# Transfer Function

## DC/DC Converters: Sharing of Transfer Functions among Control Systems

### Slope Transfer Functions: Voltage-Mode Transfer Functions

2017/01/12

From here, we consider slope transfer functions. Below is reproduced Fig. 2 from [Basic Approach to Transfer Functions and Their Derivation], but from this section, we will explain the part G_{2(S)}, consisting of a PWM component and driver, that follows the error amplifier G_{1(S)} of the previous section. In representative PWM control, there are a voltage mode and a current mode. We begin with transfer functions for the voltage mode.

**Voltage-mode transfer functions**

Voltage-mode control in a DC/DC converter is the most basic system. Only the output voltage is fed back through a feedback loop. By using the voltage difference obtained by comparison with a reference voltage using an error amplifier, and further comparing with a slope wave (triangular wave), the pulse width of a PWM signal is determined and the output voltage is controlled.

Fig. 1 shows a block in which the output V_{c} from the error amplifier and the slope voltage are input to the PWM generator to generate the PWM output.

Here, by comparing a fixed slope wave with V_{c}, the transfer function of the block F_{m}, which is PWM controlling the duty D = T_{on}/T, is derived.

Fig. 2 shows the input waveform and the PWM output waveform. The slope wave rises with a constant slope by V_{ramp} over one period. Through comparison with V_{c}, the on and off times of the driver output V_{Dout}, that is, the duty cycle, is determined. The control conditions are as follows.

When V_{c} ＞ V_{slope} , V_{Dout} = ON

When V_{c} ＜ V_{slope} , V_{Dout} = OFF

Moreover, in Fig. 2:

When V_{slope} = V_{c} = ０ , D = 0

When V_{slope} = V_{c} = V_{ramp} , D = 1

Equation 3-1 is obtained from similarity of the triangle formed from V_{slope} and V_{c}, and by modifying this equation using D = T_{on}/T, we obtain equation 3-2.

Moreover, from perturbation of equation 3-2 （ D → D + ∆D , V_{c} → V_{c} + ∆_{VC}） equation 3-3 is obtained.

## Key Points:

・The transfer function of the block that generates PWM output is derived from the error amplifier output and a slope wave.

・In this section, we first consider transfer functions for voltage-mode control.