# Circuit theory -- Conditions to balance a resistive bridge circuit

• Engineering
CG139
1. Homework Statement

Establish the condition required to make the current through Ze in Figure 1 zero (i.e. For the voltage Vx to equal Vy).

## The Attempt at a Solution

I'm not sure if I'm on the right line here, hopefully someone can point me in the right direction if it's wrong...

If Zb and Zc were the same value then the voltage drop across them would be equal, likewise Za and Zd would also have to be the same value to have voltage Vx equal to Vy. In this condition there will be no current flowing through Ze.

Is my attempt correct or at least on the right path?

Staff Emeritus
Hi CG139. http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif [Broken]

If Zb and Zc were the same value then the voltage drop across them would be equal, likewise Za and Zd would also have to be the same value to have voltage Vx equal to Vy. In this condition there will be no current flowing through Ze.
You've indicated here a special case of the more general right answer. :)

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2021 Award
To add to NO's statement, you have selected not only a special case but a VERY special case ... that is, it is NOT enough that Zb=Zc or that Za=Zd it would be necessary that BOTH of those conditions occur at the same time. This is a very special case indeed.

Think about the relationships between Za to Zb and Zc to Zd

If Zb and Zc were the same value then the voltage drop across them would be equal, likewise Za and Zd would also have to be the same value to have voltage Vx equal to Vy. In this condition there will be no current flowing through Ze.

Is very limited. Specifically, "If Zb and Zc were the same value then the voltage drop across them would be equal" is a very limited case because it then REQUIRES that Za=Zd

Staff Emeritus
Think about the relationships between Za to Zb and Zc to Zd
It is better to allow the poster some opportunity to think about the response, rather than leaping in and doing the thinking for him. In the long run, depriving a poster of the opportunity to review his own work is performing a disservice.

A little restraint can provide a poster the necessary time to process what his attention has been drawn to.

I know it's tempting to scramble to fill in what a previous poster has left unsaid. It was alluded to but left unstated for good reason.

Gold Member
2021 Award
It is better to allow the poster some opportunity to think about the response, rather than leaping in and doing the thinking for him. In the long run, depriving a poster of the opportunity to review his own work is performing a disservice.

A little restraint can provide a poster the necessary time to process what his attention has been drawn to.

I know it's tempting to scramble to fill in what a previous poster has left unsaid. It was alluded to but left unstated for good reason.
Yeah, you are right. I normally do that and in fact I sometimes chide others for not doing it. Thanks.

CG139
Thanks NascentOxygen and Phinds, this subject has had me scratching my head a lot and I'm not sure I've fully understood the replies. Do I just need to expand my statement to include that both conditions need to occur at the same time? When I wrote my original statement I was thinking that, i just didn't make it clear- will try harder.

Homework Helper
Dear CG,

Can you write down an expression for Vx using the variables in this figure ?

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• V_divider.jpg
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CG139
Hi BvU,

Vx = (Za / (Zb+Za)) *Vin

I struggled to get the format how I wanted it. I've tried to use Latex but don't think I can use it as I'm using a tablet rather than a laptop.

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Homework Helper
Very well. Now Vy in this picture:

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• V_div2.jpg
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CG139
Vy = (Zd / (Zc+Zd)) *Vin

Staff Emeritus
Going back to your original diagram. Given these values, calculate ZC for the balance condition (i.e., no current through Ze)

ZA = 5k,
ZB = 3k
ZD = 12k
ZC = ?

Homework Helper
Dear CG,

As O2 isn says more or less, we now go towards vx = vy.
Simply grab together vx = vy, #8 and #10. You see that Vin can be divided out.
That already gives you an adequate answer for the exercise. But it can be made more compact and more elegant by inverting on both sides and subtracting 1.

CG139
NascentOxygen,

Zc = (Zb * Zd) / Za

Zc = (3k * 12k) / 5k

Zc = 7.2k

These values would balance the circuit (Vx=Vy).

BvU,

Sorry, but I don't fully understand #12. Apologies for my poor knowledge, I'm a long way from my comfort zone with this and the material from my Uni doesn't seem to help (not helped by the fact my maths is also poor).

By 'grab together' do you mean simplify to create one expression?

Thanks to you both for your help so far, it is very much appreciated!

Homework Helper
No need to apologize. Yes, grab together is an undiplomatic way to say: if x=u and y=v and x=y then u = v. Mathematicians might say "use symmetry and transitivity", physicists just grab :)

Vx = (Za / (Zb+Za)) *Vin (your post),
Vy = (Zd / (Zc+Zd)) *Vin (your post), and from the exercise wording:
Vx = Vy (we're not interested in either of these two, they just have to be equal)

"Grab together" is then luring you to write (Za / (Zb+Za)) *Vin = (Zd / (Zc+Zd)) *Vin
Vin in itself isn't all that interesting either, bringing us to (Za / (Zb+Za)) = (Zd / (Zc+Zd))

This time you have a general answer, not just something for a special case.
(a few pathological cases have to be excluded, though: Vin = 0, Zb+Za = 0, Zc+Zd=0; no big deal)

That (Za / (Zb+Za)) = (Zd / (Zc+Zd)) already is an adequate answer for the exercise. But it can be made more compact and more elegant by inverting on both sides and subtracting 1. That way you get an expression the exercise is really asking for.

Staff Emeritus