Current sources and voltage sources connection

Question

We know that every voltage source can be transformed into a current source, and vice versa. We also know that voltage sources are connected in series and current sources are connected in parallel. Isn't this paradoxical, because the same set of sources should be connected in both series and parallel, depending on how we choose to view them?

Answer 1

They are the same

The simple truth about voltage and current sources in electronics is that they are the same network of two elements in series - voltage source and varying "resistor" (transistor). The only difference is in the behavior of the dynamic "resistor": in the first case, it keeps the voltage across the network constant; in the second case, it keeps the current through the network constant.

Current stabilizers

Initially, transistors are current stabilizing "resistors". An example is the common emitter stage with a constant voltage applied to the base and a load included in the collector.

Voltage stabilizers

But their behavior can be reversed by applying а negative feedback. An example is the common collector stage (emitter follower) with a constant voltage applied to the base and a load included in the emitter. As a result, the "current-stabilizing" transistor becomes "voltage-stabilizing".

Conflicts between homogeneous sources

When we connect such made voltage sources in parallel or current sources in series, their internal dynamic "resistors" begin to compete and finally they reach their resistance limits - zero or infinity. From this point on, these sources cease to act as such (voltage or current). There is no such problem in electrical circuits when connecting simple sources made by "static" resistors.

Applications

However, these "incorrect" connections of homogeneous sources can be useful and they are widely used in electronic circuits.

Voltage conflicts. For example, in differential amplifier stages ("long-tailed pair") two voltage sources (emitter followers) are connected in parallel and they compete. As a result, the currents through them vigorously change (high gain).

Current conflicts. In amplifier stages with the so-called "dynamic load", two current sources (transistors, current mirrors) are connected in series and they compete. As a result, the voltage across them vigorously change (high gain).

See also my paper.

Mutual help between heterogeneous sources)

When a voltage source is connected in parallel to a current source, there is no conflict between them - the former keeps up the voltage constant while the latter keeps up the current constant. They even mutually help - each of them serves as an ideal load for the other.

Cascode circuits are typical example of such connection.

Answer 2

We also know that voltage sources are connected in series and current sources are connected in parallel.

We don't necessarily know that. Ideal voltage sources cannot be connected in parallel, because the circuit theory they belong to doesn't allow that. That's not the physical reality though. Ideal current sources also cannot be connected in series, because of how circuit theory deals with them. Again, physical current sources can be connected in series no problem.

Let's expand on that:

1. Physical voltage sources have non-zero source impedances.

2. Physical voltage sources may have an inherent diode within - e.g. any unipolar DC lab power supply behaves that way.

3. Physical current sources have a range of compliance voltages where they have high impedance. Outside of this range, their impedance may fall dramatically - by orders of magnitude.

So, if you connect two lab power supplies in parallel, the one set to a lower voltage is essentially out of the circuit at DC. If you connect two alkaline batteries in parallel, a charge-equalizing current will flow between them, based on the disparity between cell voltages. This current may be high, but it's certainly finite and can be modeled reasonably well with accurate electrochemical cell models.

If you connect two physical current sources in series, one of them will prevail, depending on the orientation of their connection, and the compliance voltage ranges.

Let's look at a practical examples of current sources in series:

^{simulate this circuit – Schematic created using CircuitLab}

The two sources are set to provide relatively stable 1mA and 4mA over a decade-wide voltage range - from 3V to 30V, with rudimentary matching of the transistors. As you can see, the sources operate normally when connected in parallel. But when you connect them in series, it turns out that the 4mA source pulls down its input voltage to about 1V, and the 1mA source operates normally.

Answer 3

You seem to be confused by Thevenin/Norton transformations which state that the following two configurations are equivalent:

^{simulate this circuit – Schematic created using CircuitLab}

So in order these to be equivalent the following must hold: $$ R_{Thevenin}=R_{Norton} $$ $$ V_{Thevenin}=I_{Norton} \cdot R_{Norton}$$

Assuming the resistance are non-zero, these configurations can be freely connected both in series or in parallel and still stay equivalent, that is the following (parallel connection) is equivalent (assuming the equations above hold).

^{simulate this circuit}

and the following series connections is equivalent:

^{simulate this circuit}

Now, what you can't do is to perform the above transformation to ideal sources, when \$R_{Thevenin} = R_{Norton}=0\$. In that case the second equation will always yield \$V_{Thevenin}=0\$, or if transforming the other way around - \$I_{Norton}=\infty\$ - which is not physical. So we established, that ideal sources can't be transformed.

Now, we can also see that ideal voltage sources cannot be connected in parallel (unless equal voltage), because they will force two conflicting voltages on the same points, and also two ideal current sources can't be connected in series - as they will force two conflicting currents in the same wire. We can also see contradictions if we replace the resistances in the above diagrams with shorts.

So the above story was to address the preposition in the question:

"If we can transform current and voltage sources to each other, then we should be able to connect them both in series and in parallel" - which is a true proposition.

Another one (reworded) is "Current sources can't be connected in series and voltage sources can't be connected in parallel" - is true only for ideal sources, which cannot be transformed to each other as shown above.

Therefore both propositions are true and not conflicting, because the first one is speaking about non-ideal sources, the second one is about ideal ones.