Magnitude is returning a complex number if I precede it with Complex Expand?

Question

I define a function called VoutCoin:

VoutCoin[Mut_, \[Omega]_, Lc_, Rc_] = 
ComplexExpand[Abs[I1coin* (j \[Omega] L1 + (j \[Omega] Mut)^2/(j \[Omega] Lc + Rc))]] /. values

I have the suffixes and the values defined:

M = 10^6;
k = 10^3;
m = 10^-3;
u = 10^-6;
n = 10^-9;
pf = 10^-12;


values = {
R1 -> 180,
C1 -> 9.4 n,
L1 -> 1.7 m,
Vin -> LaplaceTransform[Cos[2*\[Pi]*138 k*t - \[Pi]], t, s]
};

j = I;
s = j*\[Omega];

if I invoke the function as defined above with the following arguments, I get:

In[1030]:= VoutCoin[.5, 2*\[Pi]*10 k, 10 u, .1]

Out[1030]= {0. + 8.40133*10^-8 I}

if I remove the ComplexExpand preceding the Abs in the function, I get:

VoutCoin[Mut_, \[Omega]_, Lc_, Rc_] = Abs[I1coin* (j \[Omega] L1 + (j \[Omega] Mut)^2/(j \[Omega] Lc + Rc))] /. values // Simplify

In[1095]:= VoutCoin[.5, 2*\[Pi]*10 k, 10 u, .1]

Out[1095]= {8.40133*10^-8}

What is going on? Why would ComplexExpand[Abs[]] return a complex numnber?

In case anyone wants plugs this into Mathematica, I'm including I1coin below

In[1092]:= I1coin = I1 /. sol1

Out[1092]= {(Vin (Rc + I Lc \[Omega]))/(Mut^2 \[Omega]^2 - (-R1 + I/(C1 \[Omega]) - I L1 \[Omega]) (Rc + 
 I Lc \[Omega]))}

Thank you in advance for your help....

--------Edit Per Request_----------------------

M = 10^6;
k = 10^3;
m = 10^-3;
u = 10^-6;
n = 10^-9;
pf = 10^-12;

values = {
R1 -> 180,
C1 -> 9.4 n,
L1 -> 1.7 m,
Vin -> LaplaceTransform[Cos[2*\[Pi]*138 k*t - \[Pi]], t, s]
};

j = I;

s = j*\[Omega];

eq1 = Vin - I1 (R1 + 1/(j \[Omega] C1) + j \[Omega] L1) -j \[Omega] Mut I2 == 0;
eq2 = I2 (j \[Omega] Lc + Rc) + j \[Omega] Mut I1 == 0;

sol1 = Solve[eq1 && eq2, {I1, I2}]

I1coin = I1 /. sol1

VoutCoin[Mut_, \[Omega]_, Lc_, Rc_] = ComplexExpand[Abs[I1coin* (j \[Omega] L1 + (j \[Omega] Mut)^2/(j \[Omega] Lc + Rc))]] /. values // Simplify

 In[1159]:= VoutCoin[.5, 2*\[Pi]*10 k, 10 u, .1]

 Out[1159]= {0. + 8.40133*10^-8 I}

Question is about result out[1159] --- I don't understand why Mathematica returns a complex number for a magnitude. It seems to happen when I precede Abs with ComplexExpand

Answer 1

Consider your values rule:

Vin /. values

-((I ω)/(76176000000 π^2 - ω^2))

So, values is replacing Vin with a pure imaginary number. Now, your VoutCoin function is basically:

f[ω_] = ComplexExpand[Sqrt[Vin^2]] /. values

Sqrt[-(ω^2/(76176000000 π^2 - ω^2)^2)]

Here, ComplexExpand[Sqrt[Vin^2]] evaluates to just Sqrt[Vin^2], and then you replace Vin with a pure imaginary number, hence you get the square root of a manifestly negative number. So, naturally:

f[1.]

(* 0. + 1.33009*10^-12 I *)

is imaginary.

If you move the closing bracket of the ComplexExpand so that it include the values replacement, then you will get a real answer:

VoutCoin[Mut_,ω_,Lc_,Rc_] = ComplexExpand[
    Abs[I1coin*(j ω L1+(j ω Mut)^2/(j ω Lc+Rc))] /.values
] //Simplify;

VoutCoin[.5,2*π*10 k,10 u,.1]

{8.40133*10^-8}