Bod Erlc

8/22/2019 Bod Erlc

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Gary Hecht

RLC Low-Pass Filter Transfer Function

Consider the second-order RLC low-pass filter circuit below:

1 V 1s

V = I • ────── = ─────────────────── • ───── o

j2πfC 1 j2πfCR + j2πfL + ─────

j2πfC

Vo 1─── = Gain = ─────────────────────────────

Vs j2πfRC + (j2πfL)(j2πfC) + 1

Note that the denominator in the above transfer function is not in standard Bode form and,furthermore, it may not be immediately obvious how to manipulate it into standard Bode form. Theequation below is the same as the equation just above except that the denominator has been

reorganized so as to make it clear that the denominator is a quadratic equation in terms of j2πf.

1 1

Gain = ────────────────────────────── = ────────────────────────

(j2πf)(j2πf)LC + (j2πf)RC + 1 (j2πf)2LC + (j2πf)RC + 1

As such, the denominator can be viewed as follows:

(j2πf)2LC + (j2πf)RC + 1 = LCx2 + RCx + 1 where x = j2πf

The well-known quadratic-root equation can be used to determine the “roots” of a quadratic equationas documented below:

If ax2 + bx + c = 0 then the values of x for which this is true (known

as the roots r1,2) are given by:

-b ± (b2 – 4ac)1/2

r1,2 = ─────────────────

2a

However, in our situation we do not desire to set the quadratic equation to zero – instead it is ourdesire to factor the quadratic equation. But note that the quadratic-root equation can be used tofactor a quadratic equation if the quadratic equation is first divided by the coefficient of the squaredterm (e.g., ‘a’). Symbolically this is illustrated below:

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[ax2 + bx + c]/a = x2 + (b/a)x + c/a and then

x2 + (b/a)x + c/a = (x - r1)•(x – r2) where r1,2 are given by the

quadratic-root equation (using

a’ = 1, b’ = b/a, and c’ = c/a)

Before illustrating the actual factoring process for the gain equation derived earlier, examine belowhow the denominator of our gain equation, after applying the process just above, will be almost instandard Bode form:

[(j2πf)2LC + (j2πf)RC + 1]/(LC) = (j2πf)2 + (j2πf)R/L + 1/(LC)

= (j2πf - r1)•(j2πf – r2)

The process of using the quadratic-root equation for factoring the denominator of the gain equationpreviously derived will now be illustrated:

1 1/(LC)

Gain = ──────────────────────── • ────── (j2πf)2LC + (j2πf)RC + 1 1/(LC)

1/(LC)Gain = ────────────────────────────

(j2πf)2 + (j2πf)R/L + 1/(LC) ◄ —— view as x2 + (R/L)x + 1/(LC)and also as (x - r )•(x – r2)1

where r ,r are the roots1 2

1/(LC) of the quadratic equationGain = ───────────────────────

(j2πf - r1)•(j2πf – r2)

where

-b ± ( b2 – 4ac)1/2

r = ───────────────── with a = 1, b = R/L, and c = 1/(LC)1,2

2a

Specifically:

r1,2 = [-R/L ± (R 2/L2 – 4/(LC))1/2]/2

The roots r 1,2 will have the following characteristics:

► If R 2/L2 – 4/(LC) > 0 then the roots r1,2 will both be real and

negative in value (they will be negative since –R/L ± (a value less

than R/L) = a negative value)

► If R 2/L2 – 4/(LC) = 0 then the roots r1,2 will be equal, real and

negative in value (r1,2 = -R/(2L))

► If R 2/L2 – 4/(LC) < 0 then the roots r1,2 will both be complex (due to

the square root of a negative number)

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The last version of the gain equation will now be manipulated into standard Bode form:

1/(LC) 1/[(-r1)(-r2)]

Gain = ─────────────────────── • ───────────────

(j2πf - r1)•(j2πf – r2) 1/[(-r1)(-r2)]

1/(LCr1r2)

Gain = ────────────────────────────────

(j2πf/(-r1) + 1)•(j2πf/(-r2) + 1)

1/(LCr r )1 2

Gain = ──────────────────────────────── (1 + j2πf/(-r1))•(1 + j2πf/(-r2))

(and note that r1 and r2 will both be negative in value making the terms

-r1 and -r2 both positive (unless the roots are complex))

The general form of the Bode gain (magnitude) graph will now be made for each of the threesituations in regards to the roots of the denominator's quadratic equation.

► If the roots r 1 and r 2 are real and unequal in value (and they will be negative for our quadraticequation) this will result in two unequal break-point frequencies in the denominator of the gainequation:

fBP1 = R/(4πL) – (R2/L2 – 4/(LC))1/2/(4π)

fBP2 = R/(4πL) + (R2/L2 – 4/(LC))1/2/(4π)

► If the roots r 1 and r 2 are real and equal in value (and they will be negative for our quadraticequation) this will result in two equal break-point frequencies in the denominator of the gainequation:

fBP = R/(4πL)

freq.

Gain

-40db/decade1 = 0db

f BP

freq.

Gain

-20db/decade

1 = 0db

f BP1 f BP2

-40db/decade

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► If the roots r 1 and r 2 are complex this will result in a gain greater than 1 in the vicinity of adouble break-point frequency:

-40db/decade

Gain

1 = 0db

f BP freq.

Although it seems counterintuitive that a passive circuit can have a gain greater than 1 – it is indeedpossible if the circuit contains capacitance and inductance as well as sufficiently low resistance. Tofurther illustrate this, we will begin by reexamining an earlier version of the transfer function for theRLC low-pass filter circuit and viewing it from a slightly different perspective than before:

1 1Gain = ────────────────────────────── = ────────────────────────

(j2πf)(j2πf)LC + (j2πf)RC + 1 -4π2f2LC + (j2πf)RC + 1

Note that j•j = -1 and this substitution is used in the first term of the denominator to make anegative value (with no j). Doing this does not lead to transforming the transfer function intostandard Bode form – but it will allow us to more easily see why a gain greater than 1 can occur.

With this substitution we can see that there will be a frequency where the -4π2f2LC term will be

equal to -1 which, when added to the +1 term in the denominator, equals to 0 leaving only the

(j2πf)RC term in the denominator. Furthermore, if the value of R is sufficiently low, then the

(j2πf)RC term in the denominator can have a magnitude that is less than 1 making the gain

greater than 1 (because 1 divided by a number that is less than 1 will result in a value that is greaterthan 1).

The preceding discussion noted that there is a frequency when the -4π2f2LC term in the

denominator will be equal to -1 which, when added to the +1 term in the denominator, equals to 0

leaving only the (j2πf)RC term in the denominator. Note that there will actually be a small range

of frequencies where the -4π2f2LC term will be near the value -1 thereby creating a relatively

small range of frequencies where the gain can be greater than 1 (as illustrated in the aboverepresentative Bode gain graph).

Finally, note that the following changes can be made to the original second-order RLC low-pass filtercircuit (see page 1) resulting in the outcomes noted below:

► The inductor (L) and capacitor (C) can have their positions swapped

(i.e., the voltage across the inductor then creates Vo) resulting in a

second-order high-pass filter

► The resistor (R) and capacitor (C) can have their positions swapped

(i.e., the voltage across the resistor then creates Vo) resulting in a

second-order bandpass filter

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