Calculating the switching loss IRF44n
Driving with TTL is a bad idea. At 4D drive the IRFZ44 is only guaranteeing to pass 250 micro amps. You need to follow the guidelines in the spec: -



The spec says it has a test to check rise times and fall times - they use a 10V pulse with an output impedance of 12 ohms.

You cannot expect to get anywhere near this performance from TTL at 4 or 5V. The gate input capacitance is 1.5nF and this needs something like a 1 or 2A drive (off the top of my head) to get the device to switch on and off at the rate you are likely wanting.

EDIT to include drive current into the gate.

It's easiest to start off with Q = CV then differentiating we get \$\dfrac{dQ}{dT} = C\dfrac{dV}{dT}\$ where

\$\dfrac{dQ}{dt}\$ equals charging current into the gate capacitor of 1.5nF.

The voltage on the gate needs to change about 10V in 20 ns hence \$\dfrac{dV}{dt}\$ = 500,000,000.

Therefore charging current (to be supplied by gate drive) is 1.5 \$\times 10^{-9} \times 500,000,000 = 0.75A\$. This means your driver ought to be able to deliver 1 or 2A as previously mentioned.