Buck regulator operation and derivation

A Buck regulator is a switching DC-DC converter whose average output voltage Va is less than the input Vs: Va = k·Vs.

Circuit

        Q (switch)     L    iL
 +Vs ---/ ----+------uuuu----+----+------+
              |              |    |      |
              Dm (↑)         C   LOAD   Va
              |              |    |      |
 −   ---------+--------------+----+------+
  control circuit drives Q (PWM, duty k)

Operation

Mode 1 (Q ON, 0 < t < t1 = kT): Dm is reverse biased. Inductor current rises linearly from I1 to I2; L stores energy and current flows to C and the load.

Vs − Va = L·ΔI/t1   →  t1 = L·ΔI/(Vs − Va)

Mode 2 (Q OFF, t1 < t < T): The inductor current continues through freewheeling diode Dm. Inductor voltage = −Va; current falls from I2 to I1.

Va = L·ΔI/t2   →  t2 = L·ΔI/Va

Waveforms

vQ gate |‾‾‾|____|‾‾‾|____
        0  kT    T
iL   I2 |  /\    /\
     I1 | /  \__/  \__
vC/Va   |~~~~~~~~~~~~~~  small ripple about Va

Derivations

Output voltage: equate the two ΔI expressions:

(Vs − Va)·t1/L = Va·t2/L
(Vs − Va)·kT = Va·(1 − k)T
Va = k·Vs

Ripple current of inductor:

T = t1 + t2 = L·ΔI·Vs / (Va(Vs − Va))
ΔI = Va(Vs − Va)/(f·L·Vs) = Vs·k(1 − k)/(f·L)

Ripple voltage of capacitor: the AC part of iL (triangular) flows into C. Charge added in half a cycle:

ΔQ = (1/2)(T/2)(ΔI/2) = ΔI/(8f)
ΔVc = ΔQ/C = ΔI/(8fC)
ΔVc = Va(Vs − Va)/(8·L·C·f²·Vs)

Critical values for continuous inductor current and voltage:

Lc = (1 − k)R/(2f),   Cc = (1 − k)/(16·L·f²)

Input current Is = k·Ia (ideal). Advantages: high efficiency (about 90 %), only one switch. Disadvantages: input current is pulsating, and no short-circuit protection is inherent.