# Circuit Idea/Parallel Voltage Summer

Building formula: Parallel voltage summer = V-to-I converters + current summer + I-to-V converter.

## A problem: the common ground

Kirchhoff's Voltage Law has already given us an idea how to create the simplest series voltage summer. Unfortunately, it had a lot of disadvantages; maybe, the most crucial of them was the problem of the common ground.
Do you remember? First, we grounded the common point between the input source V$_{IN1}$ and the load; as a result, the sources V$_{IN2}$ and V$_{IN3}$ became flying. Then, we tried to ground the common point between the input sources V$_{IN1}$ and V$_{IN2}$; now, the source V$_{IN3}$ and the load became flying. Finally, we were forced to use only two grounded sources and a differential load.
Only, Kirchhoff has formulated another law - Kirchhoff's Current Law (KCL). Maybe, it can help us to create a perfect voltage summer having no problems with the common ground? Let's try this speculation!

## Deriving the simplest current summer from KCL

First, we can apply directly KCL to create a current summer. For this purpose, we have just to connect in parallel the input current sources I$_i$ (let's for concreteness assume again that we have three sources) to the current load L$_I$ (Fig. 3). Let's first assume that it is a perfect current load having zero resistance (e.g., just a piece of wire). The output current I$_{OUT}$ flowing through the load is the sum of the input currents: I$_{OUT}$ = I$_{IN1}$ + I$_{IN2}$ + I$_{IN3}$.
We can ask ourselves again, "Only, what is actually the summer here?", "Where is it?" The input current sources and the load are external components; so, the rest (the bare node or the junction point) serves here as a summer!
Wonderful! We have another "ideal" device - the current summer. It is apparent device because there is not actually a device:)!

## Building a parallel voltage summer

Only, we want to sum input voltages; so, we have to convert them into currents. For this purpose, we connect voltage-to-current converters between the input voltage sources and the inputs of the current summer (Fig. 4). Also, we need a voltage output; so, we have to connect the opposite current-to-voltage converter at the current summer output. In this way, we have assembled a composed voltage summing circuit:
Let's compare the two viewpoints: From the classical viewpoint, the parallel voltage summer consists of a few resistors; from our viewpoint, it contains voltage-to-current converters, a current summer and a current-to-voltage converter. The use of applying such a system approach is that we see the function of the resistors in the circuit; we see the forest for the tree!

## Exploring the circuit

Let's now see how the circuit operates. Attractive voltage bars will help us to visualize the invisible voltages and voltage drops (Fig. 5); current loops will show us where the currents flow (remember: every current returns where it has begun flowing).
The input voltage sources V$_{IN}$ produce voltages, the resistors R$_{IN}$ convert them into currents, the junction point sums the currents and finally, the resistor R converts back the current sum into an output voltage V$_{OUT}$.
Only, the output voltage introduces an error because it subtracts from the input voltages. Now, the effective voltages V$_{R1}$, V$_{R2}$ and V$_{R3}$ create the currents instead the whole input voltages V$_{IN1}$, V$_{IN2}$ and V$_{IN3}$:
I$_{IN1}$ = (V$_{IN1}$ - V$_{OUT}$)/R$_1$ = V$_{R1}$/R$_1$
I$_{IN2}$ = (V$_{IN2}$ - V$_{OUT}$)/R$_2$ = V$_{R2}$/R$_2$
I$_{IN3}$ = (V$_{IN3}$ - V$_{OUT}$)/R$_3$ = V$_{R3}$/R$_3$
How do we remove the error?

## Do we really need an I-to-V converter?

Maybe, you have already noted the contradictory role of the resistor R.
In order not to disturb the input sources (see above), we want the resistance R to be as small as possible (preferably R = 0). As a result, the input voltage-to-current converters become each other independent; only, the output voltage decreases:( We will use this technique later when we build an active parallel voltage summer.
But why do not we increase the resistance R up to infinity (i.e., just to remove the resistor R)? In this case, the input voltage-to-current converters become each other absolutely dependent but we obtain a maximal output voltage. We can use this solution when the load has an infinite internal resistance (e.g., when we have buffered the passive summer by a non-inverting amplifier).
Only, there is a sound reason to connect the "unnecessary" resistor R - it can add the input coefficients up to 1 (see more explanations). We will use this technique below to build a simple digital-to-analog converter.

## Applications

The parallel voltage summer exists in many analog circuits considered in electronics books; only, authors do not discern and do not pay attention to it. As a result, it is presented rather implicitly than manifestly. In this section, we will do our best to show its presence in various electronic circuits. This famous circuit deserves our attention.

### Op-amp inverting voltage summer

Maybe, the most important application of the passive summer is building an op-amp active summer. Only, what is the idea behind such an op-amp inverting summing circuit (Fig. 7)?
From the classical viewpoint, an op-amp inverting summer consists of input resistors Ri, a negative feedback resistor R and an op-amp. Only, thinking of the active circuit in this way we can't discern the basic idea behind it; we do not see the forest for the tree.
From our fresh viewpoint, an op-amp inverting summer consists of a passive parallel voltage summer and an op-amp:
More precisely speaking, an n-input op-amp inverting summer needs an (n+1)-input passive parallel voltage summer. For example, in order to build a 3-input op-amp inverting summer (our case) we have to add an additional fourth input IN$_4$. A properly supplied op-amp serves as an additional input voltage source that adjusts its output voltage V$_{OUT}$ (the "input" voltage V$_{IN4}$) so that to zero the passive summer's output voltage V$_A$ (the virtual ground). As a result, the op-amp's output voltage represents the sum of the input voltages V$_{IN}$. Let's repeat again:
We (i.e., the op-amp) introduce(s) an additional "compensating" input and make its voltage equal to the sum of the rest "true" input voltages. Then, we abandon the "genuine" summer's output voltage V$_A$ (now, it serves only as an indication of an equilibrium) and begin using the compensating voltage V$_{IN4}$ as an output.

### Digital-to-analog converter

What is a digital-to-analog converter (DAC)? It is just a circuit that materializes an abstract digit (most frequently, converting it into a voltage). A parallel voltage summer having binary-weighted inputs can do this work.
In this example (Fig. 8), a 3-bit digital device drives the DAC. Here we suppose that the high voltage levels of the three digital outputs are relatively equal; this voltage serves as a reference V$_{REF}$. The output voltage is:
V$_{OUT}$ = 0.1 x b0 x V$_{REF}$ + 0.2 x b1 x V$_{REF}$ + 0.4 x b2 x V$_{REF}$
For example, you can connect such a simple DAC to the parallel computer port (such as a printer port).
The resistor R is absolutely necessary in this application; it adds the sum of the input coefficients up to 1 (according to Daisy's theorem). You can use the nice Brandy's formula to calculate the resistances.

### Audio mixer

Another popular application of the parallel summing circuit is audio mixing of analog signals from a few voltage sources.
Note again that the parallel summer not just sums the signals; it also attenuates them. In this case, this feature is useful.

## Parallel versus series summer

We can do a few remarks about the parallel voltage summing circuit in comparison with the series one.
* The output voltage of the parallel voltage summer is a weighted sum of the input voltages; the output voltage of the series voltage summer is a sum of the whole input voltages.
* The input sources and the load of the parallel voltage summer are grounded; only two devices (sources or a source and a load) of the series configuration can be grounded, the rest remain flying.

## Further reading

Parallel voltage summer builds the circuit by using the simpler voltage-to-current and current-to-voltage converters (Flash animated; Ruffle plugin is needed).