Ming Sun – Silicon achitect, design lead, researcher.

Background

The most import parameter or performance metric for any switching regulator is efficiency, because that is the main purpose switching regulators are designed for. The question is we should ask is that when calculating or measuring switching regulator's efficiency, should average value or the RMS (root mean square) value be used?

For example, we all know for a buck converter, its input current is pulsating. Now, the question is we should ask is that when testing efficiency, should we use RMS value or the average value for the input power calculation?

Most often, when test engineers or designers measure the efficiency during the bench measurement, the average value is being used. So what is the reason behind the scene?

Math derivation

The efficiency can be calculated by its definition:

`eta = {E_{out}}/{E_{"in"}} = {int_{0}^{T_{s}}P_{out}}/{int_{0}^{T_{s}}P_{"in"}}`

(1)

Where, E_out is the output energy, E_in is the input energy, P_out is the output power and P_out is the input power.

To calculate the power, three methods can be used:

`P=V*I=I^2R={V^2}/{R}`

(2)

When to use average value:

`eta = {int_{0}^{T_{s}}P_{out}}/{int_{0}^{T_{s}}P_{"in"}} = {int_{0}^{T_{s}}V_{out}I_{out}}/{int_{0}^{T_{s}}V_{"in"}I_{"in"}}`

(3)

Typicall when measuring the efficiency of the switching regulator, V_in is supplied with a DC source, which means V_in is a constant. I_out is typically supplied from a DC current source, which can be considered as a constant as well. Therefore, Eq (3) can be rewritten as:

`eta = {int_{0}^{T_{s}}V_{out}I_{out}}/{int_{0}^{T_{s}}V_{"in"}I_{"in"}} = {I_{out}int_{0}^{T_{s}}V_{out}}/{V_{"in"}int_{0}^{T_{s}}I_{"in"}} = {V_{out,avg}*I_{out}}/{V_{"in"}*I_{"in,avg"}}`

(4)

Where, V_out,avg is the average value of switching regulator's output voltage and I_{in, avg} is the average value of the inptu current.

Conclusion:
During the bench test or simulation, if we bias V_in with a DC voltage source and I_out with a DC current source, when measuring efficiency, we should use average value of I_in and V_out.

When to use RMS value:

Now, we do not always use a DC current source at the switching regulator's output. Sometimes, we use a resistive load as well. In such case, Eq. (3) can be rewritten as:

`eta = {int_0^{T_{s}}V_{out}^2/Rdt}/{int_0^{T_{s}}V_{"in"}*I_{"in"}dt} = {V_{out,rms}^2/R}/{V_{"in"}*I_{"in,avg"}}`

(5)

Where, the RMS value of V_out is defined as:

`V_{out,rms} = sqrt(1/T_{s}*int_0^{T_{s}}V_{out}^2dt)`

(6)

However,

Since, switching regulator's output ripple is usually very small, the RMS value and average value of Vout can be roughly the same. As a result, Eq (4) and Eq (5) may give roughly the same efficiency results.

Conclusion

From the above discussion and math equations, it is clear that when one of the parameter is set to be constant when using V*I to calculate the efficiency, we should use average value for the other term.
If we use V²/R or I²*R, we should use RMS value for the other term.

Using Average or RMS in Efficiency Calculation

Background

Math derivation

When to use average value:

When to use RMS value:

Conclusion