State-Space Averaging

by Brad Suppanz


State-Space Averaging (SSA) is one of the essential techniques for analyzing switched mode power conversion circuits.  Whether we are using a computer to run simulations or doing hand calculations, SSA allows us to extend our standard DC and AC circuit analysis techniques to switching circuits.  Transient analyses can also be run much faster by using SSA models.

The trade-offs are, by applying SSA, we are ignoring the cycle-by-cycle switching and looking at the average characteristics of the circuit at frequencies below the Nyquist frequency (1/2 the switching frequency).  We then loose the ability to see switching ripple or actual switching waveforms, but we gain the ability to quickly determine many things about a circuit such as:

Finding the state-space averaged equations for a circuit is rather straightforward.  These are the required steps:
  1. Draw the circuit in each state (e.g. state 1 = transistor ON, state 2 = diode ON).
  2. Write the desired nodal, mesh, or element equations for each state.
  3. Determine the ratio of time spent in each state (per cycle).
  4. Multiply each state equation by it's time ratio and sum to form a "weighted average" of the state equations.

Example

Find the SSA equation for the inductor's current-voltage relationship in a continuous mode boost converter as shown below:
 

Step 1.  Draw the circuit in each state:
 
 
State 1:  Switch D is ON and D' is OFF.
 
State 2:  Switch D' is ON and D is OFF.
Step 2.  Write the inductor equation for each state:
 
In State 1:  Vin = L*dIL/dt
In State 2:  (Vin - Vout) = L*dIL/dt
Step 3.  Determine the ratio of time spent in each state (per cycle).
 
In State 1:  The ratio in this state is simply D by definition.  Where, D = Ton/Tsw.
In State 2:  The ratio in this state is simply D' by definition.  Where, D' = ( 1 - D ).
Step 4.  Multiply each state equation by it's time ratio and sum to form a "weighted average" of the state equations.

Combining state equations:

where the first term on each side of the equation corresponds to switch D being ON, and the second term on each side of the equation corresponds to switch D' being ON.

Simplifying, we get:

Now that we have the SSA equation for the inductor current-voltage relationship, what can we do with it?  Well, we can use it to study the converter operation as we choose.  One thing we can easily do is find the steady-state input to output transfer function of the boost converter.  In steady-state, the inductor current does not change, so we get: This is the familiar input-output relationship as seen from the SSA perspective.

The real power of the State-Space Averaging technique goes far beyond this simple example.  Once the basics of SSA are understood, they can be applied to any switched mode power supply, or used to derive generic switch element models.  The next important technique that is often used is that of finding the small-signal equations by "perturbing" the SSA equations.

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Written 9/2/97
Last updated 7/20/04