The Hemmi 301 and Boonshaft & Fuchs Slide RulesA Brief Introduction to Transfer Functions

Richard Smith Hughes

This brief introduction on transfer functions presents the basic equations discussed in my JOS Hemmi 301and Boonshaft & Fuchs article. The basic theory is really quite simple, basically just solving for thehypotenuse and phase angle of a right triangle. Let us solve for the transfer function (or gain), G, of thesimple RC low pass for the filter shown below (the output voltage Vout is equal to the input voltage Vin forlow frequencies, but heads to zero at high frequencies).

The impedance of the capacitor is frequency dependent; there is also a 90o phase shift between the voltageacross the capacitor and the current; electronic types refer to this 90o phase shift as j. Do not worry abouthow we get the phase shift, assume it is valid-which it is. Anyway, the impedance of the capacitor, ZC =1/j(2πfC), where f is the frequency in Hz (or cycles per second) and j is +90o. The transfer function for thecircuit is:

(Vout/Vin)(f) = ZC/(R + ZC) = G(f) = [1/j(2πfC)]/[R + 1/j(2πfC)] = 1/[1 + j(2πfRC)]

The (f) in this equation means that G(f) is a function of frequency. We electronic engineers like to makeequations as simple appearing as possible; let ω (angular frequency in radians/sec.) ≡ 2πf and s ≡ j ω. The transfer function may now be given as:

(Vout/Vin)(f) = G(f) = 1/[1 + j(2πfRC)] = (Vout/Vin)(ω) = G(ω) = 1/[1 + j(ωRC)] = (Vout/Vin)(s) = G(s) = 1/[1 + sRC]

These equations are equivalent, only with different symbols. Looking at the denominator, remember j isjust +90o, we have the simple right triangle shown below;

Point B is given, in vector form, as; B = 1 + j(2πfRC) = 1 + j(ωRC) side b = 1, side a = (2πf RC) = (ωRC), the hypotenuse, c, is

c = √[1 + (2πfRC)2] = √[1 + (ωRC)2].The phase angle, A ≡ θ = tan-1(2πfRC) = tan-1(ωRC)

Our transfer function is: j2πf = jω = s

G(f) = 1/[1 + j(2πfRC)] = G(ω) = 1/[1 + j(ωRC)] =G(s) = 1/[1 + sRC] With a magnitude, M; 2πf = ω

M(f) = 1/√ [1 + (2πfRC)2] = M(ω) = 1/√ [1 + (ω RC)2]and phase, θ; remember that 1/tan = -tan

θ = -tan-1(2πfRC) = -tan-1(ωRC)

Often you will see the magnitude, M expressed in dB; MdB = 20Log10(M). Our magnitude equations, indB, may now be given as (remember that Log10(1/x) = Log10(x

-1) = - Log10(x)):

Magnitude, MdB; 2πf = ωM(f)dB = -20Log10{√ [1 + (2πfRC)2]} = M(ω)dB = -20Log10{√ [1 + (ωRC)2]}

Phase, θ θ = -tan-1(2πfRC) = -tan-1(ωRC)

We are almost there. Let us see what happens when 2πfRC = ω RC = 1; M(f)dB = M(ω)dB = -20Log10{√ [1 + (1)2]} = -20Log10√ [2] = -3dB

θ = -tan-1(1) = -45o

This is called the 3dB frequency (f3dB or ω3dB) and it is a constant for a given RC; f3dB = 1/(2πRC) and ω3dB= 1/RC. OK, RC may be given as; RC = 1/(2πf3dB) = 1/ ω3dB. Substituting this into our magnitude andphase equations:

Magnitude, MdB; f3dB= 1/(2πRC), ω3dB= 1/RCM(f)dB = -20Log10{√ [1 + (f/f3dB)2]} = M(ω)dB = -20Log10{√ [1 + (ω/ω3dB)2]}

Phase, θ θ = -tan-1(f/f3dB) = -tan-1(ω/ω3dB)

Electronic engineers also like to use time constants, τ, where τ = RC; remember RC = 1/2πf3dB = 1/ω3dB, thus τ = 1/2πf3dB = 1/ω3dB. Substituting τ = RC in our magnitude and phase equations:

Magnitude, MdB; τ = RC M(f)dB = -20Log10[√ [1 + (2πfτ)2] = M(ω)dB = -20Log10[√ [1 + (ωτ)2]]

Phase, θ θ = -tan-1(2πfτ) = -tan-1(ωτ)

That’s it! Below is a summary of our transfer function journey:

Transfer function; j2πf = jω = s G(f) = 1/[1 + j(2πfRC)] = G(ω) = 1/[1 + j(ωRC)] =G(s) = 1/[1 + sRC]

Magnitude, MdB; 2πf = ω M(f)dB = -20Log10{√ [1 + (2πfRC)2]} = M(ω)dB = -20Log10{√ [1 + (ωRC)2]}

Phase, θ θ = -tan-1(2πfRC) = -tan-1(ωRC)

Magnitude, MdB; f3dB = 1/(2πRC) and ω3dB= 1/RCM(f)dB = -20Log10{√ [1 + (f/f3dB)2]} = M(ω)dB = -20Log10{√ [1 + (ω/ω3dB)2]}

Phase, θ θ = -tan-1(f/f3dB) = -tan-1(ω/ω3dB)

Magnitude, MdB; τ = RC M(f)dB = -20Log10[√ [1 + (2πfτ)2] = M(ω)dB = -20Log10[√ [1 + (ωτ)2]]

Phase, θ θ = -tan-1(2πfτ) = -tan-1(ωτ)

NOTEThese equations can be solved using a slide rule, but it really time-consuming for the solution for a largenumber of frequencies. Enter the Hemmi 301 and Boonshaft & Fuchs; real time savers. The Hemmi 301uses ωτ equations, and the Boonshaft & Fuchs uses u = ω and udB = ωdB equations.