Introduction#

We can connect two-ports in ways similar to one-ports.

Series-Series Connection#

Definition: Series-Series Connection

Two two-ports are series-series connected when their input ports are connected in series with one another and their output ports are also connected in series with one another:

Series-Series Two-Ports

In a series-series connection, we have the following:

\[ \boldsymbol{i}_{\mathcal{F}_1} = \boldsymbol{i}_{\mathcal{F}_2} \]

Theorem: Series-Series Equivalent

Two two-ports \(\mathcal{F}_1\) and \(\mathcal{F}_2\) in a series-series connection are equivalent to a single two-port \(\mathcal{F}\):

Series-Series Two-Ports Equivalence

We have the following:

\[ \boldsymbol{i}^{\mathcal{F}} = \boldsymbol{i}^{\mathcal{F}_1} = \boldsymbol{i}^{\mathcal{F}_2} \qquad \boldsymbol{v}^{\mathcal{F}} = \boldsymbol{v}^{\mathcal{F}_1} + \boldsymbol{v}^{\mathcal{F}_2} \]

If \(\mathcal{F}_1\) and \(\mathcal{F}_2\) have impedance representations \(\boldsymbol{v}^{\mathcal{F}_1} = R_1 (\boldsymbol{i}^{\mathcal{F}_1})\) and \(\boldsymbol{v}^{\mathcal{F}_2} = R_2 (\boldsymbol{i}^{\mathcal{F}_2})\), then \(\mathcal{F}\) also has an impedance representation

\[ \boldsymbol{v}^{\mathcal{F}} = R(\boldsymbol{i}^{\mathcal{F}}), \]

where \(R\) is the sum of the functions \(R_1\) and \(R_2\):

\[ R = R_1 + R_2 \]

Proof

TODO

Series-Parallel Connection#

Definition: Series-Parallel Connection

Two two-ports are series-parallel connected when their input ports are connected in series with one another and their output ports are connected in parallel with one another:

Series-Parallel Two-Ports

In a series-parallel connection, we have the following:

\[ i_1^{\mathcal{F}_1} = i_1^{\mathcal{F}_2} \qquad v_2^{\mathcal{F}_1} = v_2^{\mathcal{F}_2} \]

Theorem: Series-Parallel Equivalent

Two two-ports \(\mathcal{F}_1\) and \(\mathcal{F}_2\) in a series-parallel connection are equivalent to a single two-port \(\mathcal{F}\):

Series-Parallel Two-Ports Equivalence

We have the following:

\[ \begin{aligned}i_1^{\mathcal{F}} &= i_1^{\mathcal{F}_1} = i_1^{\mathcal{F}_2} \\ \\ v_1^{\mathcal{F}} &= v_1^{\mathcal{F}_1} + v_1^{\mathcal{F}_2} \end{aligned} \qquad \qquad \begin{aligned} i_2^{\mathcal{F}} &= i_2^{\mathcal{F}_1} + i_2^{\mathcal{F}_2} \\ \\ v_2^{\mathcal{F}} &= v_2^{\mathcal{F}_1} = v_2^{\mathcal{F}_2}\end{aligned} \]

If \(\mathcal{F}_1\) and \(\mathcal{F}_2\) have hybrid representations \(\begin{bmatrix}v_1^{\mathcal{F}_1} \\ i_2^{\mathcal{F}_1}\end{bmatrix} = H_1 \left(\begin{bmatrix}i_1^{\mathcal{F}_1} \\ v_2^{\mathcal{F}_1}\end{bmatrix}\right)\) and \(\begin{bmatrix}v_1^{\mathcal{F}_2} \\ i_2^{\mathcal{F}_2}\end{bmatrix} = H_2 \left(\begin{bmatrix}i_1^{\mathcal{F}_2} \\ v_2^{\mathcal{F}_2}\end{bmatrix}\right)\), then \(\mathcal{F}\) also has a hybrid representation

\[ \begin{bmatrix}v_1^{\mathcal{F}} \\ i_2^{\mathcal{F}}\end{bmatrix} = H \left(\begin{bmatrix}i_1^{\mathcal{F}} \\ v_2^{\mathcal{F}}\end{bmatrix}\right), \]

where \(H\) is the sum of the functions \(H_1\) and \(H_2\):

\[ H = H_1 + H_2 \]

Proof

TODO

Parallel-Series Connection#

Definition: Parallel-Series Connection

Two two-ports are parallel-series connected when their input ports are connected in parallel with one another and their output ports are connected in series with one another:

Parallel-Series Two-Ports

In a parallel-series connection, we have the following:

\[ v_1^{\mathcal{F}_1} = v_1^{\mathcal{F}_2} \qquad i_2^{\mathcal{F}_1} = i_2^{\mathcal{F}_2} \]

Theorem: Parallel-Series Equivalent

Two two-ports \(\mathcal{F}_1\) and \(\mathcal{F}_2\) in a parallel-series connection are equivalent to a single two-port \(\mathcal{F}\):

Parallel-Series Two-Ports Equivalence

We have the following:

\[ \begin{aligned}i_1^{\mathcal{F}} &= i_1^{\mathcal{F}_1} + i_1^{\mathcal{F}_2} \\ \\ v_1^{\mathcal{F}} &= v_1^{\mathcal{F}_1} = v_1^{\mathcal{F}_2} \end{aligned}\qquad \qquad\begin{aligned}i_2^{\mathcal{F}} &= i_2^{\mathcal{F}_1} = i_2^{\mathcal{F}_2} \\ \\ v_2^{\mathcal{F}} &= v_2^{\mathcal{F}_1} + v_2^{\mathcal{F}_2}\end{aligned} \]

If \(\mathcal{F}_1\) and \(\mathcal{F}_2\) have inverse hybrid representations \(\begin{bmatrix}i_1^{\mathcal{F}_1} \\ v_2^{\mathcal{F}_1}\end{bmatrix} = H_1' \left(\begin{bmatrix}v_1^{\mathcal{F}_1} \\ i_2^{\mathcal{F}_1}\end{bmatrix}\right)\) and \(\begin{bmatrix}i_1^{\mathcal{F}_2} \\ v_2^{\mathcal{F}_2}\end{bmatrix} = H_2' \left(\begin{bmatrix}v_1^{\mathcal{F}_2} \\ i_2^{\mathcal{F}_2}\end{bmatrix}\right)\), then \(\mathcal{F}\) also has an inverse hybrid representation

\[ \begin{bmatrix}i_1^{\mathcal{F}} \\ v_2^{\mathcal{F}}\end{bmatrix} = H' \left(\begin{bmatrix}v_1^{\mathcal{F}} \\ i_2^{\mathcal{F}}\end{bmatrix}\right), \]

where \(H'\) is the sum of the functions \(H_1'\) and \(H_2'\):

\[ H' = H_1' + H_2' \]

Proof

TODO

Parallel-Parallel Connection#

Definition: Parallel-Parallel Connection

Two two-ports are parallel-parallel connected when their input ports are connected in parallel with one another and their output ports are also connected in parallel with one another:

Parallel-Parallel Two-Ports

In a parallel-parallel connection, we have the following:

\[ \boldsymbol{v}^{\mathcal{F}_1} = \boldsymbol{v}^{\mathcal{F}_2} \]

Theorem: Parallel-Parallel Equivalent

Two two-ports \(\mathcal{F}_1\) and \(\mathcal{F}_2\) in a parallel-parallel connection are equivalent to a single two-port \(\mathcal{F}\):

Parallel-Parallel Two-Ports Equivalence

We always have:

\[ \boldsymbol{v}^{\mathcal{F}} = \boldsymbol{v}^{\mathcal{F}_1} = \boldsymbol{v}^{\mathcal{F}_2} \qquad \boldsymbol{i}^{\mathcal{F}} = \boldsymbol{i}^{\mathcal{F}_1} + \boldsymbol{i}^{\mathcal{F}_2} \]

If \(\mathcal{F}_1\) and \(\mathcal{F}_2\) have admittance representations \(\boldsymbol{i}^{\mathcal{F}_1} = G_1(\boldsymbol{v}^{\mathcal{F}_1})\) and \(\boldsymbol{i}^{\mathcal{F}_2} = G_2(\boldsymbol{v}^{\mathcal{F}_2})\), then \(\mathcal{F}\) also has an admittance representation:

\[ \boldsymbol{i}^{\mathcal{F}} = G(\boldsymbol{v}^{\mathcal{F}}), \]

where \(G\) is the sum of the functions \(G_1\) and \(G_2\):

\[ G = G_1 + G_2 \]

Proof

TODO

Cascade Connection#

Definition: Cascade Connection

Two two-ports are cascade connected when the output port of the first is connected to the input port of the second:

Cascaded Two-Ports

We have the following:

\[ v_2^{\mathcal{F}_1} = v_1^{\mathcal{F}_2} \qquad i_2^{\mathcal{F}_1} = - i_1^{\mathcal{F}_2} \]

Theorem: Cascade Equivalent

Two two-ports \(\mathcal{F}_1\) and \(\mathcal{F}_2\) in a cascade connection are equivalent to a single two-port \(\mathcal{F}\):

Cascaded Two-Ports Equivalence

We have the following:

\[ \begin{aligned} v_1^{\mathcal{F}} &= v_1^{\mathcal{F}_1} \\ \\ i_1^{\mathcal{F}} &= i_1^{\mathcal{F}_1} \end{aligned} \qquad \qquad \begin{aligned} v_2^{\mathcal{F}_1} &= v_1^{\mathcal{F}_2} \\ \\ i_2^{\mathcal{F}_1} &= -i_1^{\mathcal{F}_2} \end{aligned} \qquad \qquad \begin{aligned} v_2^{\mathcal{F}} &= v_2^{\mathcal{F}_2} \\ \\ i_2^{\mathcal{F}} &= i_2^{\mathcal{F}_2} \end{aligned} \]

If \(\mathcal{F}_1\) and \(\mathcal{F}_2\) have forwards transmission representations \(\begin{bmatrix}v_1^{\mathcal{F}_1} \\ i_1^{\mathcal{F}_1}\end{bmatrix} = T_1 \left(\begin{bmatrix}v_2^{\mathcal{F}_1} \\ -i_2^{\mathcal{F}_1}\end{bmatrix}\right)\) and \(\begin{bmatrix}v_1^{\mathcal{F}_2} \\ i_1^{\mathcal{F}_2}\end{bmatrix} = T_2 \left(\begin{bmatrix}v_2^{\mathcal{F}_2} \\ -i_2^{\mathcal{F}_2}\end{bmatrix}\right)\), then \(\mathcal{F}\) also has a forwards transmission representation

\[ \begin{bmatrix}v_1^{\mathcal{F}} \\ i_1^{\mathcal{F}}\end{bmatrix} = T \left(\begin{bmatrix}v_2^{\mathcal{F}} \\ -i_2^{\mathcal{F}}\end{bmatrix}\right), \]

where \(T\) is the composition of the functions \(T_1\) and \(T_2\):

\[ T = T_1 \circ T_2 \]

If \(\mathcal{F}_1\) and \(\mathcal{F}_2\) have backwards transmission representations \(\begin{bmatrix}v_2^{\mathcal{F}_1} \\ i_2^{\mathcal{F}_1}\end{bmatrix} = T_1' \left(\begin{bmatrix}v_1^{\mathcal{F}_1} \\ -i_1^{\mathcal{F}_1}\end{bmatrix}\right)\) and \(\begin{bmatrix}v_2^{\mathcal{F}_2} \\ i_2^{\mathcal{F}_2}\end{bmatrix} = T_2' \left(\begin{bmatrix}v_1^{\mathcal{F}_2} \\ -i_1^{\mathcal{F}_2}\end{bmatrix}\right)\), then \(\mathcal{F}\) also has a backwards transmission representation

\[ \begin{bmatrix}v_2^{\mathcal{F}} \\ i_2^{\mathcal{F}}\end{bmatrix} = T' \left(\begin{bmatrix}v_1^{\mathcal{F}} \\ -i_1^{\mathcal{F}}\end{bmatrix}\right), \]

where \(T\) is the composition of the functions \(T_2\) and \(T_1\):

\[ T' = T_2' \circ T_1' \]

Proof

TODO