Graphically Determine The Output Signal Level Of An RC Filter

where τ is the filter’s time constant and τ₁ is the pulse duration (Fig. 1).

For simplicity, define:

τ₁ = DT                                   (6)

where D is the duty cycle of the input pulse train;

τ = kT                                                       (7)

where k is the number of periods for the filter time constant. So Equations 1 and 2 become:

Vh_n = V_Aβ +Vlow_nγ          (8)

Vlow_n = Vh_(n–1)α                                       (9)

To derive the equation for the limit value of the filter output, you must use the recurrent equations for the low and high values and calculate the average, which is based on the initial parameters only. The equations for six high/low pairs are:

Vh₀ = V_Aβ

Vlow₀ = 0

Vh₁ = V_Aβ(γα + 1)

Vlow₁ = V_Aβα

Vh₂ = V_Aβ(γ²α² + γα + 1)

Vlow₂ = V_Aβα(γα + 1)

Vh₃ = V_Aβ(γ³α³ + γ²α² + γα + 1)

Vlow₃ = V_Aβα(γ²α² + γα + 1)

Vh₄ = V_Aβ(γ⁴α⁴ + γ³α³ + γ²α² + γα + 1)

Vlow₄ = V_Aβα(γ³α³ + γ²α² + γα + 1)

Vh₅ = V_Aβ(γ⁵α⁵ + γ⁴α⁴ + γ³α³ + γ²α₂ + γα + 1)

Vlow₅ = V_Aβα(γ⁴α⁴ + γ³α³ + γ²α² + γα + 1)

Note from the above equations that:

and:

The average for the output voltage is:

You can calculate the limit for the normalized output voltage using MathCAD tools:

or:

(See the derivation of Equations 13 and 14 in “Normalized Output Voltage Limit Derivation,” below.)

Equation 14 is a transcendent equation with two variables, which is hard to solve symbolically. But you can find the limit value graphically by fixing the value of k and plotting the limit as a function of the duty cycle, D. If k = 100.0:

A plot of this function shows that the limit of the normalized RC filter output is the duty cycle, D (Fig. 2).

To create an example, assign values to the variables in Figure 1:

V_A = 1 V

D = 0.6

T = 20 µs

τ₁ = DT

k = 50

τ = kT

Then:

α = e^{–(T – τ1)/τ} = 0.992

β = 1 – e^–τ1/τ = 0.012

τ = e^–τ1/τ = 0.988

Then:

To plot the graph for the limit, assign n = 1…400. Then:

Figure 3 shows the resulting plot.

Gregory Mirsky is a principal electrical engineer at Atlas Materials Testing Company. He holds an MS from the St. Petersburg Baltic Technical University, Russia, and a PhD from the Moscow State Pedagogical University.

NORMALIZED OUTPUT VOLTAGE LIMIT DERIVATION

Equation 12 can be interpreted as:

Its solution is:

Only the lower expression is of interest:

Its solution is:

Again, only the lower expression is of interest:

which can be converted into Equation 13.