1

Figure 1: Basic 2-port network.

IV. Linear N-Port Networks

ECE 420 — Fall 2001 — Prof. Frey

Thus far, we have studied the fundamental characterization of linear circuits. Such an

approach is of particular value in understanding the intrinsic properties of circuits. However, it is

common in Electrical Engineering to consider networks as having inputs and outputs that are applied

and measured at ports, respectively. As a result, an entire theory of electrical networks has been

developed for the characterization of electrical networks viewed as N-ports. We will now explore

some of this theory and use it to say some interesting things about electronics.

2-Ports--Modeling

By far the most common N-port is that

where N=2--namely, the 2-port. Figure 1 depicts a

general 2-port network. Each port of the network

comprises 2 wires across which we may measure a

voltage as indicated by V and V , corresponding1 2

to ports 1 and 2, respectively. We also define

corresponding port currents, I and I , flowing into1 2

the ports at the positive voltage reference. This is,

of course, the natural extension of a 1-port where only one pair of wires is considered. It is worth

noting that in general the wires shown explicitly to describe the ports correspond to nodes within the

linear 2-port network and, therefore, the ports could share the same internal nodes. For example,

ground could be common to both ports.

The characterization of 2-ports is a natural generalization of that for 1-ports. While a 1-port

network can be described as a single branch in some electrical network, a 2-port can be represented

as a pair of branches. The interesting difference, however, is that the pair of branches associated

with a 2-port may be coupled. Recall that Thevenin’s Theorem tells us that there is in general an

affine relationship between voltage and current is a one port--that is, the port voltage (current) is a

linear function of the port current (voltage) plus a source term due solely to internal sources. The

V ' Z I

V ' /0000/0000

V1

V2

; I ' /0000/0000

I1

I2

; Z ' /0000/0000

Z11 Z12

Z21 Z22

2

(1)

port voltage and current are not intrinsically functions of any other voltages and currents outside of

this 1-port. On the other hand, in a 2-port, a port voltage or current may be a function of not only

that port’s current or voltage, but also the other port’s voltage or current, in addition to any internal

sources.

Another natural extension of 1-port ideas to 2-ports is that regarding power. The port current

and voltage references in 1-ports is chosen so that the product of the port voltage and current yields

the power dissipated by, or within, the 1-port. By choosing the referencing similarly for 2-ports we

obtain a natural generalization. Specifically, the sum of the products of voltage and current at each

of the ports equals the power dissipated by, or within, the 2-port. Hence, V I + V I = P, where P1 1 2 2

denotes the power dissipated in the 2-port. Notice that, unlike in 1-ports, it is possible for the power

to be zero, despite the fact that the port voltages and currents may all be nonzero. This allows for

some very useful possibilities.

Before continuing, let us note that it is customary in the context of the following discussion

to assume that N-ports contain no internal independent sources. Such a restriction is not necessary,

but allows us to focus on the intrinsic coupling properties of 2-ports without the clutter due to

possible internal sources. Besides, due to the superposition principle, we can always turn off internal

sources for the present analysis and turn them on again at a later point in the analysis if we want to

determine their effect on the system. This will be done later, but for now let us assume that we can

characterize 2-ports by purely linear (as opposed to affine) relations, making the port voltages and

currents dependent upon only one another.

There are many choices for the possible relations between port variables. One popular choice

is the so-called z-parameter model. In this case the port voltages are expressed as functions of the

port currents. These relations are typically expressed in matrix form as shown below:

The elements of the Z matrix are referred to as the z-parameters for the 2-port, and they explicitly

I ' Y V ; Y ' /0000/0000

Y11 Y12

Y21 Y22

V1 ' (R1%R2) I1 % R2 I2 ; V2 ' R2 I1 % R2 I2

Y Z ' /0000/0000

(R1%R2) R2

R2 R2

3

(2)Figure 2: Example 2-port network.

(3)

tell how the port voltages are controlled by the port currents. Clearly, the z-parameters each have

units of impedance, which explains the choice of letters. Note that if Z and Z are both zero then12 21

the 2-port reduces to a pair of uncoupled 1-ports, which are simply impedance elements since

internal sources are assumed to be zero. It is the cross coupling expressed through nonzero values

for the off diagonal elements, Z and Z , that give a 2-port its character, as we shall see later.12 21

An alternative characterization for a

2-port is given by its so-called y-parameter

representation. In this case the port currents

are given as linear functions of the port

voltages. Such a characterization is given

mathematically as,

In this case, each y-parameter has units of admittance,

explaining the choice of letters. Again, it is the nonzero off diagonal elements that give a 2-port its

character. Looking at (1) and (2), it seems clear that there must be a simple relationship between the

z-parameter matrix, Z, and the y-parameter matrix, Y. Specifically, Y must be the inverse of Z. This

is indeed the case. Anytime that both z- and y-parameter descriptions exist for a 2-port, the Z and

Y matrices will be inverses of one another.

Let us consider an example to help clarify the discussion. Consider the 2-port network of

Figure 2, where current sources have been attached to the ports for the purpose of determining the

network’s z-parameters. It is a simple matter to solve for the port voltages in terms of the port

currents which are necessarily equal to the applied current sources. The result of these calculations

is given below.

I1 '1R1

V1 &1R1

V2 ; I2 ' &1R1

V1 % (1R1

%1R2

)V2

Y Y ' /0000/0000

G1 &G1

&G1 (G1%G2)

/0000/0000

V1

I2

' H /0000/0000

I1

V2

' /0000/0000

h11 h12

h21 h22

/0000/0000

I1

V2

4

(4)

(5)

The application of voltage sources instead of current sources to the 2-port of Figure 2 allows us to

solve for the port currents in terms of the applied port voltages, yielding y-parameters as follows:

It is a simple matter to see that Z and Y in (3) and (4) are inverses of one another. The only time this

property will not technically hold is when the 2-port in question fails to possess both z- and y-

parameter characterizations. This could happen, for example, if R were replaced by a short in the1

circuit of Figure 2. In this case the z-parameters exist and are given by the result in (3) with R = 0.1

However, the z-parameter matrix is now singular and, hence, its inverse fails to exist. But inspection

of the new circuit reveals that with R replaced by a short the port voltages are now forced to be1

equal so that a pair of independent port voltages may not be specified. This means that y-parameters

may not be found for such a network, which is suggested by the lack of an inverse for the z-

parameter matrix. The dual situation occurs if the resistor, R , is replaced by an open circuit. Now,2

y-parameters may be found, but z-parameters may not, since the port currents are no longer

independent. This fact is anticipated by the fact that the y-parameter matrix is singular this time.

Another popular characterization for 2-ports is the so-called h-parameter model, where now

one port voltage, V , and one port current, I , are assumed to be dependent upon the remaining port1 2

variables--namely, I and V . In this case, the general form of the 2-port equations is as given below.1 2

Equation (5) defines the so-called h-parameters for a 2-port. These may be found for the network

of Figure 2 by attaching a current source at port 1, forcing the current, I , and a voltage source at port1

2, forcing the voltage, V . The complementary variables may be easily computed with the result,2

V1 ' R1 I1 % V2 ; I2 ' & I1 %1R2

V2

Y H ' /0000/0000

R1 1

&1 G2

/0000/0000

I1

V2

' G /0000/0000

V1

I2

' /0000/0000

g11 g12

g21 g22

/0000/0000

V1

I2

5

(6)

(7)

As might be expected, the h-parameter matrix, H, may be computed using either the z- or y-

parameter matrices; however, the calculations are a little more messy.

The last of this group of 2-port characterizations is given by the so-called g-parameter model,

which is the complement to the h-parameter characterization. Specifically, for g-parameters, we

have,

Observe that the g-parameters must each have different units, since they do not all relate one type

of quantity--for example, a current--to another type--for example, a voltage. This is the case for h-

parameters as well. Comparing the h- and g-parameter definitions of (5) and (7), it becomes clear

that the g-parameter matrix, G, must be the inverse of the h-parameter matrix, H. As a result, we can

find the g-parameters by either attaching a voltage source to port 1 and a current source to port 2, and

computing the complementary variables, or by inverting H. While g-parameters have been defined

for completeness, they find little use in Electrical Engineering. Historically, z-, y-, and h-parameters

have been used almost exclusively for practical circuits.

Another benefit of the 2-port characterizations introduced thus far is in their ability to suggest

equivalent circuits for 2-ports. In particular, just as the affine relation between port current, port

voltage, and internal sources suggest equivalent circuits for 1-ports--namely, Thevenin and Norton

equivalents--so do the different 2-port characterizations given above suggest equivalent circuits. To

see this, consider the z-parameter characterization for a 2-port given in (1). Writing out the z-

parameter equations individually yields,

V1 ' Z11 I1 % Z12 I2 ; V2 ' Z21 I1 % Z22 I2

6

(8)

Figure 3: Z-parameter 2-port model.

Figure 4: Y-parameter model for a 2-port.

A reasonable interpretation for the equations in (8) is that each port voltage may be found by adding

two voltages. These two voltages are given by an impedance times the respective port current and

a transimpedance times the other port current. Electrically, this is equivalent to an impedance

element carrying the port current in series with a current controlled dependent voltage source. The

circuit of Figure 3 shows this idea. This is

an equivalent circuit that may be used to

model any 2-port that possesses a z-

parameter characterization. For example,

by using the z-parameters given in (3) in the

2-port model of Figure 3, we obtain the

equivalent z-parameter model for the circuit

of Figure 2.

Following similar logic to that

above in finding the z-parameter model, we may determine the y-parameter model for a 2-port.

Specifically, by writing out the y-parameter characterization given in (2) in a way analogous to that

shown in (8), we may observe that each of the port currents is given by the sum of two separate

currents. These two currents are given by an admittance times the respective port voltage and a

transadmittance times the other port

voltage. Because of this interpretation, we

may derive an equivalent circuit at each

port of a 2-port to be the parallel

combination of an admittance and a voltage

controlled dependent current source.

Putting these ideas together yields the y-

parameter model for a 2-port shown in

Figure 4.

It is interesting to compare the

/0000/0000

V1

I1

' /000 /000A B

C D/0000

/0000V2

& I2

; T / /000 /000A B

C D

7

Figure 5: h-parameter model for a 2-port.

(9)

models of Figures 3 and 4. Notice that the z-parameter model resembles a pair of coupled Thevenin

equivalent circuits. On the other hand, the y-parameter model of Figure 4 resembles a pair of

coupled Norton equivalent circuits. It is

then a simple matter to understand that an

h-parameter model is a hybrid of the z- and

y-parameter models had by using a

combination of Thevenin and Norton type

circuits at the ports. In particular, one can

easily show that the h-parameter model for

a 2-port is given by the circuit in Figure 5.

Clearly, the g-parameter model is that

circuit with the Thevenin type circuit at port 2 and the Norton type circuit at port 1. Having found

the various 2-port equivalent circuits, one may freely employ them in replacing any linear 2-port

(with internal sources inactive) for the purpose, for example, of simplifying some larger system.

There are other types of models which have been used to characterize 2-ports. One in

particular that has found use in cascading networks is the so-called transmission matrix, which is

also referred to as the ABCD parameter, characterization. Unlike the z-, y-, h-, and g-parameter

models, the transmission matrix characterizes the 2-port as a relationship between the ports-that is,

the port 1 variables are consided as dependent upon the port 2 variables. Specifically, we have,

T is defined to be the transmission matrix, composed of ABCD-parameters. Notice that, in this

characterization, the output (port 2) controls the input (port 1), and that the output port current (-I )2

is measured leaving port 2. With this slight change in perspective it is easy to calculate the

composite T matrix of a cascade of 2-ports by multiplying the T matrices for each of the individual

networks.

One other popular 2-port characterization is that involving the so-called S-parameters. This

8

Figure 6: 2-port driven by a source at port 1.

model for characterizing networks is best suited to high frequency networks where the propagation

and reflection of signals must be taken into account. Specifically, S-parameters relate the reflected

signals at the ports of the network to the incident signals. All incident and reflected signals will be

of the same type--i.e., voltages or currents--in this type of characterization. When considering

lumped networks (as we have been doing), the S-parameter characterization becomes unwieldy and

a bit contrived. The best appreciation of S-parameters, especially in the context of lumped networks,

is had by thinking in terms of the power delivered to the various ports of a network by external

sources, where each source has some nonzero source impedance.

Reciprocity

Having discussed the basics of 2-port representations, it is of interest to note an important

generic classification typically given to them. Namely, 2-ports are typically referred to as being

either reciprocal or non-reciprocal. The idea of reciprocity is an old one and arises from a very

simple, and sometimes very useful,

property that 2-ports may possess. In order

to understand this, consider the 2-port

network of Figure 1, where we have

attached a current source at port 1, as

shown in Figure 6. We may calculate the

response, V , due to the applied current2

source that specifies I . Given the z-1

parameters for the 2-port, and recognizing that I = 0 (by inspection), it is a simple matter to verify2

that V = Z I . Hence, Z is the transfer function from the source, I , to the response, V . Now2 21 1 21 1 2

suppose that we were to remove the current source from port 1 and connect it to port 2, thereby

specifying I . This time port 1 would be open, and if we were to find V in response to this source,2 1

we would find that V = Z I . Now Z is the transfer function from the source, I , to the response,1 12 2 12 2

V . Clearly, if Z = Z , then the network response voltage to the current source will be the same1 12 21

for both cases. This result seems quite unusual to most people who see it for the first time, because

networks that are quite asymmetrical looking often have the property that Z = Z . For example,12 21

9

Figure 7: 2-port driven by a voltage source atport 1.

the circuit of Figure 2 possesses this property as shown in (3), despite the fact that it looks different

at the two ports. Networks possessing this property--namely that, Z = Z --are called reciprocal12 21

networks.

Considering the fact that the different 2-port characterizations given earlier must all be

related, reciprocity must specify more than just the properties above. For example, we have already

observed that the y-parameter matrix is the inverse of the z-parameter matrix. Whenever a 2-port

is reciprocal, its z-parameter matrix must be symmetric, since the off-diagonal elements, Z and Z ,12 21

are equal. A basic property of matrices is

that the inverse of a symmetric matrix is

symmetric. Hence, the off-diagonal

elements of the y-parameter matrix, Y and12

Y , must also be equal. (We assume, of21

course, that both the z- and y-parameter

matrices exist.) Just as the equality of Z12

and Z implies a circuit property, so does21

the equality of Y and Y . To see this,12 21

consider the network of Figure 7, where a 2-port is being driven by a voltage at port 1, and a short

has been placed across port 2. Observe that the port 1 voltage is set by the source and the port 2

voltage is zero. We may look at the response, I , to the source making V . It is a simple matter to2 1

verify that I = Y V , making Y the transfer function from input, V , to output, I . Now suppose2 21 1 21 1 2

that we replace the source at port 1 with the short and the short at port 2 with the voltage source.

This time V is equal to the voltage source and V is equal to zero. Now the transfer function from2 1

the input, V , to the response, I , is given by Y . Therefore, if the 2-port is reciprocal, then these2 1 12

transfer functions must be equal and the response measured in the two cases is the same.

We can say something about the h- and g-parameters regarding reciprocity as well. As

suggested above, the h- and g-parameters may be derived from the z- or y-parameters (assuming they

exist.). In particular, it can be easily proven that,

h12 'z12

z22

; h21 ' &z21

z22

; g12 'y12

y22

; g21 ' &y21

y22

10

(10)

Figure 8: 2-port driven by current sources.

From these relations it is clear that a reciprocal network will have h- and g-parameters obeying the

constraints, h = -h and g = -g .21 12 21 12

It is reasonable at this point to ask what kind of circuits are going to be reciprocal. With the

theoretical tools developed so far, we are in a position to answer this question. As suggested earlier,

let us view a linear 2-port as being a pair of branches possessing a joint constitutive law given by one

of the 2-port characterizations discussed above. To make things less abstract, assume that the pair

of branches are characterized by a known z-

parameter matrix. Let us assume that the 2-

port being considered is driven by a pair of

current sources, as shown in Figure 2, for

example. We have specified a network

having 4 branches, two of which are current

sources, and two of which comprise the

linear 2-port. The situation is shown

schematically in Figure 8.

Now let us consider the response of this network to the source at port 1, labeled I , measured3

as the voltage across the current source at port 2, labeled I . Despite the orientation of I , however,4 4

let us measure this voltage from top to bottom, making it equal to the voltage labeled V in Figure2

8. We assume that I is set to zero during this measurement. As usual in a linear system, this4

response will be some transfer function times the input source, I . Now let us recall the adjoint3

network concept discussed earlier, and apply it to this system. Specifically, the adjoint to the circuit

of Figure 8 is that circuit having the same topology but with the input and output reversed. That is,

we now assume the input to be applied via I and the output to be measusred as the voltage across4

I from top to bottom, equal to V , as can be seen from Figure 8. It is a straightforward matter to3 1

verify that the adjoint network for this very simple circuit is that network where the coupled branches

have a joint constitutive law that is the transpose of the original. This follows directly from the fact

/0000/0000

V1

V2

' /0000/0000

sL1 sM

sM sL2

/0000/0000

I1

I2

; M ' k L1 L2

11

Figure 9: Ideal transformer representation.

(11)

that the modified nodal matrix of an adjoint network must be the transpose of that of the original.

Therefore, if the z-parameter matrix of the 2-port in the circuit of Figure 8 is symmetric, then it is

self-adjoint, and the response measured using I as the source will match that when using I as the4 3

source in the above scenario. In summary, we have proven that a self-adjoint network is reciprocal,

at least for the case of a z-parameter model. Following the above reasoning, you can prove with a

little extra effort that self-adjoint networks are reciprocal in general in the context of 2-port analysis.

Moreover, this idea of self-adjoint networks shows how to extend the idea of reciprocity to N-port

networks in general.

Special 2-port Networks

Having discussed 2-ports in the abstract, we are now ready to consider the more well known

specific types of 2-ports. We begin with the most widely used reciprocal 2-port--namely, the

transformer. A transformer is simply a pair

of coupled inductors and must, as a dierct

consequence of the physics, be a reciprocal

network. Figure 9 shows the basic idea of

a transformer. Of course, the transformer is

only that part of the circuit within the

dotterd lines. In a transformer, we assume

that the inductors, L and L , are coupled1 2

via some form of flux linkage so that there

is a mutual inductance that can be measured

between them, given by M. The usual dot notation has been used to indicate the orientation of the

flux linkage between the inductors. The equations relating voltage and current in the ideal

transformer of Figure 9 are given by,

V1 ' NV2 ; I2 ' &NI1 Y /0000/0000

V1

I2

' /000 /0000 N

&N 0/0000

/0000I1

V2

12

(12)

where k is defined as the coupling coefficient which may take on values from 0 to 1. Typically, k

is a number quite close to one in practice--for example, in the range from 0.95 to 0.999--and is

assumed equal to 1 for an ideal transformer. Notice that the transformer has been characterized with

a z-parameter representation. This is standard practice and quite convenient in light of the

constitutive laws specified by the physics of a transformer.

It is common, however, to use an even simpler description for an ideal transformer. This

description takes into account the so-called turns ratio, N, of the transformer. Specifically, suppose

that for the transformer of Figure 9, the primary winding--that is, the coil of wire comprising L --has1

N times as many turns as the secondary of the transformer--that is, the coil comprising L . Then we2

have the common idealized description for a transformer given in (12) below.

This characterization of a transformer is certainly idealized, since it predicts no frequency

dependence whatsoever and, in particular, that DC signals may be coupled between the ports, which

is, of course, impossible using a physical transformer. Nevertheless, the characterization given in

(12) is quite useful is understanding the operation of real transformers as an approximation. Note

that the representation given in (12) is an h-parameter representation, unlike that of (11). This is

because the ideal relations between current and voltage using the turns ratio do not permit either a

z-parameter or a y-parameter representation. Also notice that the h-parameter matrix is skew-

symmetric, which implies that an ideal transformer is a reciprocal 2-port. Finally, it is important to

observe that an ideal transformer is a lossless network--that is, the total power dissipated in the 2-

port is identically zero. This is easily determined by using the relations in (12) to compute P = V1

I + V I = V I + (V /N) (-NI ) = 0. 1 2 2 1 1 1 1

It is an interesting exercise to find the range of inductance and frequency that one must

assume in the actual transformer representation of (11) to get the approximation of (12). The

problem is not well posed unless one considers the impedance at the primary and the secondary of

the transformer that would result from attaching a source, having a nonzero source impedance, and

I2 ' gV1 ; I1 ' &gV2

I ' /0000/0000

I1

I2

' /000 /0000 &g

g 0/0000

/0000V1

V2

' YV

V ' /0000/0000

V1

V2

' /000 /0000 &r

r 0/0000

/0000I1

I2

' Z I ; r ' 1/g

13

(13)

(14)

(15)

a load. This would make a good homework problem. Another interesting perspective on the ideal

transformer described by (12) is had by considering whether a circuit can be created to implement

these 2-port relations. Even though coupled inductors could never truly implement the equations of

(12), there is no fundamental reason why a circuit could not be designed to do the job. Perhaps you

could think of how to do this (Another good homework problem.). The discussion below will offer

some possibilities.

Gyrators

We now turn to another special 2-port, called a gyrator, that has received much attention over

the years. Unlike the transformer, this 2-port is non-reciprocal, although it is anti-reciprocal, which

is a special property in its own right. In the context of the above, a gyrator is simply a special case

of a linear 2-port. While there are many possible 2-port descriptions, the most common way of

writing the basic equations relating the port parameters in a gyrator is as follows:

Using these equations it is simple to write the y-parameter 2-port description for a gyrator as,

This suggests that a gyrator can be implemented with voltage controlled current sources, having

gains of g and -g, respectively. By inverting the relations in equation (14), one obtains a gyrator

formulation based upon current controlled voltage sources. Specifically,

P ' V T I ' V1 V2/0000

/0000I1

I2

' V1 V2 /000 /0000 &g

g 0/0000

/0000V1

V2

' V2 V1 & V1 V2 ' 0

14

(16)

Figure 10: The gyrator circuitsymbol.

Figure 11: Gyrator with a load.

where the z-parameter matrix, Z, is just the inverse of the y-parameter matrix, Y. Since it is most

convenient to realize practical voltage controlled current source networks, as opposed to current

controlled voltage source networks, the formulation in equation (14) is generally preferred. For

theoretical purposes, of course, both formulations are useful. Using equation (14) it is simple to

show that a gyrator is a lossless electrical network. Specifically,

The fact that gyrators are, in theory, lossless makes them

attractive in filter synthesis. Recall that ideal transformers

are also lossless.

Let us take a closer look at the properties which

gyrators possess that make these circuits interesting for use

in electronics. A first property is that these 2-ports are not

reciprocal networks, since their y-parameter matrices are

not symmetric. In fact, these matrices are skew symmetric--

that is, they equal the negative of their transpose. It is well

known to circuit theorists that non-reciprocal networks cannot be realized with only passive

components--that is resistors, capacitors, and

inductors. This means that gyrators are strictly

active networks that must, therefore, be realized

with active components, such as transistors or

operational amplifiers. 2-port gyrators have been

given their own circuit symbol which is shown in

Figure 10. The gyration constant, g, is built into

the symbol.

Perhaps the most important property of a gyrator is its ability to transform admittances into

impedances. Specifically, when an admittance is connected to one port of a gyrator, the impedance

Zload 'V2

&I2

; Zinput 'V1

I1

'I2 /g

&gV2

'1

g 2

1Zload

'1

g 2Yload

Zinput '1

g 2sC ' sLeq ; Leq '

C

g 2

15

(17)

(18)

Figure 12: RLC bandpass filter.

Figure 13: Bandpass filter using a gyrator.

looking into the other port is exactly a scaled version of that admittance. The derivation can be

accomplished with the help of Figure 11, where Z is the impedance attached to port 2, Y is itsload load

reciprocal--that is, the admittance attached to port 2--and Z is the impedance seen looking intoinput

port 1. We have,

The gyration constant, g, determines the scale factor, but the nature of the input impedance is

determined by the admittance attached to port 2. Therefore, if a capacitor, C, is attached to port 2,

then we have,

This simple relation explains the vast majority of

the gyrator’s popularity in electronic design. It

shows that a capacitor can be used to replace an

inductor in a circuit with the help of a gyrator.

Since inductors are rarely desirable in electronic

circuits operating below about 1GHz, this idea is

quite appealing. Capacitors and gyrators

are convenient to realize as part of

integrated circuits.

Filter synthesis based on the inductor

simulation described above is usually done

by starting with an RLC prototype, and

replacing the inductors with

capacitor/gyrator combinations. Consider

the following example of a very simple second order bandpass filter shown in Figure 12. After

replacing the inductor with a gyrator/capacitor combination, the filter is realized solely using

Yload ' jN

k'1Yk Y Zinput '

Yload

g 2' j

N

k'1

Yk

g 2

Zload ' jN

k'1Zk Y Yinput '

1Zinput

'1

Yload /g 2' g 2 Zload ' j

N

k'1g 2 Zk

16

(19)

Figure 14: Gyrator realization usingtransconductance amplifiers.

capacitors as the reactive elements, as shown in Figure 13. Furthermore, the filter can be tuned

electronically if the gyration constant can be varied electronically.

A byproduct of the property discussed above is that series and parallel circuits may be

interchanged with the help of a gyrator. Suppose one port, say port 2, of a gyrator is loaded with a

parallel combination of elements. The admittance of this combination is the sum of the admittances

of each of the elements. At the other port, port 1, the input impedance will be a scaled version of

this admittance; hence, a sum of impedances. Since the composite input impedance seen at port 1

is given by a sum of impedances, it must be equivalent to a series combination of elements.

Therefore, the gyrator converts a parallel network into a series network. Using similar logic, it

becomes clear that a series network connected to port 2 will be reflected as a parallel network

looking into port 1. These results are summarized below.

The realization of gyrators in

electronic form is quite simple; however, as

usual, different circuit realizations are

preferable to others, depending on the

application. To begin, consider the

simplest generic realization comprised of a

pair of transconductance amplifiers, as

shown in Figure 14. Each transconductance

amplifier is assumed to have infinite input

and output impedance, with an output

current equal to the transconductance, G = g, times the input voltage applied between the + and -m

terminals. The circuit shown in the figure satisfies the basic 2-port relations for a gyrator, given by

equation (14).

Z 'V2

&I2

; &Z 'V1

I1

V1 ' kV2 ; I2 ' kI1 YV1

I1

'kV2

I2 /k' &k 2

V2

& I2

' &k 2 Z

17

(20)

(21)

The circuit of Figure 14 does not implement the most general form of a gyrator, since both

ports of the gyrator realization in the figure have ground in common. Therefore, only ground

referenced impedances may be transformed as described above. This limitation stems from the fact

that the transconductors in Figure 14 have single-ended outputs. If differential input/differential

output transconductors are used then a general “floating” gyrator realization is created--that is, a

gyrator whose ports need not be referenced in any way to ground. On the other hand, if you get

clever, you can realize a floating gyrator with a pair of grounded gyrators. Can you see how this

could be done?

Negative Impedance Converters

Now that we have introduced the gyrator, it is interesting to see what other creative 2-port

networks we might find. The gyrator, as shown above, can be used to make impedances at one port

appear differently at the other port. Specifically, a gyrator can make a capacitor look like an

inductor. A negative impedance converter (NIC) is a 2-port, as its name suggests, that can reflect

the negative of a given impedance at its opposite port. Rather than just give the definition of this 2-

port, let’s see if we can invent it. Suppose we start with the assumption that an impedance, Z,

connected to port 2 of this network will reflect an impedance of -Z at port 1. Then we have,

Suppose, for the sake of generality, that we let the port 2 current follow the port 1 current with a gain

of k, and that the port 1 voltage follows the port 2 voltage also with a gain of k. Then we have,

Clearly, if k = 1, then the impedance seen looking into port 1 is the negative of that attached to port

2. Hence, a network following the relations in (21) is an NIC. Putting this into the matrix form that

we have been using gives us this general representation for a negative impedance converter.

/0000/0000

V1

I2

' /000 /0000 k

k 0/0000

/0000I1

V2

18

(22)

Note that the above matrix representation gives the h-parameter description of this 2-port.

Furthermore, since the off-daigonal terms are equal, this is a non-reciprocal network; hence, it cannot

be realized with only passive elements. Also notice that this description is identical to the ideal

transformer except that the off-diagonal terms have the same sign. Recall that an ideal transformer

can be used to reflect different impendances as well; however, a transformer is a positive impedance

converter, in light of the current discussion. The idea of a network that creates negative impedances

is intriguing. Now the only challenge is in finding a use for such an unusual network. Certainly one

use might be to cancel the effects of a positive impedance, which is exactly what is necessary in an

oscillator circuit, for example.

As a final thought on transformers, gyrators, and NICs, notice that by cascading a pair of

gyrators or a pair of NICs, one obtains an ideal transformer. Does this give you any ideas about

possible circuit realizations for an ideal transformer?

Ideal Op Amps

Operational amplifiers certainly represent an important component in modern electronics.

While they possess many characteristics that are interesting, it usually requires an extensive look at

the nonideal properties of op amps to be able to use them effectively in practical designs.

Nevertheless, the idea of ideal op amps still is used to introduce students to the concept, and then

to help model the behavior of circuits incorporating op amps. This is because nonideal op amps can

always be modeled as being ideal op amps with surrounding components. Therefore, the study of

ideal op amps remains of interest even to the expert in the field, as well as to the circuit theorist.

One of the most intriguing features of ideal op amps is that they are 2-ports with unusually

degenerate constitutive laws. Recall that for an ideal op amp we assume that the input differential

voltage is zero and that the output voltage is finite. Actually, we assume that the output voltage is

an infinite gain times the differential input, but that the output is finite. This leads immediately to

V1 ' 0 ; I1 ' 0 ; V2 , I2 are unknown

19

(23)

Figure 15: Circuit symbols for thenullator and norator.

the assumption of zero input voltage. On the other hand, we typically assume that an ideal op amp

is a 2-port having an open circuit at port 1 and an ideal voltage controlled voltage source (VCVS)

at port 2. Thus, its input current is zero. This leaves us with the curious characterization that both

the input voltage and current for the ideal op amp are zero. Furthermore, this complete knowledge

of the input port variables is accompanied by a complete lack of knowledge of the output port

variables. Specifically, we cannot say what the output voltage or current are for an ideal op amp

solely from the knowledge of the input voltage and current (which are both zero!). The 2-port

equations for an ideal op amp are summarized below.

Despite the unusual nature of these relations they qualify as a perfectly acceptable set of constitutive

laws for a 2-port. That is, they impose exactly 2 constraints on the four network variables, which

is exactly what every other 2-port constitutive law does. Therefore, larger networks incorporating

ideal op amps admit to all of the formalism we have

developed thus far to characterize general circuits. In

addition, we can expect that linear networks including ideal

op amps will possess unique solutions, and that such

networks will have linear transfer functions and have

topological duals and adjoints.

Unlike the other 2-ports we have considered, ideal

op amps cannot be given 2-port models that use known

branches. Specifically, what branch has identically zero

voltage and current? It looks like a short and an open

simultaneously! Also, what branch has absolutely no constraints on voltage and current? Because

no such branches exist in our experience, they have been created by circuit theorists. In particular,

a branch having identically zero voltage and current is called a nullator. A branch with absolutely

no constraint on its voltage and current is called a norator. The circuit symbols for these fictitious

branches have been assigned as well and are shown in Figure 15. Using these symbols, we can

V ' (V1 ,V2 , @@@ ,VN )T ; I ' ( I1 , I2 , @@@ , IN )T

20

Figure 16: General N-port network.

(24)

describe an ideal operational amplifier as being the 2-port having a nullator at port 1 and a norator

at port 2. Such a characterization can be used to do some interesting transformations on circuits

including operational amplifiers. We discussed some of these in class.

General N-port networks

Having described 2-ports in detail, we are

now in a position to generalize the concept of a 2-

port to that of the N-port. As the name suggests, N-

ports are networks having N ports defined by pairs

of wires, across which we may measure port

voltages, and into which we may measure port

currents. The basic idea is given in Figure 16. As

in the case of 2-ports, the wires defining each port

of this network may not all connect to independent nodes within the network. For example, all of

the ports may have ground in common. Notice that the currents are all defined as flowing into the

wire attached to the positive voltage reference. This makes the definition of power delivered to the

N-port a natural extension of that for 2-ports. Specifically, the sum of the products of voltage and

current measured at all ports equals the power delivered to the N-port. To give a more concrete

mathematical basis for the discussion, let us define the port voltage and port current vectors as

follows:

The power, P, delivered to the N-port is now simply the inner product of these vectors.

We may generalize the characterizations for 2-ports given earlier is a natural way.

Specifically, the generalization of the z-parameter matrix is the open circuit impedance matrix, Z .OC

This N x N matrix of elements each having units of impedance yields the port voltage vector in terms

of the port current vector. The short circuit admittance matrix, Y , generalizes the y-parameterSC

matrix by allowing us to express the port current vector in terms of the port voltage vector.

V ' ZOC I ; I ' YSC V

21

(25)

Specifically,

We can define a hybrid matrix, H, along the lines of h-parameters by defining port vectors having

a mix of voltages and current and using H to form a relation between them. We will see this idea

below. Finally, note that we may generalize the idea of reciprocity. An N-port is reciprocal if its

open circuit impedance matrix and (assuming both matrices exist)/or its short circuit admittance

matrix are symmetric. An interesting consequence of this generalization is that all 1-ports must be

reciprocal. Another interesting result is that every N-port composed of exclusively reciprocal

elements--that is, 1-ports, reciprocal 2-ports, etc.--is itself reciprocal. Can you see how to prove this

result?

Memoryless and reactive N-ports

Let us now consider a special subset of N-port networks--namely, memoryless N-ports.

These are N-ports containing no reactive elements. Any circuit made up of only resistors, for

example, would be memoryless. In addition, any circuit containing only resistors, ideal transformers,

ideal op amps, gyrators, and NICs would also be memoryless. It should also be noted that the idea

of a memoryless N-port is not restricted to linear networks. Any nonlinear network not containing

reactive elements is also considered memoryless. (For that matter, the idea of reciprocity is not

limited to linear networks either.) Note that in general we must exclude any subnetwork with

memory, meaning that memoryless N-ports may not contain transmission lines, or any other circuitry

with delay, such as a sample and hold. Such components have not been considered thus far, so we

will not worry about them in our present discussion.

You might ask why the idea of a memoryless N-port is introduced. The answer lies in the

fact that such a network relates all port variables through purely instantaneous relations. As a result,

the N-port description, such as the open circuit impedance matrix, may be used to relate the port

variables in either the time or the frequency domain. Furthermore, a memoryless N-port has no

dynamics associated with it. In particular, it has no internal states and its complete response,

V ' Zoc I % VS ; I ' Ysc V % I S

22

(26)

measured at any of its ports, to an input applied to any of its ports will contain no transient. Note,

however, that since a memoryless N-port may contain memoryless active components, it may be

inherently unstable. A simple example would be that of a negative resistor, which constitutes a

memoryless 1-port which adds power to any other network to which it is connected. Note that

memoryless N-ports dissipate or contribute exclusively “real” power at each port.

The complementary network to a memoryless N-port is that of a purely reactive N-port. Such

an N-port contains purely reactive components. As opposed to memoryless N-ports, reactive N-ports

dissipate or contribute zero “real” power to a network. While power associated with a memoryless

N-port would be measured in Watts, power associated with a reactive N-port would be measured in

VARs. A capacitor or an inductor provides the simplest example of a reactive N-port (N=1). A pair

of coupled inductors (assuming no winding resistance or core losses) represents an interesting

reactive 2-port. It is also interesting to note that a capacitive 2-port--that is, a reactive 2-port

comprising only capacitors--may look like a pair of coupled capacitors if the internal capacitive

branches form a loop. Coupling between the ports of an N-port is evidenced by off-diagonal terms

in its open circuit impedance or short circuit admittance matrix.

As a final matter, it is sometimes useful to allow N-ports to include internal independent

sources. Up until now, we have been assuming that all internal independent sources were absent,

or at least turned off in the spirit of the superposition principle. The inclusion of sources introduces

an affine relation between port variables of the type considered in discussing Thevenin and Norton

Equivalents. To make things clearer in the discussion below, let us formally write down the N-port

voltage/current relations discussed thus far assuming the presence of internal independent sources.

We have,

In this formulation, the source vectors, V and I , are analogous to the Thevenin and Norton sourcesS S

in 1-ports. In fact, for the special case where N=1, the source vectors, V and I , are scalars exactlyS S

equal to the Thevenin and Norton sources, respectively. The hybrid formulation of these equations

will be given below.

23

Figure 17: RLC circuit.

Figure 18: Decomposed version of thecircuit of Figure 17.

State Equation Formulation

An elegant formulation for state equations in general circuits can be written using the ideas

introduced above. Suppose that we have a circuit composed of arbitrary components which includes

a number of capacitors and inductors. It is always possible to partition this network into a

memoryless (often referred to as “resistive”) N-port attached to a pair of reactive N-ports,

specifically one capacitive N-port and one inductive N-port, where the “N” will be different in

general for these reactive networks than the “N” for the main resistive network. It is further possible

to define the resistive N-port in such a way that the subset of ports attached to the capacitive N-port

is completely exclusive of the remaining ports attached to the inductive N-port. To be more specific,

let there be exactly N ports associated with the capacitiveC

N-port, and exactly N ports associated with the inductiveL

N-port. Then it follows from the above statements that it

is always possible to characterize the resistive N-port with

a description incorporating N = N + N ports.C L

Since this decomposition may seem a bit confusing,

let us consider a simple example. Figure 17 shows an RLC

circuit that we would like to decompose along the

lines of the above discussion. We will include the

capacitor in a capacitive N-port with N = N = 1.C

We will include the inductor in an inductive N-port

with N = N = 1. Finally, the resistive N-port willL

comprise the voltage source and the resistor and

will have N = 2 ports. The RLC circuit has been

redrawn in Figure 18 to reflect these definitions.

Note that the two ports of the resistive N-port are in

parallel. This is allowed since one of these ports

has a capacitor attached, and the other has an

inductor attached. Note that while the port voltages

of the resistive 2-port will be equal, the port

/0000/0000

V1

I2

' H /0000/0000

I1

V2

% VS ; H ' /000 /0000 1

&1 G; VS ' /000 /000

0

&GVS

V1 ' VL ' Lddt

IL ' &Lddt

I1

I2 ' & IC ' &Cddt

VC ' &Cddt

V2

/0000/0000

V1

I2

' & /000 /000L 0

0 Cddt

/0000/0000

I1

V2

' H /0000/0000

I1

V2

% VS

ddt

/0000/0000

IL

VC

' /000 /0001 /L 0

0 &1/C/000 /000

0 1

1 G/0000

/0000IL

VC

% /000 /0001 /L 0

0 &1/CVS

24

(27)

(28)

(29)

(30)

currents will not in general be equal. Continuing with the example, let us characterize the various

N-ports defined. Suppose we choose to characterize the resistive N-port by attaching a current

source to the port (call it port “1") where the inductor is connected and by attaching a voltage source

to the port (call it port “2") where the capacitor is connected. Defining these sources to be I and V1 2

, respectively, we may easily write an h-parameter characterization for the resistive 2-port, where it

will be augmented by a term due to the internal independent source. For this case we have,

Now let us continue by writing the characterization for the reactive 1-ports. Consulting Figure 18

to make sure we get the right orientation for currents, we get,

Putting together the above results yields the following set of differential equations:

These equations may be easily solved for the derivatives of I and V , yielding state equations whose1 2

state variables are I and V . Note that these variables are just the negative of the inductor current1 2

and exactly the capacitor voltage, respectively. Alternatively, we may write these equations directly

in terms of inductor current and capacitor voltage, with the result,

VL '

/000000000000000

/000000000000000

VL1

VL2

!

VLNL

' Lddt

/000000000000000

/000000000000000

IL1

IL2

!

ILNL

' Lddt

I L ; I C '

/000000000000000

/000000000000000

IC1

IC2

!

ICNC

' Cddt

/000000000000000

/000000000000000

VC1

VC2

!

VCNC

' Cddt

VC

25

(31)

It is a simple matter to verify that these are the same equations that would result from our earlier

approach to getting the state equations.

Having seen this example, it is a fairly straightforward matter to decide how to extend the

approach to the general case. Using the preliminaries introduced above, let us suppose that a

collection of inductors are connected to the first N ports of a resistive N-port. This collection ofL

inductors may include couplings and inductor cutsets within it. In any event, it constitutes a reactive

(inductive) N-port, where N = N . Let us further suppose that a collection of capacitors, some ofL

which may form loops, are connected to the last N ports of the resistive N-port. This collection ofC

capacitors constitutes a reactive (capacitive) N-port, where N = N . (Please note: In class, IC

attached the capacitive N-port to the first set of ports. There is no problem in doing this, of

course, but it will cause some confusion in comparing to your notes if you miss this point. I

chose to do it this way when I originally wrote this section of the notes because I thought it

flowed better.) Let us define the port relations associated with these reactive multi-port networks

in the following way, where all port variables are defined with the usual relative reference

orientations:

In this case, L and C are N x N and N x N matrices, respectively. Off-diagonal terms in theseL L C C

matrices will be due to internal coupling, inductor cutsets, and capacitor loops. We now continue

by looking at the resistive N-port to which these reactive multi-ports are connected. Without loss

of generality, we may assume that each of the port voltages of the resistive N-port is oriented to

match the voltage orientation of the respective ports of the reactive multi-port networks. Then, each

of the port currents of the resistive N-port will be equal to the negative of the respective port currents

of the reactive multi-port networks. Assuming that we characterize the resistive N-port using a

hybrid model where each of the first N ports are driven by current sources and each of theL

/0000000000000000000000000000000000000

/0000000000000000000000000000000000000

V1

V2

!

VNL

INL%1

INL%2

!

IN

'/000000

/000000VL

& I C

' H

/0000000000000000000000000000000000000

/0000000000000000000000000000000000000

I1

I2

!

INL

VNL%1

VNL%2

!

VN

% VS ' H/000000

/000000& I L

VC

% VS

/000000/000000

VL

& I C

' /0000/0000

HLL HLC

HCL HCC

/000000/000000

& I L

VC

% VS

/000000/000000

VL

I C

' /0000/0000

&HLL HLC

HCL &HCC

/000000/000000

I L

VC

% /000 /0001 0

0 &1VS ; /000000

/000000VL

I C

' /000 /000L 0

0 Cddt/000000

/000000I L

VC

26

(32)

(33)

(34)

remaining N ports are driven by voltage sources, we have the general description given below.C

In order to deal with the minus signs more carefully, let us rewrite the above hybrid characterization

showing a rather obvious partitioning of the H matrix in consideration of the partitioning of the port

voltage/current vectors. Specifically, we have,

Now let us rewrite and put together all of the multi-port equations above in a way that allows them

to be easily combined in a final step. This is done below.

In the formulation above, the “0"s and the “1"s appearing in the matrices represent appropriately

dimensioned blocks of zeros or identity matrices, respectively. Also notice that the capacitive and

inductive multi-port networks have been combined into a single reactive N-port, characterized by

a block diagonal N x N matrix. Combining the resistive and reactive N-port equations above easily

yields this general state space representation for the overall system:

ddt/000000

/000000I L

VC

' /0000/0000

L &1 0

0 C &1/0000

/0000&HLL HLC

HCL &HCC

/000000/000000

I L

VC

% /0000/0000

L &1 0

0 &C &1VS

27

(35)

This formulation is elegant for several reasons. First, it shows how the state equations for

a general circuit are related to its resistive and reactive N-port characterizations. Second, it naturally

handles the problem of determining the correct number of state variables, regardless of the

topological nature of the network. This is because the definitions of the capacitive and inductive

multi-ports can only be made in ways that specify the right number of independent ports. This is

easily seen by observing that there must always be enough ports to fully characterize the connectivity

of the reactive and resistive multi-ports. However, there may never be too many reactive ports

defined without precluding a proper characterization of the reactive multi-ports.

Another benefit of the formulation above is that it provides excellent insight into the structure

of the state matrix for general circuits. Notice that if the resistive N-port is reciprocal, its H matrix

must be symmetric or skew symmetric on a block basis--that is, H and H are symmetric and HLL CC LC

= -H . This is the natural generalization of our earlier result, and may be easily proven from theCLT

fact that the open circuit impedance matrix and the short circuit admittance matrix of a reciprocal

N-port must be symmetric. All purely reactive N-ports are reciprocal, which follows directly from

the fact that they are self-adjoint. Therefore, the matrices, C and L, defined above are symmetric.

It now follows immediately that the state matrix of an arbitrary passive RC or RL network must be

symmetric and, coupled with some basic passivity arguments--that is, that Z and/or Y must beOC SC

“positive definite”-- its eigenvalues must be real. This proves that no passive RC or RL filter may

have complex poles. Either there must be active elements in the network, yielding an H matrix with

asymmetric blocks on the diagonal, or the network must have both capacitors and inductors. Until

now, we could not easily prove this fundamental circuit design result.