Boost converter transfer function with non-ideal components

Discussion in 'Homework Help' started by J_Rod, Mar 1, 2016.

  1. J_Rod

    Thread Starter Member

    Nov 4, 2014
    109
    6
    Hi, I am stuck trying the derivation for a boost converter transfer function in continuous conduction, open-loop operation mode to find Vout/Vin with non-ideal components, that is, a Rsw, the MOSFET switch on-state resistance, ESR, an equivalent series resistance with the capacitor, RD, the on-state resistance of the diode, and RL, the inductor's resistance.

    When the switch is ON:
     L \frac{di_L}{dt} = V_{in} -i_L (R_L +R_{DS,ON})

     C \frac{dV_C}{dt} = \frac{-V_{out}}{R + ESR}

     C \frac{dV_C}{dt} = \frac{V_C - V_{out}}{ESR}

    When the switch is off:
     L \frac{di_L}{dt} = V_{in} -V_{out} -i_L(R_L + R_D)

     C \frac{dV_C}{dt} = i_L -\frac{V_{out}}{R +ESR}

     V_C = V_{out} -i_CESR

    Then
     \Delta i_{Lon} + \Delta i_{Loff} = 0

    But to get  \Delta i_L terms I would have to integrate the equations with the inductor voltage, which both contain a term with  i_L . How can that be done? Thanks.
     
  2. wayneh

    Expert

    Sep 9, 2010
    12,157
    3,064
    Had diffy Q yet? I believe that's a first order differential equation. I'm pretty rusty on the terminology. It's been a long time but I believe almost any text on the subject will cover the solution.
     
  3. J_Rod

    Thread Starter Member

    Nov 4, 2014
    109
    6
    That's how to solve for  i_L , but I want to find the expression for  \Delta i_{Lon} to add to the expression for  \Delta i_{Loff} to sum to 0, and then rearrange to find  \frac {V_{out}}{V_{in}} . Would it be incorrect to set  i_L = I_L , the average value, on the right side of the equations, if I assume a small ripple?
     
  4. wayneh

    Expert

    Sep 9, 2010
    12,157
    3,064
    Well if you have i as a function of time, can't you integrate to get ∆i ? (I'm just hoping that asking questions helps you see a solution. I haven't worked this out.)
     
  5. anhnha

    Active Member

    Apr 19, 2012
    774
    48
    There are some principles you should use here to solve for the result:
    1. Inductor volt-second balance
    2. Capacitor charge balance
    3. Small ripple approximation
    Refer to this for detail.
    With small ripple approximation, you can assume that Vout and iL are constant (and let's call it IL).
    With #1 and #2, you have set of two equations and you can now solve for Vout and IL.
    With small approximation, iL in the right hand side can be considered as a constant.
    However, if you want an exact calculation, then I think it is much easier to use Laplace transform.
     
Loading...