Switching Power Supplies A - Z

Chapter 7

Returning to the RC network, we can ask — how did we actually arrive at the transfer

function stated above? For that, we first use Kirchhoff’s voltage law to generate the

following differential equation:

v

i

= v

res

(t ) + v

cap

(t ) = i(t)R +

q(t)

C

where i(t) is the charging current, q(t) is the charge on the capacitor, v

res

(t) is the voltage

across the resistor, and v

cap

(t) is the voltage across the capacitor (i.e. v

o

(t), the output).

Further, since charge is related to current by dq(t)/dt = i(t), we can write the preceding

equation as

v

i

= R ×

dq(t)

dt

+

q(t)

C

or

dq(t)

dt

+

1

RC

q(t) =

v

i

R

At this point we actually “cheat” a little. Knowing the properties of ...