Computing Differential Pairs

To obtain various differential pair quantities, we first define differential and common voltages v_d and v_c in terms of the singled-ended terminal voltage pairs v₁and v₂, (see Setting up Differential Pairs).

The differential and common voltages v_d and v_c are defined by

		(1)

Consistent with power conservation, the corresponding differential and common currents, represented as i_d and i_c respectively, are defined by

		(2)

Equations (1) and (2) can be concisely represented as

where

• Q is the real, non-singular matrix defined by

  (4)

• Q^-T is the inverse transpose of Q defined by

  (5)

Using eq. (3), we may easily transform between single-ended and differential quantities.

Differential Admittance and Impedance Matrices

The terminal admittance (Y) and impedance (Z) matrices discussed in the previous topics in the Technical Notes relate single-ended voltages and currents as i = Yv and v = Zi, respectively. By first making differential voltage and current definitions, e and u eq. (3) can be used to derive Y_dand Z_d matrices that relate those qualities.

For example, if i = Yv, then substituting eq. (3) yields

Q^-Tu = YQe.

Solving for u yields

u = Q^TYQe

and the matrix Y_drelating differential quantities e and u is defined by

A similar procedure applies to the terminal Z_d.

Differential S-Matrices

It is clear that an S-matrix can be computed for differential signals because it is possible to compute admittance and impedance matrices for differential signals. The differential S-matrix can be envisioned as relating in-going and out-going waves on imaginary transmission lines attached to the differential ports. Characteristic impedances must be specified for these lines unless they are considered to be matched loaded which is automatically the case when the wave port(s) that contain the single-ended terminals used to define the differential pair is not renormalized.

In the single-ended case, the characteristic impedance for a pair of transmission lines may be written in the form of a matrix relating the voltages and currents on the two (uncoupled) lines,

(7)

where Z⁽¹⁾ref and Z⁽²⁾ref are the user-specified renormalizing impedances. In the differential case, the matrix equation relating differential and common currents and voltages is written as:

(8)

In this case, Z^(d)ref and Z^(c)ref denote the user-specified differential and common renormalizing impedances, respectively.