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**Building a Parallel Voltage Summer**

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

## A problem: *the common ground*Edit

*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 KCLEdit

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!

*The simplest current summer is just a node.*

Wonderful! We have another "ideal" device - the current summer. It is apparent device because there is not actually a device:)!

## Building a parallel voltage summerEdit

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:

*Parallel voltage summer*=

*V-to-I converters*+

*current summer*+

*I-to-V converter*

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 circuitEdit

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?Edit

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.

## ApplicationsEdit

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 summerEdit

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:

*Op-amp inverting summer = parallel voltage summer + 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 converterEdit

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 mixerEdit

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 summerEdit

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 readingEdit

Parallel voltage summer builds the circuit by using the simpler voltage-to-current and current-to-voltage converters (animated).

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