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One-Port Interconnections#

One-Ports in Parallel#

Definition: One-Ports in Parallel

A parallel circuit is an electrical element which consists of one-ports whose terminals are connected at the same nodes.

Parallel Circuit

We also say that the one-ports are connected in parallel.

Theorem: Voltage across Parallel Circuits

The voltage across all one-ports in a parallel circuit is the same.

Voltage across Parallel Circuit

Proof

TODO

Theorem: Equivalence of Parallel Circuits

Every parallel circuit of one-ports \(\mathcal{F}_1, \dotsc, \mathcal{F}_n\) is equivalent to a single one-port with I-V characteristic \(\mathcal{F}\):

\[ \mathcal{F} = \{(V, I) \mid \exists I_1, \dotsc, I_n: (V, I_1) \in \mathcal{F}_1, \dotsc, (V, I_n) \in \mathcal{F}_n \text{ with } I = I_1 + \cdots + I_n\} \]

Equivalent Parallel Circuit

In other words, the voltage across the equivalent one-port is still \(V\), but the current \(I\) flowing through it is the sum of the currents flowing through \(\mathcal{F}_1, \dotsc, \mathcal{F}_n\):

\[ I = \sum_{k = 1}^n I_k \]
Proof

TODO

Algorithm: Graphical Determining of the I-V Characteristic

We want to determine \(\mathcal{F}\) graphically:

  1. Draw \(\mathcal{F}_1, \dotsc, \mathcal{F}_n\) on the same I-V graph.
  2. Pick some voltage \(v\). Sum up the corresponding currents \(i_1, \dotsc, i_n\) of \(\mathcal{F}_1, \dotsc, \mathcal{F}_n\). The point \((v, i_1 + \cdots + i_n)\) is then part of \(\mathcal{F}\).
    • If any of \(\mathcal{F}_1, \dotsc, \mathcal{F}_n\) is not defined for \(v\), then \(\mathcal{F}\) is also not defined for \(v\)!
  3. Repeat step 2 a few times to get a few points of \(\mathcal{F}\). From these points you can roughly draw the graph of \(\mathcal{F}\).
Example

TODO

Theorem: Duality of Parallel Circuits

Every parallel circuit with \(n\) components is one-ports to a series circuit with \(n\) components.

Proof

TODO

One-Ports in Series#

Definition: Series Circuits

A series circuit is an electrical element which consists of one-ports whose terminals are sequentially connected to one another.

Series Circuit

Theorem: Current through Series Circuit

The current flowing through each one-port in a series circuit is the same.

Current through Series Circuit

Proof

TODO

Theorem: Equivalence of Series Circuits

Every series circuit of one-ports \(\mathcal{F}_1, \dotsc, \mathcal{F}_n\) is equivalent to a single one-port with V-I characteristic \(\mathcal{F}\):

\[ \mathcal{F} = \{(V, I) \mid \exists V_1, \dotsc, V_n: (V_1, I) \in \mathcal{F}_1, \dotsc, (V_n, I) \in \mathcal{F}_n \text{ with } V = V_1 + \cdots + V_n\} \]

Equivalent Series Circuit

In other words, the current flowing through the equivalent one-port is still \(I\), but the voltage across it is the sum of the voltages across \(\mathcal{F}_1, \dotsc, \mathcal{F}_n\):

\[ V = \sum_{k=1}^n V_k \]
Proof

TODO

Algorithm: Graphical Determining of the I-V Characteristic

We want to determine \(\mathcal{F}\) graphically:

  1. Draw \(\mathcal{F}_1, \dotsc, \mathcal{F}_n\) on the same I-V graph.
  2. Pick some current \(i\). Sum up the corresponding voltages \(v_1, \dotsc, v_n\) of \(\mathcal{F}_1, \dotsc, \mathcal{F}_n\). The point \((v_1 + \cdots + v_n, i)\) is then part of \(\mathcal{F}\).
    • If any of \(\mathcal{F}_1, \dotsc, \mathcal{F}_n\) is not defined for \(i\), then \(\mathcal{F}\) is also not defined for \(i\)!
  3. Repeat step 2 a few times to get a few points of \(\mathcal{F}\). From these points you can roughly draw the graph of \(\mathcal{F}\).
Example

TODO

Theorem: Duality of Series Circuits

Every series circuit with \(n\) components is dual to a parallel circuit with \(n\) components.

Proof

TODO