It is possible to solve this using actual relationships between the circuit's components.
Most of the given answers have it right without going through such trouble though. The only possibilities to have Vo/Vs = 1/2 are unsatisfying solutions:
- A 0 Ohm value for Rx wich leads to a short circuit
- Removing Ro (the 10-Ohm resistor) which changes the terms of the problem
It is indeed possible to quickly reach such a conclusion either with "common" physical sense or a more rigorous approach.
A first method infers a conclusion directly from the exercise's constraint : Vo/Vs = 1/2
In other words, this voltage relationship means the circuit divides the source's voltage in two halves :
- A 1st "Vs/2" voltage across the isolated Rx resistor terminals
- A 2nd "Vs/2" voltage across the two resistors Rx and Ro in parallel
From there, it is easy to conclude the impossibility to solve this, since two different resistive circuits with different values (the single Rx and the paralleled Rx and 10-Ohm) would have an equal voltage at their terminals. As a reminder, the paralleled resistors have an equivalent resistance equal to 1/(1/Rx + 1/Ro), obviously different from Rx (unless Ro has an infinite resistance, ie. is equivalent to an open circuit, or simply said no branch at all)
It is therefore a voltage divider problem, and the only way to divide Vs in two equal halves with the same Rx resistance is as follows :
simulate this circuit – Schematic created using CircuitLab
The solution above is actually equivalent to having an infinite value for Ro, which is like simply removing it.
The other possibility would be to not use the same Rx resistor in the two branches. It is then funny (perverse?) your teacher framed the problem through adjusting Rx's value when it is actually the fixed 10-Ohm resistor value which should be changed...?
- A second method describes the physical relationship the components have between each other, through Kirchhoff's laws. From this diagram :
simulate this circuit
You can infer the following relationships :
- (1) Vs = Vx - Vo (mesh rule)
- (2) Vx/Rx = Vo/Rx + Vo/Ro (sum of currents from Ohm law and junction rule)
- (3) Vo/Vs = 1/2 (constraint your teacher introduced)
The problem has two parameters : the resistance value of Ro (10 Ohm in this case) and the ratio between Vo and Vs (let's call it "alpha")
For the sake of genericity, I replaced the fixed 10 Ohm value by Ro. The answer can thus be generalized for any values.
Using the three equations above, and substituting Vx with Vs - Vo you arrive at the following result :
Rx/Ro = Vs/Vo - 2
Rx = Ro(Vs/Vo - 2)
Now we have a direct relationship between Ro and Rx so that the coefficient between Vs and Vo is respected.
As you can see, the value of "2" you teacher gave for Vs/Vo is the trivial solution, which yields "Rx = 0"
Had your teacher asked the same question for a Vo/Vs ratio of 1/5, then we'd have :
Vs/Vo = 5
Rx = Ro (5 - 2) = 3 x Ro
With our 10 Ohm value for Ro, Rx would require a 30 Ohm resistance.
Finally, here's a plot of Rx against alpha values :
> It has the form of an invert function
> It yields positive (Rx) values only for alpha ratios between 0 and 1/2 (both of which we know are exluded : the former implies a 0V value for Vo, the latter we proved was a short circuit)