathrm{.}2$ V at $700$ mA $.$

i$.$ Calculate the value of the resistor $R$ required to limit current through the LEDs to $700$ mA $.$

ii$.$ Calculate the power dissipated by each resistor $R.$

c$.$ A classmate proposes an alternative circuit shown below.

i$.$ Calculate the value of resistor $R_A$ to limit current through the LEDs to $700$ mA $.$

ii$.$ Calculate the power dissipated by $R_A.$

iii. Discuss tradeoffs between the two circuit designs.