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Q Adder
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Appendix section of Phil's transmission line notes (Q Adder) that states and proves Facts 5 to 8 about Z0 = sqrt((R+jωL)/(G+jωC)). It derives the real/imaginary decomposition using magnitude and phase arguments, with a geometric branch-cut sketch left as a reader exercise. It then gives asymptotic expansions of Re(Z0) and Im(Z0) for high frequency, low frequency with G>0, and G=0, using Maple. Many equations are lost in extraction.
AI-written summary; may contain errors. This description is approximate.
Extracted text (machine-read; may contain errors)
Fact 5: The real + imaginary decomposition of Z0 is given by
Z0 = = = (a/) [ - jσ ]
where a = ()1/4 = |Z0| α =
σ = sign(RC-LG) β ≡ RG + ω2LC
Proof of Fact 5: The proof is similar to the proof of Fact 1. Let
q = = = |q| ejθ -π < θ < π ( but see few lines below)
Then
|q|2 = | | 2 = ≡ a4 => |q| = = a2
Next,
q = = = =
so
Re(q) = > 0 => -π/2 < θ < π/2
Im(q) = sign[Im(q)] = sign(LG-RC) = sign(θ) ≡ Σ .
Then
cosθ = Re(q)/ |q| = =
≡ β/α where α ≡ and β ≡ RG + ω2LC
Now write
s ≡ = = |s| eiφ |s| = = a = |Z0| φ = θ/2 -π/4 < φ < π /4
Re(s) = |s| cosφ = a cos(θ/2) = a = (a/)
Im(s) = |s| sinφ = a sin(θ/2) = Σ a = Σ (a/)
giving the result
s = (a/) [ + jΣ ]
where a = ()1/4 = |Z0| α =
Σ = sign(LG-RC) β ≡ RG + ω2LC
Letting σ = -Σ = sign(RC-LG) one gets the final form,
s = (a/) [ - jσ ]
where a = ()1/4 = |Z0| α =
σ = sign(RC-LG) β ≡ RG + ω2LC
Our Maple verification of this result is a bit ugly so we omit the code.
An alternate geometric derivation giving the same results begins as follows:
Z0 = = = where r = (R/L) and g = (G/C) .
The drawing shows the complex z plane for the function Z0(z) = in the particular case that r > g, in which case the z-plane has a branch cut from -r to -g. The z values of interest are only those on the positive imaginary axis where z = jω.
Reader Exercise: Finish this derivation and obtain the same results shown in **
Hint: cos(β-α) = (rg+ω2)/(AB) and sin(β-α) = ω(r-g)/(AB) where A = and B = .
Fact 6: In the high frequency limit,
Re(Z0) ≈ + (1/8ω2) (RC+3GL)(RC-GL) / (C5/2L3/2)
Im(Z0) ≈ - (1/2ω)(RC-GL) / (L1/2C3/2) ω >> R/L and ω >> G/C
Proof of Fact 6:
We first enter in Maple the basic parameters, omitting the σ in Im(Z0) just for the moment,
We then request an expansion of Re(Z0) for large ω:
The first term is the expected . The 1/ω2 term can be simplified as follows:
and then the result is
Re(Z0) = + (1/8ω2) (RC+3GL)(RC-GL) / (C5/2L3/2) + order(1/ω4)
For the imaginary part we do a similar set of steps:
with the result
Im(Z0) = - (1/2ω)(RC-GL) / (L1/2C3/2)
Fact 7: In the low frequency limit with G > 0,
Re(Z0) ≈ - (ω2/8) (3RC+GL)(RC-GL) / (C5/2L3/2)
Im(Z0) ≈ - (ω/2) (RC-GL) / ( R1/2G3/2) ω << R/L and ω << G/C
Proof of Fact 7:
The Maple expressions are entered as in Fact 6, and then:
so that
Re(Z0) = - (ω2/8) (3RC+GL)(RC-GL) / (C5/2L3/2) + order(ω4)
Im(Z0) = - (ω/2) (RC-GL) / ( R1/2G3/2) + order(ω3)
Fact 8: In the low frequency limit with G = 0, both components diverge as:
Re(Z0) ≈ (1/) 1/
Im(Z0) ≈ - (1/) 1/
Proof of Fact 8:
We first enter the general forms, setting G = 0,
The expansions are then,
with the final results,
Re(Z0) = (1/)
Im(Z0) = - (1/) .
These results are obvious without using Maple,
Z0 = = (j)-1/2 = e-jπ/4 = [ 1-j]/,
but the expansions provide more terms if needed.