Limited reciprocity of passive circuits is developed long before [1] - [9] . The development is not limited only to circuit theory but also in other fields such as electromagnetics and microwaves. In fact, reciprocity of electrical circuits is a special case of electromagnetics, going way back to Lorentz reciprocity. What we are going to propose in this presentation is to make a general approach to circuits, both passive and active. It is going to be an extended reciprocity for circuits with all types of dependent sources as well as all types of I/O ports. The active components in circuits typically consist of four types of controlled sources: Voltage Controlled Voltage Sources (VCVS or e), Current Controlled Current Sources (CCCS or f), Voltage Controlled Current Sources (VCCS or g), and Current Controlled Voltage Sources (CCVS or h). The pair of I/O ports can also be one of the four types: Input Voltage and Output Voltage (VIVO or e), Input Current and Output Current (CICO or f), Input Current and Output Voltage (CIVO or g), and Input Voltage and Output Current (VICO or h). Similar, but limited, approaches are also reported in literature [6] - [9] for active circuits. Typical I/O ports are considered on pair at a time here. Dealing with I/O ports, particularly with more than one pair, is also investigated in [10] . However, this presentation takes the lead and combines all cases into consideration and applies them to the reciprocal issue.

Reciprocity in linear circuits is defined as a property that cross-relates voltages and currents at two ports in a circuit, where the position of ports (input and output) is interchanged between the circuit and its reciprocal. It states as the following. Given circuit N, assume a current I₁ applied to a port P₁ generates a voltage V₂ at port P₂. Now, if we apply the same current I₁ to P₂ in N the same voltage V₂ is produced at P₁. Similarly, assume a voltage V₁ applied to port P₁ in N which generates a current I₂ at another port P₂. With reciprocity, if voltage V₁ is applied to P₂ it produces a current I₂ at P₁.

For proof we can simply consider the Nodal Admittance Matrix (NAM) of circuit N which is symmetrical about the main diagonal. Evidently, we can always reduce the NAM to a 2 × 2 matrix containing only ports P₁ and P₂. The new NAM is still symmetrical, and the reciprocity is quite apparent. Here is the reduced NAM with only I/O ports, P₁ and P₂ present.

$\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{21}& Y_{22}\end{array}\right]\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right]=\left[\begin{array}{c}{I}_1\\ I_2\end{array}\right]$(1)

For I₂ = 0 we get

$\frac{V_2}{I_1}=\frac{Y_{21}}{Y_{21}{Y}_{12}-Y_{11}{Y}_{22}}$(2)

For the reciprocal circuit we interchange the inputs and for I₁ = 0 we get

$\frac{V_1}{I_2}=\frac{Y_{12}}{Y_{21}{Y}_{12}-Y_{11}{Y}_{22}}$(3)

Now, since Y₂₁ = Y₂₁ for passive circuits we get

$\frac{V_1}{I_2}=\frac{V_2}{I_1}$(4)

which proves reciprocity in passive circuits. This simply shows that a passive circuit is its own reciprocal. Whereas, as we will see, the reciprocal of an active circuit is different from the original circuit. In addition, the NAM method hardly works for active circuits when we include a variety of dependent sources in the circuit and apply the reciprocity to different types of I/O ports. Therefore, we need to take a different path, but before describing a new path let us start with a generalized reciprocity theorem first.

Theorem 1: General Reciprocity For an LTI active circuit N₁ with all four types of controlled sources and assuming a pair of I/O ports of any type, we can always find another LTI active circuit N₂, as the reciprocal of N₁, with the following properties.

Assume an excitation signal E₁ applied to a port P₁ in N₁ generates an output response R₁ in another port P₂ resulting in the transfer function F₁ = R₁/E₁. Next, let us assume N₂ to be the reciprocal of N₁ with P₂ as its input port and P₁ its output port. Then an excitation signal E₂ applied to port P₂ in N₂ is generating an output response R₂ in port P₁ so that we get the new transfer function F₂ = R₂/E₂. Provided that E₂ and R₂ are selected according to the rules (discussed later) we get F₂ = F₁. This also proves that, if N₂ is the reciprocal of N₁ then N₁ is also the reciprocal of N₂.

Starting from the last step, it is evident that, if N₂ is the reciprocal of N₁ then, if we apply the same procedure on N₂ we get to N₁, the reciprocal of N₂. To prove the theorem itself, we will try to construct the reciprocal circuit N₂, but before that, let us consider some simple cases first.

Example 1 Let us take a two-stage amplifier with its linear equivalent circuit shown in Figure 1(a) . The circuit has two VCCS devices and a pair of I/O ports with an input voltage V_i and an output voltage V_o. The reciprocal circuit of the amplifier is also constructed and given in Figure 1(b) . As we can see, the passive elements in the reciprocal stay unchanged. The VCCS devices are turned around, so that in each device the controlling element and the controlled element are swapped. However, in the case of the I/O ports, not only are the input and output ports swapped but the voltages are changed to currents as well. So, the voltage gain V_o/V_i in the amplifier is matched with the current gain I_i/I_o in the reciprocal circuit. After the buildup, both circuits are simulated for the AC analysis, and the frequency responses of the gains (V_o/V_i and I_i/I_o) are plotted in Figure 2 ¹. The s-expanded transfer functions of both the circuit and its reciprocal are also given in (5), where the frequency f is in MHz scale. Notice that the TF has three poles and two zeros, and this can be easily verified from the output plots. Finally, the Amplifier bandwidth is given as BW = 93 KHz, and the gain bandwidth product is GB = 4.88 dB-MHz.

$\frac{V_o}{V_i}=\frac{I_i}{I_o}=\frac{640+125s+0.19s^2}{3+5.6s+0.6s^2+8.7\ast {10}^{-3}{s}^3}$(5)

Example 2 Consider an amplifier with feedback, as shown in Figure 3(a) . The linearized circuit of the amplifier is shown in Figure 3(b) . To construct the reciprocal, we follow the same rules we did in example 1, that is, the passive circuit stays unchanged while the active components and the I/O ports swap their elements while considering the types. Namely, the VCVS in the amplifier is replaced with a CCCS, and the VIVO is replaced with a CICO ports, as shown in Figure 3(c) . After simulating the amplifier and its reciprocal we get the gains as F₁ = V_i/V_o = −29.3 for the amplifier, and F₂ = I_i/I_o = −29.3 for the reciprocal circuit, as expected.

A nullor consists of a nullator and a norator. A nullor is either three-terminal or four-terminal. In a three-terminal, the nullator and the norator have a common node. A nullor also carries a sign depending on the direction in the actual component it represents. In a three-terminal the sign is positive if the two arrows merge at the common node or depart. Otherwise, the sign is negative. For four-terminal nullors we are undecided until the process gets through, and they change to three-terminal nullors [10] . A nullor also carries a value. The value of a nullor is 1. When it replaces a controlled device the gain of the controlled source is multiplied with the coefficient-multiplier p_k, for the sub-circuit k. For example, if a nullor replaces a VCVS device with e = 20 V/V, then the nullor selected is of e type with, and the value e = 20 V/V is multiplied by p_k. Another property of a nullor is its adaptability when removed, as illustrated in Figure 4 . This can be stated as follows. If a nullor, which has replaced an active device or a pair of I/O ports, is removed from a circuit the status of the circuit will be the same as if the original device or the pair of I/O ports is removed [12] .

When all active devices and all I/O ports in a circuit are replaced with nullors the circuit is called a nullor circuit. That is, a nullor circuit is built up of passive elements and nullors only. Next, when all the passive elements are eliminated through the Admittance Method (AM)² the remaining nullor circuit is called all-nullor circuit. Now, we need to explain how this nullor circuit is built.

To create the reciprocal of a nullor circuit we first need to know what we expect from a reciprocal circuit. As stated in Theorem 1, both the circuit and its reciprocal must produce the same transfer function with the I/O ports swapped. This means the same circuit NAM determinant and cofactors. In topological formula this is translated into having the same STP, and one way this can happen is when for each nullor in the circuit the elements (nullator and norator) are swapped. There is, however, one condition attached to this and that is: when one or more nullors are removed from both, the circuit and its reciprocal, they must remain identical, except for the nullor elements being swapped. This means that some types of nullors must be replaced with other types when moving from the main circuit to its reciprocal. Figure 7 shows how the nullors might change their types when moving from the circuit to its reciprocal. Notice that nullors of types g (VCCS and CIVO ports) and h (CCVS and VICO ports) stay unchanged, whereas types e (VCVS and VIVO ports) and f (CCCS and CICO ports) get interchanged.

Finally, to construct the reciprocal of an active circuit we need to take the following steps.

1) Copy the passive circuit without any changes.

2) Swap controlled sources with their corresponding controlling elements. Do the same with I/O ports.

3) In doing part 2, make device exchanges if needed, according to Figure 7 , and do the same with the I/O ports.

Notice in Figure 7 that the second column of nullors and the reciprocals are shown to be identical.

Appendix B provides more details on the computation of the determinants, cofactors, and the transfer functions for both the circuit and its reciprocal. To clarify, let us take the following example.

Example 3 Consider an active circuit represented in its nullor equivalent graph, shown in Figure 8(a) . The circuit consists of four active devices with one (e₄) replacing a pair of I/O ports. Two devices are of h type and the other two are of e type. To read a nullor, for example, h₃, the nullator is specified by h₃ and the corresponding norator by 3. Referring to Figure 7 , it follows that the h type nullors remain unchanged in the reciprocal, whereas the e types are replaced with f devices. The reciprocal is shown in Figure 8(b) . So, the points to notice here are as follows: a) the passive circuit remains unchanged with both y and r present in the circuit, b) while the original circuit has two e and two h types of active devices, the reciprocal has two f and two h types of devices. The circuits are simulated and the responses of the two are identical (not shown), as expected.

Example 4 In another example, we assume a MOS amplifier for high frequencies. Figure 9(a) shows the linearized equivalent circuit of the amplifier with the component values: R₁ = 5 KOhms, R₂ = 10 KOhms, C₁ = 10 pF, C₂ = 1 pF, and g₁ = 2 mA/V. The purpose here is to get the voltage gain A_v = V_o/V_i in s-expanded format through the topological formula by using STP methodology. Figure 9(b) is the graphic representation of the amplifier with nullors replacing the dependent source as well as the I/O ports. Notice that the I/O port is of e (VIVO) type with the coefficient e₂ = 1 (see Appendix A). Because of the two nullors we need to have four sub-circuits with the graphs given in Figures 9(b)-(e) . Two graphs, (c) and (d) are to find the determinant and the other two are for getting the specific cofactor. Let us assume that T_c, T_d, T_e, and T_b denote the STPs associated with the graphs of Figures 9(c)-(e) , and (b) respectively. Then we find

$T_c=5\ast {10}^{-16}\ast s^2+65\ast {10}^{-9}\ast s+1$ (6)

$T_d=5\times {10}^{-8}\ast g_1\ast s$ (7)

$T_e={10}^{-8}\ast s$ (8)

$T_b=-10\ast g_1$ (9)

The desire transfer function (gain) is found as

$A_v=\frac{V_o}{V_i}=\frac{T_e+T_b}{T_c+T_d}=\frac{{10}^{-8}\ast s-10\ast g_1}{5\ast {10}^{-16}\ast s^2+65\ast {10}^{-9}\ast s+1+5\ast {10}^{-8}\ast g_1\ast s}$

$A_v=\frac{T_e+T_b}{T_c+T_d}=\frac{{10}^{-8}\ast s-20}{5\ast {10}^{-16}\ast s^2+165\ast {10}^{-9}\ast s+1}$ (10)

Note that for low frequency we get the expected value

$A_v=-R_2\ast g_1=-20\text{V}/\text{V}$. (11)

As was shown, reciprocity works for any nullor circuit, matching the transfer functions between a nullor circuit and its reciprocal. Apparently, reciprocity must also work for modern active devices [12] [14] since they are represented in nullor circuits. Within those reported, we select a CCII+ circuit to work with.

CCII+ Figure 10(a) is a CCII+ in nullor representation [14] . The model circuit shows two pairs of nullors that must always be treated together. Figure 10(b) shows a conceptual CCII+ connected as an inverted integrator with the voltage transfer function given as.

$\frac{V_o}{V_x}=-\frac{R_o}{R_x}\frac{1}{s{R}_2{C}_1}$(12)

We now try to create the reciprocal of the integrator. To do this, we need to make two types of changes. First, swap the nullator and norator in each nullor, and second, get the transfer function to represent the ratio between the current in the input port I_x and the current source I_o connected to the output port, as shown in Figure 10(c) . The current transfer function is simply found as.

$\frac{I_x}{I_o}=-\frac{R_o}{R_x}\frac{1}{s{R}_2{C}_1}$(13)

As we expect, the two transfer functions are identical.

Up to now we have established reciprocity rules for circuits with one pair of I/O ports. The idea that a pair of I/O ports can be converted to a controlled source, and then to a nullor, was the key factor in uniformity of active circuits, consisting of passive and active components. It so happens that this conversion cannot generally continue for multiple I/O ports, which comes from the fact that, despite dependent sources, pairs of I/O ports are not exclusively related. What it means is that, for an input port P₁, any other port (or node) in the circuit can be paired as an I/O port.

We usually have no problem with multiple output ports if there is only one source of input. In this situation, the input can be paired separately with each output port. Then, how does the reciprocal circuit look like? Considering the superposition for linear circuits the reciprocal circuit is simply constructed exactly like what we did for a single pair of I/O ports, except the single input source, in the main circuit, repeatedly appears at the multiple output ports, and the input port shows the summation of the outputs in the original circuit. For example, let us assume that we are stimulating a circuit N₁ with an input current I_i = 10 mA and we get two output responses, V_o1 = 3 V at port P₁ and V_o2 = 5 V at port P₂. We then construct N₂, the reciprocal of N₁. Now, if we apply I_o1 = 10 mA and I_o2 = 10 mA to the two output ports P₁ and P₂ we will receive a voltage of V_i = 3 + 5 = 8 V at the input port. Note that if the output ports are different in types (I or V) we must refer to Figure 7 and make the right selections. Let us take another example. Assume we stimulate a circuit N₁ with an input current X_i applied to the input port. Then we get two output responses, one output voltage X₂ at port P₁ and the other one an output current X₂ at port P₂. We then construct N₂, the reciprocal of N₁. Now, if we apply a current X_i at P₁ and a voltage X_i at port P₃ we receive a current X₃ at the input port in the reciprocal circuit, so that X₃ = X₁ + X₂. This is shown on Figure 11(a) and the reciprocal in Figure 11(b) . Example 5 is another case of one input and two outputs.

Example 5 Consider the same two-stage amplifier given in Example 1 with its linear equivalent circuit shown in Figure 12(a) . The difference here is that we assume two output V_o and V_x here, still with a single input V_i. The reciprocal circuit is given in Figure 12(b) , where I_o and I_x are both equal to V_i in terms of the values. The circuit and its reciprocal are simulated, and the resulting plots for the transfer functions are given in Figure 13 . In Figure 13 one plot represents V_out/V_i, where V_out = V_o + V_x, and the second plot shows I_i/I_o, with I_x = I_o.

In the case of multiple inputs and a single output, just like the previous case, we can pair each input with the output port separately and repeat it for the other inputs. In fact, this case precisely matches the previous case, the only difference is that we need to exchange the circuit with its reciprocal. For multiple input sources that are proportional we may utilize the superposition applicable to linear circuits. To explain this, let us start with a circuit N with two input voltage sources V₁ and V₂, and an output port V₃ applied to ports P₁, P₂, and P₃ respectively, as shown in Figure 14(a) . Now, let us assume that with only V₁ applied, a voltage V_3,1 resulted at P₃, and similarly with only V₂ applied a voltage V_3,2 has resulted at P₃. So, we are facing two transfer functions A₁ = V_3,1/V₁, and A₂ = V_3,2/V₂. Next consider the proportionality between V₁ and V₂ such that V₂ = k*V₁, where k can be any multiplier, real or complex. Then with both V₁ and V₂ applied to the circuit we can get the output as

$V_3=V_{3,1}+V_{3,2}=\left(A_1+k\ast A_2\right)\ast V_1=A_3\ast V_1$(14)

where, $A_3=A_1+k\ast A_2$ can be considered as the pseudo-transfer function between the three ports, P₁, P₂, and P₃.

How then is the reciprocal circuit constructed for this case? As it turns out, the core for the reciprocal circuit stays the same. Figure 14(b) shows the reciprocal circuit of N with an input current I₃ at port P₃ and two output ports P₁ and P₂, representing output currents I₁ and I₂. Now, if a current I₃ = V₁ (in value) at P₃ generates two output currents I_1,3 and I_2,3 at P₁ and P₂ then according to Equation (14) we get

$I_{1,3}+k\ast I_{2,3}=V_3$ (in value)(15)

This, of course, can be conditionally extended to n ports if a) we consider one output at a time, and b) The input sources are proportional.

The situation must be similar for any number of input sources. For example, let us consider a circuit N with n input ports P₁, P₂, …, P_n, and an output port P_o. Further, assume that current sources I₁, I₂, … I_n (this, of course, can be mixed current and voltages) are assigned to ports P₁, P₂, …, P_n, respectively, and let I_m= k_m*I₁ for all m = 2, 3, …, n, where k_m can be any arbitrary multiplier. When all the input sources are applied simultaneously, they collectively generate an output voltage V_o at port P_o. Now, if N_r is the reciprocal circuit of N then removing all n input sources and instead applying a current source I₁ to the output port P_o in the circuit will generate n output voltages V_1,o, V_2,o, …, V_n,o at ports P₁, P₂, …, P_n. Then, the reciprocity rule states that.

${\displaystyle {\sum }_{m=1}^n{k}_m\ast V_{m,o}}=V_o$, for $k_1=1$.(16)

Notice that Equation (16) also works for the case of having only one input source.

One may realize that this solution for multiple input sources still is not comprehensive. The major drawback is the way we have adapted the multiplier $k_m$. There are usually unrelated sources applied to a circuit in multiple input cases. For example, signals with carriers, modulated signals in communication, or mixed signals are not necessarily related by a multiplier.

In general, the reciprocal circuit studied in this discussion is mainly for a single pair of I/O ports. In case of multiple I/O ports, however, we can adopt the superposition for linear circuits and handle the case but only if the inputs are proportionally related. In case we have multiple and unrelated input sources the suggestion is to use one I/O pair at a time. Any other possible solution to the problem is beyond the scope of this presentation.

To clarify the case, let us consider the active circuit shown on Figure 13(a) . The circuit has two inputs, a voltage source V₁ and a current source I₂. It also has an output voltage V₃, or multiple outputs if desired.

Example 6 Consider an active circuit N with two input sources: a voltage source V₁ at port P₁ and a current source I₂ at P₂. The overall response V₃ is received at Port P₃, as shown in Figure 15(a) . The reciprocal circuit N_r is generated and displayed in Figure 15(b) . According to the I/O rules, as shown in Figure 14 , the reciprocal circuit N_r has only one input source I₃ at Port P₃ and two output ports I₁ and V₂ at P₁ and P₂. Both N and N_r circuits are simulated by SPICE and the results are displayed in Figure 16 along with the SPICE listing of both circuits.

Notice that the response V₃ = 10.76307 V in circuit N is equal to the addition of the two responses I₁ (in mA) and V₂ (in V) in reciprocal circuit N_r. Now, since the applied input sources are V = 5 V and I₂ = 15 mA then the multiplier factor selected here must be k = 3.

Generalized reciprocity without nullor equivalents

Overall, we can summarize the reciprocity issue in an algorithm, describing how we can directly construct the reciprocal circuit of a given active circuit without going through the process of forming the nullor equivalent circuit. However, this may raise the following question. If we can directly find the reciprocal of a circuit without going through the nullor circuit, then why did we go through it at first place? The answer to this question is, yes, to get the reciprocal we can go directly to construct the reciprocal circuit without resorting to nullors, as the following Algorithm offers. The nullor circuits, however, provide a more uniform and simplified process for the circuit analysis that we need to proceed, as explained in Appendices A and B. To show one example regarding the simplification in using nullors is in determining T_a, discussed in Section A, Appendix B. So, the bottom line is that, if we are only interested in finding the reciprocal of a circuit, then there is no need to go through converting the circuit to a nullor circuit. In this situation reciprocity is applied to both the dependent sources and I/O ports. But in case we need to test the results, and through the analysis show the proof of the reciprocity, then we offer to convert the circuit into its nullor equivalent. In short, nullors are good for explanation and analysis but in terms of writing efficient programs for reciprocity the conversion to nullors is avoided. I have also avoided using nullors in my program “Circuit Analysis, Simulation, and Design” (CASD).

Algorithm To find the reciprocal of an active circuit we need to go through the following steps:

We consider two cases in practical situations: 1) linear circuits with the actual devices replacing the models, and 2) reciprocity in non-linear circuits.

This presentation aims at generalizing the reciprocity for any active circuit with any type of I/O ports. Reciprocity for passive components is already well-established. For active circuits, particularly when a variety of components are used, still work needs to be done. This work is trying to exactly do this. It demonstrates that for a given linear active circuit N₁ with all four types of controlled sources and assuming an I/O port of any of the four types, there is always a reciprocal circuit N₂ with the following properties. The transfer functions of the two circuits, N₁ and N₂, are identical, provided that the input and output ports of the two circuits are swapped with the I/O types adjusted. The second task is to realize and construct N₂, and beyond. To find the reciprocal circuit, we try to simplify and uniform the circuit by using some new techniques. In the first step, we remove the I/O ports by converting them to dependent sources. We then convert the entire circuit to a nullor circuit. Next, we break the circuit into sub-circuits, each for finding the coefficients of a symbolic transfer function. We then reverse the process until we reach the reciprocal circuit with identical symbolic transfer function, or in values.

The cases for multiple I/O ports are also discussed. Multiple outputs can be simply handled by superposition. For multiple inputs, it is shown how the superposition gets involved for related inputs. For non-related inputs, one way to deal with it is to go for one input at a time and hence get multiple reciprocals that are different only in I/O ports.

Finally, an algorithm is provided describing how we can directly construct the reciprocal circuit of a given active circuit without going through the process of forming its nullor equivalent circuit. In short, nullors are recommended for explaining and understanding the reciprocity procedure but in terms of computation and efficient programing the conversion to nullors is avoided.

¹One plot is slightly shifted for clarity purposes.

²For brevity, the AM method is not discussed here. A full description of the method can be found in [11] .

³In practice, when the passive circuit is eliminated most 4-terminal nullors become 3-terminals. Also removing 3-terminals may change some 4-terminals to 3-terminals.

⁴Each time we use partition operation removing or short circuiting an element in a sub-circuit we are creating a new sec-circuit.