# Circuit Idea/Series Voltage Summer

Circuit idea: Connect voltage sources in series to sum their voltages according to Kirchhoff's Voltage Law.

## Deriving a series voltage summer from KVL

Ohm's law has given us an idea how to create the most elementary voltage-to-current and current-to-voltage converters. Now, Kirchhoff's Voltage Law (KVL) will help us to make the simplest voltage summer.
According to the famous law, we have just to connect in series (Fig. 2) the input voltage sources and the load (let's for concreteness assume that we have three sources). The output voltage across the load is the sum of the input voltages:
Only, what is actually the summer here? Where is it? The input voltage sources and the load are external components; so, the rest (a bare loop or just a piece of wire) serves as a summer!
It is just wonderful! The series voltage summer is an ideal "device" because there is not actually a device:)!
The input voltage sources in the circuit on Fig. 2 are connected in the same direction; so, their voltages are summed. If we reverse some source, its voltage will be subtracted from the whole sum.

## The problem of the common ground
### Floating sources

The summing circuit on Fig. 2 is "flying"; there is no connection with the common reference point referred to as ground. Usually, the input voltage sources and the load are preliminarily connected to the ground (they can be electronic devices connected to the common power supply). So, we have to choose a point from our summing circuit that to connect to the ground. Only, what point of the circuit do we connect to the ground? Let's try some connections.
First, we can ground the common point between the input source V$_{IN1}$ and the load. Only, the sources V$_{IN2}$ and V$_{IN3}$ become floating.
Note that in this arrangement, all the input voltages are positive regarding to ground.

### Floating source and load

Then, we can ground the common point between the input sources V$_{IN1}$ and V$_{IN2}$. Only, now the source V$_{IN3}$ and the load become flying. Obviously, the series voltage summer has a problem with the common ground.
Travelling the loop in clockwise direction we can note that all the input voltages have the same polarity direction (- +, - +, - +). Only, regarding to the ground, the input voltage V$_{IN1}$ is negative while the others are positive.

### Floating load

Fortunately, there is a very successful combination - two grounded input sources and a differential load. It is widely used in op-amp circuit design because, as a rule, the operational amplifiers have differential inputs. Almost all the series summer applications are based on this arrangement (see the examples below).

## Applications

The series 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.

### Series battery configuration

Is there a more suitable example of using a series voltage summer than a series battery configuration? It is invented by Volta in 1799 by placing many galvanic cells in series. From the classical viewpoint, such a battery consists only of cells; from our viewpoint, it contains cells and a series voltage summer (it sounds quite strange, doesn't it?):

### Wheatstone bridge

We may think of this legendary circuit as a system of two voltage dividers (Fig. 4). Their output voltages are connected in series with opposite polarities (- V$_{R2}$ +, + V$_{R4}$ -) so that they are subtracted (regarding to the ground, the voltages have the same polarities). In this way, the output voltage of this series voltage summer represents the difference between the voltages.
If the bridge is balanced, the result is zero
and no current flows through the load (no matter how much low-resistive it is). This phenomenon is referred to as bootstrapping; Baron Munchhausen put it in practice for the first time (the legend says that he was using his own boot straps to pull himself out of the sea:).
Note that in this arrangement, the two input voltage sources are grounded while the load (usually the so called zero indicator or galvanometer) is flying.

### Circuits with series negative feedback

Circuits with negative feedback are the great mass of devices in analog electronics. Let's, for example, consider the most elementary of them - the voltage follower. It consists of three components (Fig. 5): a source of energy E (power source), a regulating element R and a voltage subtractor. The regulating element produces an output voltage V$_{OUT}$ (by controlling the energy of the power source) and continuously compares it with the input voltage V$_{IN}$ by the voltage subtractor, in order to keep zero difference between them. As a result, the output "copy" voltage is equal to the input "original" one.
Depending on the nature of the voltage subtractor (series or parallel), there are two kinds of circuits: with series negative feedback and with parallel negative feedback. Circuits with series negative feedback are based on a series voltage summer acting as a subtractor.

#### Transistor NFB circuits

Emitter follower is the most elementary transistor circuit with series negative feedback that we can build. For this purpose, we connect the output part of the transistor (collector-emitter) in series to the power supply and the load (Fig. 6). It acts as a regulating element that doses the voltage of the power supply by changing its present resistance.
Then, to make the transistor compare the output voltage V$_{OUT}$ with the input voltage V$_{IN}$, we connect the two voltages in series and apply their difference dV = V$_{IN}$ - V$_{OUT}$ to the input part of the transistor (the base-emitter junction). As a result, the transistor produces an output voltage that is almost equal to the input one:
V$_{OUT}$ ≈ V$_{IN}$ (we have written "≈" instead "=" because of the troublesome voltage V$_{BE0}$).
According to the classical viewpoint, the emitter follower consists only of a transistor. From our viewpoint, the emitter follower contains a power supply, a transistor and a series voltage summer:
Emitter follower = power supply + transistor + series voltage summer

#### Op-amp NFB circuits

Similarly, an op-amp voltage follower is the most elementary op-amp circuit with series negative feedback that we can build. For this purpose, we supply properly the op-amp and connect the op-amp output to the load (Fig. 7). Now, the op-amp acts as a regulating element that doses the voltage of the power supply by changing the present resistance of its output transistors.
Then, in order to make the op-amp compare the output voltage V$_{OUT}$ with the input voltage V$_{IN}$, we connect as above the two voltages in series and apply their difference dV = V$_{IN}$ - V$_{OUT}$ to the op-amp input (the input voltage - to the non-inverting op-amp input and the output voltage - to the inverting op-amp input). As a result, the op-amp produces an output voltage that is almost exactly equal to the input one: V$_{OUT}$ = V$_{IN}$.
From the classical viewpoint, the op-amp voltage follover consists only of an operational amplifier. From our viewpoint, the op-amp follower consists of a power supply, an op-amp and a series voltage summer (a bare loop):
Op-amp voltage follower = power supply + op-amp + series voltage summer

## Series versus parallel summer

We can do a few remarks about the circuit of a series voltage summer comparing it with the parallel one.
* The output voltage of the series summer is a sum of the whole input voltages; if we want to sum voltages with weighting coefficients, we have to attenuate them by voltage dividers. The circuit of the parallel voltage summer is more appropriate for this case.
* The output voltage of the series summer can exceed any of the input voltages; the output voltage of the parallel voltage summer is less than the highest input voltage.
* There are not any losses in the series summer; the resistors of the parallel voltage summer dissipate power.
* The series voltage summer has a problem with the common ground; the parallel voltage summer has not such a problem.
* An interesting feature of the series voltage summer is that the current flowing through the input voltage sources and the load is the same.