2. DC ANALYSIS 16
2.9 MULTI-TERMINAL ELEMENTS
A multi-terminal element is an element with m terminals available for external connections.
After general description of multi-terminal elements, three-terminal element and four-terminal element are considered. Then, analysis of circuits containing multi-terminal elements is discussed.
ELEMENT DESCRIPTION – CONDUCTANCE MATRIX
Passive multi-terminal element
A general passive m-terminal element is presented in Fig. 2.9.1, node m is the reference one and the terminal currents and voltages satisfy the passive sign convention.
Fig. 2.9.1 Passive m-terminal element
The element can be uniquely described by equations relating external variables, . These equations can be equations expressing current by voltages:
(2.9.1)
Then, an element is described by the conductance matrix G. Its diagonal element , is a conductance between the i-th node and the reference one with all other nodes shorted to this m-th node:
. (2.9.2)
The off-diagonal element , is the so called trans-conductance, ratio of the i-th terminal current to the j-th terminal voltage, with all nodes except the j-th shorted to the reference one:
For an m-terminal circuit of resistors, the conductance matrix is symmetrical, . Then, the total number of its parameters (conductances) is:
(2.9.4) These parameters can be
a) measured,
b) calculated, if a circuit structure is known.
Before discussing in details two, three and four-terminal element term of port will be introduced.
Port is a pair of terminals at which same current may enter and leave an element.
Two-terminal element (one-port)
A general two terminal element with voltage and current that satisfy the passive sign convention is presented in Fig. 2.1.2a. For this element: and the equation relating external variables (2.9.1) is simply the Ohm’s law equation (2.1.5).
Three-terminal element
A general three terminal element is presented in Fig. 2.9.2. For this element:
and equations relating external variables are:
(2.9.5)
This element is characterized by three conductances: . n
Fig. 2.9.2 Passive three-terminal element
Measurement circuits are presented in Fig. 2.9.3a and 2.9.3b, for and , respectively.
Fig. 2.9.3 Measurement circuits for measurement of: a) G11, b) G21
The conductance is
(2.9.6a) The conductance can be measured in the similar way, by shorting 1-3 and measuring
. Finally, the conductance is
(2.9.6b) For an ideal ammeter, , and equations (2.9.6) can be simplified.
G11 G21
Example 2.9.1
Consider three resistors circuit of the T-shape structure, as presented in Fig. 2.9.4a.
Fig. 2.9.4 T-shape circuit (Example 2.9.1) and circuit for calculation of G11 and G21
The conductances and can be designated from Fig. 2.9.4b circuit. The terminal currents are
, (2.9.7a)
, (2.9.7b)
then, the conductances are
(2.9.8a)
A general two-port, presented in Fig. 2.9.5, is the special case of four-terminal element. It is described by the same set of equations as a three-terminal element (2.9.5). Port 1 between nodes 1 and 3 is the input port, port 2 between nodes 2 and 4 is the output port.
G11 G21
Active multi-terminal element
For an active multi-terminal element, terminal short-circuit currents should be added to equations (2.9.1).
+ (2.9.9)
Now, the conductances, short-circuit currents and the total number of parameters are
. (2.9.10) measurements have to be performed, five measurements for a three-terminal element.
b)
Example 2.9.1b
Consider three resistors active circuit of the T-shape structure, as presented in Fig. 2.9.6a.
After zeroing the voltage source, passive subcircuit of Fig. 2.9.4a is obtained and elements of conductance matrix can be found (2.9.8).
Short-circuit currents can be designated from Fig. 2.9.4b circuit, and they are
(2.9.14a) (2.9.14b)
Fig. 2.9.6 T-shape subcircuit (Example 2.9.1) and circuit for calculation of J
OTHER MATRICES OF MULTI-TERMINAL ELEMENT
A multi-terminal element of terminals or a multi-port of ports can be described by equations that express n external variables by other n external variables. Than, total of
(2.9.15) descriptions are possible.
The conductance matrix description, that expresses external currents by external voltages, has been discussed already.
The resistance matrix description, that expresses external voltages by external currents, is the other primary way of multi-terminal or multi-port description:
(2.9.16) All other descriptions are the so called hybrid descriptions:
(2.9.17) where,
. (2.9.17a)
For three-terminal or two-port element, four hybrid descriptions can be formulated. The so called cascade matrix description (2.9.18), that expresses output variables by input variables, is frequently used.
(2.9.18)
For a resistive m-terminal circuit, both conductance matrix and resistance matrix are symmetrical. Hybrid matrix is non-symmetrical, however it also contains M (2.9.4) linearly independent parameters. In general, having one matrix description the other one can be found.
The following relationships between the conductance matrix description and the resistance matrix description can be given:
J
Example 2.9.1a cont.
The circuit of Fig. 2.9.6a can be described by the resistance matrix:
(2.9.20)
The parameters can be designated from Fig. 2.9.7a circuit. They are
(2.9.21a)
(2.9.21b)
Fig. 2.9.7 (Example 2.5.1) Circuits for calculation of: a) R11, R21, b) E
The resistance , resistance seen from terminals 2-3 when 1-3 are opened, can be calculated in the similar way as :
(2.9.21c)
The open-circuit voltages can be designated from Fig. 2.9.7b circuit.
(2.9.21d)
ANALYSIS OF CIRCUITS WITH MULTI-TERMINAL ELEMENT(S)
Consider a circuit built of two-terminal elements, with one three-terminal element extracted, as presented in Fig. 2.9.8.
1
Fig. 2.9.8 Circuit with three terminal element extracted
The circuit nodal equations are:
(2.9.22) where: G is the circuit conductance matrix,
is vector of the circuit source currents at its nodes, internal nodes and nodes 1,2,3 I is vector of the m-terminal element currents, supplemented by zeroes:
. (2.9.22a)
A general multi-terminal element equations (2.9.9) are
*
*
*
* G V J
I . (2.9.23)
For , taking into account that:
(2.9.24) a three-terminal element equations are
I
(2.9.25)
Taking into account these equations in (2.9.22), the circuit nodal equations can be formulated.
The strategy can be generalized into arbitrary number of multi-terminal elements case and the following algorithm can be formulated.
Algorithm 2.9.1 – Nodal analysis of circuits with m-terminal element(s)
1. Disconnect (extract) multi-terminal elements, find G and of the obtained subcircuit.
2. Designate and of all multi-terminal elements; .
3. Overlap matrices onto matrix G and vectors Ji onto vector , for .
Example 2.9.2
All conductances and sources of the circuit presented in Fig. 2.9.9 are given, as well as the conductance matrix and the short-circuit currents of the three-terminal active element – its reference node is denoted by an asterisk. Find the circuit nodal equations.
Circuit nodes are numbered: 0,1,2,3,4 – node 0 is the reference one. Three-terminal element nodes are numbered: (1), (2), (3). Then, nodal equations (2.9.26) can be formulated.
Fig. 2.9.9 Circuit for Example 2.9.2
1 (1) 2 3 (2) 4 (3) (2.9.26)
1. What is the total number of parameters that characterize an active four-terminal element.
2. Find matrix R and vector E.
Fig. P.2.9.2
3. The three-terminal passive element is characterized by the following resistances:
, . Find an ideal voltmeter indication.
Fig. P.2.9.3
4. A three-terminal element is characterized by the resistances , and the open-circuit voltages: . An ideal ammeter is connected between terminals 1 and 3, an ideal voltmeter is connected between terminals 2 and 3. Find their indications.
5. Find matrix G and vector J for the subcircuits shown in Fig. P.2.9.1.
6. Find matrices R, G and C of the two-port shown. Find the expressions and values for
Fig. P.2.9.6
7. Find matrix C of the passive two-port for which in .
8. If the passive two-port shown has the conductance matrix in mS, what are the indications of ideal meters ?
Fig. P.2.9.8
12 21 1
11 R R
R
2
R22 E1 1V, E2 2V
5 , 2 20 , 3 10
1 R R
R
2 2
2 R 1
20 5
5 G 10
10 V V
A