What are the Y and Z parameters for a 2-port circuit and how do you derive them?

[magnifik]

I am solving for the Y and Z parameters for the circuit below:

For Y, I used the following equations by applying current sources at the terminals
v_a is the node at the 2v₁ dependent source
(v_a-v₁)/0.5 + (v_a-v₂)/1 = -2v₁
I₁ = v₁/1 + (v₁-v_a)/0.5
I₂ = v₂/0.5 + (v₂-v_a)/1
I got Y = [3 -2/3; 0 8/3]

For Z, I used KVL by applying voltage sources at the terminals
v₁ = I_a - I_b
0 = I_b - I_a + 0.5I_b
0 = I_c + 0.5(I_c + I_d)
v₂ = 0.5(I_c + I_d)
where I_a = I₁, I_b = 2v₁, I_c = -2v₁, and I_d = I₂
I got Z = [1/3 1/6; 0 1/3]

I am not sure what I am doing incorrectly here. Y should be equal to Z^-1, but I am not getting that. I'm not sure if my Y parameters are wrong, Z parameters, or both. :\

 

[The Electrician]

I am solving for the Y and Z parameters for the circuit below:

For Y, I used the following equations by applying current sources at the terminals
v_a is the node at the 2v₁ dependent source
(v_a-v₁)/0.5 + (v_a-v₂)/1 = -2v₁
I₁ = v₁/1 + (v₁-v_a)/0.5
I₂ = v₂/0.5 + (v₂-v_a)/1
I got Y = [3 -2/3; 0 8/3]

For Z, I used KVL by applying voltage sources at the terminals
v₁ = I_a - I_b
0 = I_b - I_a + 0.5I_b
0 = I_c + 0.5(I_c + I_d)
v₂ = 0.5(I_c + I_d)
where I_a = I₁, I_b = 2v₁, I_c = -2v₁, and I_d = I₂
I got Z = [1/3 1/6; 0 1/3]

I am not sure what I am doing incorrectly here. Y should be equal to Z^-1, but I am not getting that. I'm not sure if my Y parameters are wrong, Z parameters, or both. :\

I think the major problem is where you have:

I_b = 2v₁, I_c = -2v₁

Replace those two items with I_b - I_c = 2v₁

and see if you don't get better results.

 

[magnifik]

when using I_b - I_c = 2v₁

I got
v₁ = I_a - I_b
= I_a- I_c - 2v₁
= I_a + (1/3)I_d - 2v₁
====> v₁/I₁ = 1/3 and v₁/I₂ = 1/9
plugging in values to find the remaining parameters i got v₂/I₂ = 1/2 and v₂/I₁ = 0
so for Z parameters i got Z = [1/3 1/9; 0 1/2]

The inverse of this is [3 -2/3; 0 2] so there's still something off :\

 

[The Electrician]

You can't assume the voltage across the dependent source is zero; you have:

0 = I_b - I_a + 0.5I_b
0 = I_c + 0.5(I_c + I_d)

You need to make those:

0 = -v_a + I_b - I_a + 0.5I_b
0 = v_a + I_c + 0.5(I_c + I_d)

I'm not working out the complete solution using your method, so I may not see all the errors right away. I'm just pointing out what I see with a quick glance; you need to try to find errors also.

You might need to treat the middle two meshes together as a supermesh.

 

[magnifik]

You can't assume the voltage across the dependent source is zero; you have:

0 = I_b - I_a + 0.5I_b
0 = I_c + 0.5(I_c + I_d)

You need to make those:

0 = -v_a + I_b - I_a + 0.5I_b
0 = v_a + I_c + 0.5(I_c + I_d)

I'm not working out the complete solution using your method, so I may not see all the errors right away. I'm just pointing out what I see with a quick glance; you need to try to find errors also.

You might need to treat the middle two meshes together as a supermesh.

are my Y parameters correct? just wondering because i don't want to be trying to use them to check if my Z parameters are correct if they aren't right in the first place..

 

[The Electrician]

Yes, they are correct.

 

[The Electrician]

Have you had any luck deriving the Z parameters?

 

[magnifik]

Have you had any luck deriving the Z parameters?

yep, thanks. i just had a few algebraic errors that threw me off.