2.1 RC Filter Example

For this example, it is required to design a simple RC low-pass filter to obtain an analog output from a pulse-width modulated signal with a bandwidth of 4 kHz.

Step 1: Select the low-pass filter’s resistor and capacitor values.

Equation 2-1 shows how to calculate the values for R and C based on the cut-off frequency, ƒ_C. In this example, the resistor values were calculated based on fixed capacitor values, as shown in Table 2-1.

Equation 2-1. RC Time Constant

R C = \frac{1}{2 π \times f_{C}}

w h e r e :

R : Re s i s \tan c e

C : Capactiance

f_{C} : c u t - o f f f r e q u e n c y

Table 2-1. Calculated Resistor Values
Capacitor Value	Calculated Resistor Value
1 pF	40 MΩ
0.01 µF	4 kΩ
0.022 µF	1.8 kΩ

Step 2: Calculate attenuation at the PWM frequency.

Equation 2-2 shows the attenuation in decibels (dB) based on the RC values and the PWM frequency.

Equation 2-2. Attenuation in Decibels (dB)

A t t e n u a t i o n (d B) @ f_{P W M} = - 10 \log [1 + {(2 π \times f_{P W M} \times R C)}^{2}]

Table 2-2. Attenuation at the PWM Frequency (F_PWM)
F_PWM	R Value	C Value	Attenuation (dB) @ F_PWM
10 kHz	40 MΩ	1 pF	-8.64
10 kHz	4 kΩ	0.01 µF	-8.64
10 kHz	1.8 kΩ	0.022 µF	-8.57
100 kHz	40 MΩ	1 pF	-28.01
100 kHz	4 kΩ	0.01 µF	-28.01
100 kHz	1.8 kΩ	0.022 µF	-27.92

Figure 2-4. Bode Plot

Figure 2-5. Step Response (R = 40 MΩ, C = 1 pF, F_PWM = 10 kHz)

Figure 2-6. Step Response (R = 40 MΩ, C = 1 pF, F_PWM = 100 kHz)

Figure 2-7. Step Response (R = 4 kΩ, C = 0.01 μF, FPWM = 10 kHz)