# Circuits Analysis and Synthesis

#### nazmi107

Joined May 5, 2020
3
Hello everyone,

In my question, there is a missing resistor. I informed my lecturer about it and he said I might not need that resistor. However, I could not overcome the problem without using the resistor. I tried to apply mesh analysis but I could not get any solution. What kind of strategy can i use? Do you have any suggestions?

#### Papabravo

Joined Feb 24, 2006
14,829
Where do you think there should be a resistor where there is no resistor?

#### nazmi107

Joined May 5, 2020
3
Where do you think there should be a resistor where there is no resistor?
I mean, the resistor where stands in middle has no value. Sorry for mistelling

#### Papabravo

Joined Feb 24, 2006
14,829
I mean, the resistor where stands in middle has no value. Sorry for mistelling
Given the instructors response I'm guessing it might be irrelevant. Before the switch is thrown can you establish what the voltages and currents might be and see if it is conceivable that no current flows in that branch?

#### nazmi107

Joined May 5, 2020
3
Given the instructors response I'm guessing it might be irrelevant. Before the switch is thrown can you establish what the voltages and currents might be and see if it is conceivable that no current flows in that branch?
When I try to obtain these values before the switch is thrown, I realize that I need to use that resistor. I think I'm making a mistake about my method somewhere. I'm confused tbh

#### t_n_k

Joined Mar 6, 2009
5,455
One simple approach is to analyze for the two limiting cases where the resistance is either zero (short circuit) or infinite (open circuit). If the result is different you need to know the actual value to give a specific solution - otherwise you can only give a range of possibilities.

#### MrAl

Joined Jun 17, 2014
7,872
I mean, the resistor where stands in middle has no value. Sorry for mistelling
Hello,

One way to handle this is to simply call the ghost resistor "Rx" and then solve the problem.
Once you do that, yo have an expression or expressions for whatever you are supposed to solve for (voltage, current, etc.) that includes the ghost value Rx and then you can easily determine if Rx makes any difference or not. If it does not, then you have your result and the value Rx probably cancels out of the equation(s). If it does, then you have the expression you need so if you ever find out what the value of Rx really is you can just plug it into the final result.

For a really really simple example, say you have a voltage divider with Rx on top and 2k on the bottom fed by a DC voltage of 5 volts. The solution for a voltage divider with those resistors is:
Vout=5*2000/(2000+Rx)
and that would be the solution. In this case Rx matters but if it cancels out then it does not matter.
It is also possible that for some circuits ANY value of Rx produces the same output. It would usually cancel out but it could be that you have to think about it a little once you have the expression.

#### RBR1317

Joined Nov 13, 2010
591
In my question, there is a missing resistor.
Do you mean there is a missing resistor, or a missing resistor value? For calculation of steady-state values, the capacitor can be removed from the circuit. For t<0 the missing resistor value is not necessary, but for t>0 it is.

#### WBahn

Joined Mar 31, 2012
26,398
Do you mean there is a missing resistor, or a missing resistor value? For calculation of steady-state values, the capacitor can be removed from the circuit. For t<0 the missing resistor value is not necessary, but for t>0 it is.
While we might not need to know the value of the resistor to determine v_o(t) for t < 0, we still need to know it to determine the voltage on the capacitor prior to t = 0. We need that voltage since that is what carries across the t=0 event.

It's possible that things work out so that these offset each other and it ends up having no effect on v_o(t) even though it might affect other parts of the circuit. Doing the analysis symbolically should reveal whether or not that's the case.

#### MrAl

Joined Jun 17, 2014
7,872
While we might not need to know the value of the resistor to determine v_o(t) for t < 0, we still need to know it to determine the voltage on the capacitor prior to t = 0. We need that voltage since that is what carries across the t=0 event.

It's possible that things work out so that these offset each other and it ends up having no effect on v_o(t) even though it might affect other parts of the circuit. Doing the analysis symbolically should reveal whether or not that's the case.
Hi there,

I think the reason myself and someone else in this thread went ahead with the idea that the value may not have to be known is because in some circuits a resistor happens to be across two nodes that have exactly the same voltage. For example, a balanced resistance bridge circuit.
That doesnt mean that every circuit will have this property for one or more resistors, and it does not mean this one will either, it was just a suggestion to spark the imagination and possibly get the person to think a little more. Maybe you went ahead and solved it and found that isnt true for this particular circuit that's fine too but i would not want to tell the person who posted this question. I wanted them to find out for themselves.

#### WBahn

Joined Mar 31, 2012
26,398
Hi there,

I think the reason myself and someone else in this thread went ahead with the idea that the value may not have to be known is because in some circuits a resistor happens to be across two nodes that have exactly the same voltage. For example, a balanced resistance bridge circuit.
That doesnt mean that every circuit will have this property for one or more resistors, and it does not mean this one will either, it was just a suggestion to spark the imagination and possibly get the person to think a little more. Maybe you went ahead and solved it and found that isnt true for this particular circuit that's fine too but i would not want to tell the person who posted this question. I wanted them to find out for themselves.
I haven't solve the circuit, either. Being a dynamic circuit after the switch is thrown, it would seen unlikely that everything would stay so nicely balanced as to have no impact on the voltage across the output resistor, but I can't quite categorically rule it out.

• MrAl

#### t_n_k

Joined Mar 6, 2009
5,455
As I suggested in post #6 it is instructive to consider two limiting cases for the undesignated resistance (call it Rx) value - namely either a short circuit or an open circuit. At the same time I speculate as to the steady state values of Vo(t) for the limiting cases of Rx. There will be an assumed steady state prior to t=0 and a final steady state when the switch has been closed for a very long time - as time t tends to infinity. At either steady state condition we can treat the capacitor as an open circuit. So for the two limiting cases of Rx, what was the steady state value of Vo(t) prior to the switch closing at t=0. Also what would the steady state value of Vo(t) be for the same limiting cases of Rx be at t=infinity? Remember to treat the capacitor as an open circuit, since we are interested only in steady state conditions before and after the switch is closed at t=0. If the two steady state conditions are independent of Rx being either a short or open circuit, then Rx can be excluded altogether from the analysis. On the other hand if there are differences in the steady state conditions either side of t=0 subject to the two limiting values of Rx, then we can't ignore its presence and we need a specific value.

#### t_n_k

Joined Mar 6, 2009
5,455
My view is that we can't ignore the unknown resistance and the question is incomplete.

#### WBahn

Joined Mar 31, 2012
26,398
If the two steady state conditions are independent of Rx being either a short or open circuit, then Rx can be excluded altogether from the analysis.
I agree with everything except this statement. While this is a necessary condition to being able to ignore Rx, it is not sufficient. Even if the value of Rx doesn't affect the initial and final conditions, it could affect the time constant and thus the transient response as it transitions from one steady state condition to the other.

#### t_n_k

Joined Mar 6, 2009
5,455
I agree with everything except this statement. While this is a necessary condition to being able to ignore Rx, it is not sufficient. Even if the value of Rx doesn't affect the initial and final conditions, it could affect the time constant and thus the transient response as it transitions from one steady state condition to the other.
Thanks WBahn. Yes you are correct. In the case where the initial and final ss conditions of Vo(t) are identical for the limiting cases of Rx, I cannot just assume the unknown resistance plays no role. I would have to fully analyze the circuit with the unknown resistance in place to justify discounting it altogether. In the example at hand there is no doubt in my mind that the resistance does play a role, since it effects the final steady-state condition of Vo(t) and on that basis alone, it cannot be neglected. Thanks for pointing out the error in my reasoning.

#### MrAl

Joined Jun 17, 2014
7,872
I agree with everything except this statement. While this is a necessary condition to being able to ignore Rx, it is not sufficient. Even if the value of Rx doesn't affect the initial and final conditions, it could affect the time constant and thus the transient response as it transitions from one steady state condition to the other.
I agree, it would have to have both terminal voltages stay exactly the same over ALL time for it to be ignored completely.

#### MrAl

Joined Jun 17, 2014
7,872
As I suggested in post #6 it is instructive to consider two limiting cases for the undesignated resistance (call it Rx) value - namely either a short circuit or an open circuit. At the same time I speculate as to the steady state values of Vo(t) for the limiting cases of Rx. There will be an assumed steady state prior to t=0 and a final steady state when the switch has been closed for a very long time - as time t tends to infinity. At either steady state condition we can treat the capacitor as an open circuit. So for the two limiting cases of Rx, what was the steady state value of Vo(t) prior to the switch closing at t=0. Also what would the steady state value of Vo(t) be for the same limiting cases of Rx be at t=infinity? Remember to treat the capacitor as an open circuit, since we are interested only in steady state conditions before and after the switch is closed at t=0. If the two steady state conditions are independent of Rx being either a short or open circuit, then Rx can be excluded altogether from the analysis. On the other hand if there are differences in the steady state conditions either side of t=0 subject to the two limiting values of Rx, then we can't ignore its presence and we need a specific value.
I agree except in the case where the response is curved. In that case we could get lucky or should i say unlucky with values 0 and infinity but be wrong with a center value(s) somewhere in between.
That's probably why in most relatively simple curve fitting we should always use at least three points because two only works with a straight line.