AN542

Rev. 0.1

5

6. Closed Loop Transfer Function

The closed loop transfer function is defined as the ratio of the controlled variable to the input variable. The

controlled variables are the speaker terminals, and the input variable is the MP3 player input, the CD input, or

some other input source connected to the amplifier. Therefore, the equation for the closed-loop transfer function is

given by Equation 7.

Equation 7. Closed Loop Transfer Function

The closed loop gain describes how the output responds over the audio bandwidth to the input regulation signal. It

is understood that the output should have a specific closed loop gain with respect to the input regulation signal, and

that gain should be as flat as possible over the audio bandwidth. The inductor between the controlled variable and

the speaker terminals plays a crucial role in the performance of the amplifier as previously discussed. The closed

loop gain of the Silicon Labs Class D reference design is implemented such that approximately 1 Vpp input will

yield full output power into an 8

 load.

7. Open Loop Transfer Function

The open-loop transfer function is obtained by breaking the loop at some arbitrary point and traversing the entire

loop back to the same point. When H(s) = 1, the open loop and the forward transfer functions are identical.

Therefore, the open loop transfer function is given by Equation 8.

Equation 8.

The open loop transfer function determines whether the loop is stable, as well as determining what the overall open

loop gain of the amplifier will be over the audio bandwidth. The higher the open loop gain, the lower the error signal

and, therefore, the more easily the control loop can keep the output following the input command. Some early

Class D amplifier designs used an integrator for the error amplifier. This produced high gain at low frequencies but

low gain at high frequencies due to the pole produced by the integrator. This caused the THD to increase

dramatically above 5 kHz, destroying the high-frequency response of the amplifier. A better solution is to keep the

open loop gain constant and as high as possible throughout the audio bandwidth. This should yield a constant THD

response, and, indeed, it does, as will be shown in the performance curves in "13. Performance" on page 8.

Care should be taken in designing the open loop response of the amplifier. The three key elements are the

bandwidth, phase margin, and gain margin. In designing a Class D amplifier, the target is to have 45° of phase

margin with a bandwidth of approximately half the switching frequency. The control loop cannot compensate for the

LC filter response since the filter is outside of the loop. The entire LC filter is designed as a Bessel function with a

load resistance of 6

. Therefore, the filter is slightly underdamped at 8  and slightly overdamped at 4 . This can

be seen by placing a 100 mVpp square wave into the input and looking at the output response with an 8

 load

and a 4

 load.

Closed_Loop_Transfer_Function

G(s)

1

G(s)H(s)

+

--------------------------------

=

Open_Loop_Transfer_Function

G(s)H(s)

=