page_title: Equivalent impedance transforms
section_title: 3 and 4-terminal networks
hierarchical_section_title: Equivalent impedance transforms / 3 and 4-terminal networks
context_page_description: An equivalent impedance is an equivalent circuit of an electrical network of impedance elements which presents the same impedance between all pairs of terminals as did the given network.
  There are a number of very well known and often used equivalent circuits in linear network analysis. These include resistors in series, resistors in parallel and the extension to series and parallel circuits for capacitors, inductors and general impedances. Also well known are the Norton and Thévenin equivalent current generator and voltage generator circuits respectively, as is the Y-Δ transform. None of these are discussed in detail here; the individual linked articles should be consulted.
  The number of equivalent circuits that a linear network can be transformed into is unbounded. Even in the most trivial cases this can be seen to be true, for instance, by asking how many different combinations of resistors in parallel are equivalent to a given combined resistor. The number of series and parallel combinations that can be formed grows exponentially with the number of resistors, n.
context_section_description: When discussing 4-terminal networks, network analysis often proceeds in terms of 2-port networks, which covers a vast array of practically useful circuits.  "2-port", in essence, refers to the way the network has been connected to the outside world: that the terminals have been connected in pairs to a source or load.  It is possible to take exactly the same network and connect it to external circuitry in such a way that it is no longer behaving as a 2-port.  This idea is demonstrated in Figure 2.
  A 3-terminal network can also be used as a 2-port.  To achieve this, one of the terminals is connected in common to one terminal of both ports.  In other words, one terminal has been split into two terminals and the network has effectively been converted to a 4-terminal network.  This topology is known as unbalanced topology and is opposed to balanced topology.  Balanced topology requires, referring to Figure 3, that the impedance measured between terminals 1 and 3 is equal to the impedance measured between 2 and 4.  This is the pairs of terminals not forming ports: the case where the pairs of terminals forming ports have equal impedance is referred to as symmetrical.  Strictly speaking, any network that does not meet the balance condition is unbalanced, but the term is most often referring to the 3-terminal topology described above and in Figure 3.  Transforming an unbalanced 2-port network into a balanced network is usually quite straightforward: all series-connected elements are divided in half with one half being relocated in what was the common branch.  Transforming from balanced to unbalanced topology will often be possible with the reverse transformation but there are certain cases of certain topologies which cannot be transformed in this way.  For example, see the discussion of lattice transforms below.
  An example of a 3-terminal network transform that is not restricted to 2-ports is the Y-Δ transform.  This is a particularly important transform for finding equivalent impedances.  Its importance arises from the fact that the total impedance between two terminals cannot be determined solely by calculating series and parallel combinations except for a certain restricted class of network.  In the general case additional transformations are required.  The Y-Δ transform, its inverse the Δ-Y transform, and the n-terminal analogues of these two transforms (star-polygon transforms) represent the minimal additional transforms required to solve the general case.  Series and parallel are, in fact, the 2-terminal versions of star and polygon topology.  A common simple topology that cannot be solved by series and parallel combinations is the input impedance to a bridge network (except in the special case when the bridge is in balance).  The rest of the transforms in this section are all restricted to use with 2-ports only.
caption: Fig. 3. To be balanced, a network must have the same impedance in each "leg" of the circuit.