IV. Linear N-Port Networks


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IV. Linear N-Port Networks

  1. 1. 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 V1 and V2 , corresponding to ports 1 and 2, respectively. We also define corresponding port currents, I1 and I2 , flowing into Figure 1: Basic 2-port network. 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 1
  2. 2. 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, V1 I1 + V2 I2 = P, where P 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: ¯ ¯ V ' Z I V ' /0 /0 ; I ' /0 /0 ; Z ' /0 /0 V1 I1 Z11 Z12 (1) 0 0 00 I 00 00 Z Z 00 00 V2 00 ¯ ¯ 0 2 0 0 21 22 0 The elements of the Z matrix are referred to as the z-parameters for the 2-port, and they explicitly 2
  3. 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 Z12 and Z21 are both zero then 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, Z12 and Z21 , that give a 2-port its character, as we shall see later. 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, ¯ ' Y V ; Y ' /0 /0 Y11 Y12 00 Y Y 00 ¯ 0 21 22 0 I (2) Figure 2: Example 2-port network. 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. V1 ' (R1 % R2) I1 % R2 I2 ; V2 ' R2 I1 % R2 I2 Y Z ' /0 /0 (R1 %R2) R2 (3) 00 R2 000 0 R2 3
  4. 4. 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: 1 1 1 1 1 I1 ' V1 & V2 ; I2 ' & V1 % ( % ) V2 R1 R1 R1 R1 R2 (4) Y Y ' /0 /0 G1 & G1 00 &G (G %G ) 00 0 1 1 2 0 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 R1 were replaced by a short in the circuit of Figure 2. In this case the z-parameters exist and are given by the result in (3) with R1 = 0. However, the z-parameter matrix is now singular and, hence, its inverse fails to exist. But inspection of the new circuit reveals that with R1 replaced by a short the port voltages are now forced to be 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, R2 , is replaced by an open circuit. Now, 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, V1 , and one port current, I2 , are assumed to be dependent upon the remaining port variables--namely, I1 and V2 . In this case, the general form of the 2-port equations is as given below. /0 /0 ' H /0 /0 ' /0 /0 /0 / V1 I1 h11 h12 I1 00 I 00 00 V 00 00 h h 00 00 V 000 0 2 0 0 2 0 0 21 22 0 0 2 0 (5) 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, I1 , and a voltage source at port 2, forcing the voltage, V2 . The complementary variables may be easily computed with the result, 4
  5. 5. 1 V1 ' R1 I1 % V2 ; I2 ' & I1 % V R2 2 (6) Y H ' /0 / 00 &1 G 000 R1 1 0 2 0 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, /0 /0 ' G /0 /0 ' /0 /0 /0 / 00 I 000 I1 V1 g11 g12 V1 00 V 00 00 I 00 00 g g 00 0 2 0 0 2 0 0 21 22 0 0 2 0 (7) 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, 5
  6. 6. V1 ' Z11 I1 % Z12 I2 ; V2 ' Z21 I1 % Z22 I2 (8) 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. Figure 3: Z-parameter 2-port model. 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. Figure 4: Y-parameter model for a 2-port. It is interesting to compare the 6
  7. 7. 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. Figure 5: h-parameter model for a 2-port. 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, /0 /0 ' /0 /0 /0 /0 ; T / /0 / V1 V2 00 C D 000 A B A B 00 I 00 00 C D 00 00 &I 00 0 1 0 0 2 0 (9) 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 (-I2) 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 7
  8. 8. 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, V2 , due to the applied current Figure 6: 2-port driven by a source at port 1. source that specifies I1 . Given the z- parameters for the 2-port, and recognizing that I2 = 0 (by inspection), it is a simple matter to verify that V2 = Z21 I1 . Hence, Z21 is the transfer function from the source, I1 , to the response, V2 . Now suppose that we were to remove the current source from port 1 and connect it to port 2, thereby specifying I2 . This time port 1 would be open, and if we were to find V1 in response to this source, we would find that V1 = Z12 I2 . Now Z12 is the transfer function from the source, I2 , to the response, V1 . Clearly, if Z12 = Z21 , then the network response voltage to the current source will be the same 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 Z12 = Z21 . For example, 8
  9. 9. 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, Z12 = Z21 --are called reciprocal 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, Z12 and Z21, 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, Y12 and Y 21 , must also be equal. (We assume, of course, that both the z- and y-parameter matrices exist.) Just as the equality of Z12 Figure 7: 2-port driven by a voltage source at and Z21 implies a circuit property, so does port 1. the equality of Y12 and Y21 . To see this, 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, I2 , to the source making V1 . It is a simple matter to verify that I2 = Y21 V1 , making Y21 the transfer function from input, V1 , to output, I2 . Now suppose that we replace the source at port 1 with the short and the short at port 2 with the voltage source. This time V2 is equal to the voltage source and V1 is equal to zero. Now the transfer function from the input, V2 , to the response, I1 , is given by Y12 . Therefore, if the 2-port is reciprocal, then these 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, 9
  10. 10. z12 z21 y12 y21 h12 ' ; h21 ' & ; g12 ' ; g21 ' & (10) z22 z22 y22 y22 From these relations it is clear that a reciprocal network will have h- and g-parameters obeying the constraints, h21 = -h12 and g21 = -g12. 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 Figure 8: 2-port driven by current sources. schematically in Figure 8. Now let us consider the response of this network to the source at port 1, labeled I3, measured as the voltage across the current source at port 2, labeled I4. Despite the orientation of I4, however, let us measure this voltage from top to bottom, making it equal to the voltage labeled V2 in Figure 8. We assume that I4 is set to zero during this measurement. As usual in a linear system, this response will be some transfer function times the input source, I3 . Now let us recall the adjoint 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 I4 and the output to be measusred as the voltage across I3 from top to bottom, equal to V1, as can be seen from Figure 8. It is a straightforward matter to 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 10
  11. 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 I4 as the source will match that when using I3 as the 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, L1 and L2 , are coupled via some form of flux linkage so that there Figure 9: Ideal transformer representation. 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, /0 /0 ' /0 / / / ; M ' k L1 L2 00 V 00 00 sM sL 000 000 I 000 V1 sL1 sM I1 0 2 0 0 2 0 0 2 0 (11) 11
  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 L1--has N times as many turns as the secondary of the transformer--that is, the coil comprising L2 . Then we have the common idealized description for a transformer given in (12) below. V1 ' N V2 ; I2 ' &N I1 Y /0 /0 ' /0 /0 /0 / 00 I 00 00 & N 0 00 00 V 000 V1 0 N I1 0 2 0 0 2 0 (12) 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 I1 + V2 I2 = V1 I1 + (V1 /N) (-NI1 ) = 0. 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 12
  13. 13. 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: I2 ' gV1 ; I1 ' &g V2 (13) Using these equations it is simple to write the y-parameter 2-port description for a gyrator as, I ' /0 /0 ' /0 / / / ' YV 00 I 00 00 g 0 000 000 V 000 I1 0 &g V1 ¯ ¯ 0 2 0 0 2 0 (14) 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, V ' /0 /0 ' /0 / / / ' Z I ; r ' 1/ g 00 V 00 00 r 0 000 000 I 000 V1 0 &r I1 ¯ ¯ 0 2 0 0 2 0 (15) 13
  14. 14. 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, P ' V I ' V1 V2 /0 / ' V1 V2 /0 / / / ' V2 V1 & V1 V2 ' 0 I1 0 &g V1 00 I 000 00 g 0 000 000 V2 000 ¯ T¯ 0 2 0 0 0 (16) 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 Figure 10: The gyrator circuit not symmetric. In fact, these matrices are skew symmetric-- symbol. 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 Figure 11: Gyrator with a load. 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 14
  15. 15. 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 Zload is the impedance attached to port 2, Yload is its reciprocal--that is, the admittance attached to port 2--and Zinput is the impedance seen looking into port 1. We have, V2 V1 I2 / g 1 1 1 Zload ' ; Zinput ' ' ' ' Yload (17) &I2 I1 &g V2 g 2 Zload g2 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, 1 C Zinput ' sC ' s Leq ; Leq ' (18) 2 g g2 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 Figure 12: RLC bandpass filter. 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 Figure 13: Bandpass filter using a gyrator. 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 15
  16. 16. 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. Yload ' j Y k Y Zinput ' ' j N Yload N Yk k'1 g2 k'1 g2 (19) Zload ' j Z k Y Yinput ' ' g 2 Zload ' j g 2 Zk N N 1 1 ' k'1 Zinput Yload / g 2 k'1 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 Figure 14: Gyrator realization using amplifier is assumed to have infinite input transconductance amplifiers. and output impedance, with an output current equal to the transconductance, Gm = g, times the input voltage applied between the + and - terminals. The circuit shown in the figure satisfies the basic 2-port relations for a gyrator, given by equation (14). 16
  17. 17. 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, V2 V1 Z ' ; &Z ' (20) &I2 I1 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, V1 k V2 V2 V1 ' k V2 ; I2 ' kI1 Y ' ' &k 2 ' &k 2Z (21) I1 I2 / k &I2 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. 17
  18. 18. /0 /0 ' /0 /0 /0 / 00 I 00 00 k 0 00 00 V 000 V1 0 k I1 0 2 0 0 2 0 (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 18
  19. 19. 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. V1 ' 0 ; I1 ' 0 ; V2 , I2 are unknown (23) 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 Figure 15: Circuit symbols for the nullator and norator. 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 19
  20. 20. 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 Figure 16: General N-port network. 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: ¯ ¯ V ' ( V1 , V2 , @@@ , VN )T ; I ' ( I1 , I2 , @@@ , IN )T (24) 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, ZOC. 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, YSC , generalizes the y-parameter matrix by allowing us to express the port current vector in terms of the port voltage vector. 20
  21. 21. Specifically, ¯ ¯ ¯ ¯ V ' ZOC I ; I ' YSC V (25) 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, 21
  22. 22. 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, ¯ ¯ ¯ ¯ ¯ ¯ V ' Zoc I % VS ; I ' Ysc V % I S (26) In this formulation, the source vectors, VS and ¯S , are analogous to the Thevenin and Norton sources ¯ I in 1-ports. In fact, for the special case where N=1, the source vectors, VS and ¯S , are scalars exactly ¯ I equal to the Thevenin and Norton sources, respectively. The hybrid formulation of these equations will be given below. 22
  23. 23. 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 NC ports associated with the capacitive N-port, and exactly NL ports associated with the inductive 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 = NC + NL ports. Since this decomposition may seem a bit confusing, Figure 17: RLC circuit. 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 = NC = 1. We will include the inductor in an inductive N-port with N = NL = 1. Finally, the resistive N-port will 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 Figure 18: Decomposed version of the inductor attached. Note that while the port voltages circuit of Figure 17. of the resistive 2-port will be equal, the port 23
  24. 24. 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 I1 and V2 , 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, /0 /0 ' H /0 /0 % VS ; H ' /0 /0 ; VS ' /0 / V V1 I1 00 &G 000 S 0 1 0 00 I 00 00 V 00 00 &1 G 00 ¯ ¯ 0 2 0 0 2 0 (27) 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, d d V1 ' VL ' L IL ' & L I dt dt 1 d d (28) I2 ' & IC ' &C VC ' &C V dt dt 2 Putting together the above results yields the following set of differential equations: /0 /0 ' & /0 /0 /0 /0 ' H /0 / % VS 00 V 000 V1 L 0 d I1 I1 00 I 00 00 0 C 00 dt 00 V2 00 ¯ 0 2 0 0 0 0 2 0 (29) These equations may be easily solved for the derivatives of I1 and V2, yielding state equations whose state variables are I1 and V2. Note that these variables are just the negative of the inductor current and exactly the capacitor voltage, respectively. Alternatively, we may write these equations directly in terms of inductor current and capacitor voltage, with the result, /0 /0 ' /0 /0 /0 /0 /0 /0 % /0 /0 VS IL 1/L 0 0 1 IL 1/L 0 00 V 00 00 0 &1/ C 00 00 1 G 00 00 VC 00 00 0 &1/ C 00 d ¯ 0 C 0 0 0 (30) dt 24
  25. 25. 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 NL ports of a resistive N-port. This collection of inductors may include couplings and inductor cutsets within it. In any event, it constitutes a reactive (inductive) N-port, where N = NL . Let us further suppose that a collection of capacitors, some of which may form loops, are connected to the last NC ports of the resistive N-port. This collection of capacitors constitutes a reactive (capacitive) N-port, where N = NC . (Please note: In class, I 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: /0 L1 /0 /0 L1 /0 /0 C1 /0 /0 C1 /0 V I I V 00 V 00 0 00 00 I 00 00 V 00 ¯ ' 000 L2 00 d 000 IL2 00 ¯ ' 000 C2 000 ' C d 000 C2 00 VL 0 00 ' L 00 00 ' L I L ; I C 0 00 ' C VC 00 ! 00 dt 00 ! 00 00 ! 000 dt 000 ! 00 d ¯ d ¯ (31) 00 00 00 00 00 00 00 00 00 V 00 00 I 00 00 I 00 00 V 00 dt dt 00 LNL 00 00 LN L 00 00 CNC 00 00 CN C 00 In this case, L and C are NL x NL and NC x NC matrices, respectively. Off-diagonal terms in these 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 NL ports are driven by current sources and each of the 25
  26. 26. remaining NC ports are driven by voltage sources, we have the general description given below. /0 /0 /0 /0 V1 I1 00 00 00 00 00 00 00 00 00 00 00 00 V2 I2 00 00 00 00 00 00 00 00 ! ! 00 00 00 00 00 00 / VL /0 00 00 00 00 ' 00 00 ' H 000 00 % V ' H /00 /0 VN ¯ IN ¯ &IL 00 00 00 ¯ 00 00 ¯ 00 % VS 00 00 00 L L 00 00 00 & I C 00 ¯ ¯ 00 VC (32) 00 00 0 00 00 0 S IN VN 00 00 00 00 L%1 L%1 00 00 00 00 00 00 00 00 IN VN 00 00 00 00 L%2 L%2 00 00 00 00 00 00 00 00 ! ! 00 00 00 00 IN 0 VN 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, /0 VL /0 / LL LC /0 /0 & I L /0 ¯ ¯ 00 00 ' 00 00 00 % VS H H 00 & I 00 00 HCL HCC 00 00 V 00 ¯ (33) 0 ¯ C 0 0 0 C ¯ 0 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. /0 L /0 / /0 /0 L /0 ¯ ; /00 /0 L 0 d/ 0 /0 00 % /0 /0 VS 00 ¯ 0 0 0 C /00 dt 000 00 ' /0 ¯ ¯ ¯ ¯ 00 00 ' 00 & HLL HLC 00 0 V I VL IL ¯ 0 1 0 00 I 00 00 HCL & HCC 00 00 V 00 00 0 & 1 00 VC 000 00 I C 00 0 0 00 (34) 0 C 0 0 0 C 0 ¯ ¯ 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: 26
  27. 27. d /0 I L /0 / L /0 /0 /0 /0 I L /0 / L /0 V ¯ ¯ 0 00 ' 00 &HLL HLC 00 00 % 00 &1 &1 dt 000 V 0 0 00 00 0 C &1 00 00 H 00 00 V 00 00 0 &C &1 00 S ¯ 0 0 (35) 0 C ¯ 0 0 CL &HCC 0 0 C ¯ 0 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, HLL and HCC are symmetric and HLC = -HCLT . This is the natural generalization of our earlier result, and may be easily proven from the 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 ZOC and/or YSC must be “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. 27