Unnecessary pull down resistors on BJT and FET transistors?
The reasons become clear when you are considering not only the ideal behaviour of the transistors but also their parasitic elements.
The pull-down resistor at a npn-type BJT's base helps keeping the base "low" whenever the driving element for the base resistor should be unconnected or in a tristate mode. Without this resistor, charge entering the base via the capacitance between collector and base ("Miller capacitance") could remain there and turn on the transistor.
There are two common reasons for a series gate resistor in a MOSFET circuit. One is that the resistor limits the driving current and allows for some control of the gate charge current (think of the gate as a capacitor that needs to be dis/charged in order to turn the MOSFET off or on). With a carefully chosen resistor, you are able to get some control over the turn-on or turn-off transition times of the MOSFET. Sometimes, you even use a resistor paralleled by a diode and another resistor to have different charge and discharge currents, i.e. a chance to influence the turn-on-time in a different way than the turn-off time. The second reason for a base resistor is that the trace inductances around the MOSFET form a resonant LC tank with the MOSFET's parasitic capacitances. When all you want is a clean transition of the gate voltage (rectangular waveform), you may get a lot of ringing in reality. The ringing may be so severe that the MOSFET turns on and off a couple of times before settling and finally obeys to what the driver requests. A resistor inside of the LC resonant circuit around the gate driver is able to damp this resonance and the path between driver and gate is the easiest spot to put the resistor. For small-signal circuits, these resistors may not be necessary, but when driving power MOSFETs, you absolutely need them.
A series resistor in the gate line of a MOSFET will protect the driver (microcontroller) from ringing effects caused by parasitic inductances.
The optimum value for Rg is very application dependant. You want the MOSFET to switch as quickly as possible to minimise switching losses, but not so fast that parasitic inductances and capacitances associated with the pcb layout and any wiring to a load, will cause high di/dt voltage spiking or ringing.If you find that an optimised value of Rg controls switch on OK but slows the turn off too much, then a fix is to place a diode across Rg with its cathode towards the gate drive circuit. This will bypass Rg during turn off thereby speeding up the turn off. Placing a resistor in series with the diode will enable you to control turn off time independantly of turn on. Further reading (for all aspects of mosfet switching).
For switching small loads (like 100mA), or when a real MOSFET driver chip is used, the gate resistor is probably not needed.
(Note: these links were on the first G results page for "mosfet gate resistor")
The series resistor to the MOSFET gate is sometimes required to reduce a current peak when switching due to the gate's capacitance. Logic circuit, esp. microcontrollers allow only for a very low capacitive load. It can also be used to reduce slew-rate (the switching speed).
A pull-down on the gate is used to prevent the gate from floating if the controlling I/O is configured as input. In this case the resistor's value can be chosen pretty large (1~10M\$\Omega\$).
The base resistor on the BJT is often combined with a pull-up, and this combination is used to set a stable quiescent point. [our teacher in college, not very good at English and apparently only having seen the word in print pronounced it as "keskent". It took us a while to understand what he meant :-)]