Introduction

What is a Bridge Circuit?

Balancing a Bridge

Sensitivity of Bridge Circuit Output Voltage

Problems

        Making measurements with sensors is a common way in which many engineers and scientists encounter electrical devices.  There are many different ways in which physical variables like temperature, light intensity, pressure and numerous other physical variables can be measured electrically.  Devices used to measure a physical variable are called sensors.  Some different kinds of sensors include the following.

• Sensors which change resistance as the physical variable changes.
• Thermistors for temperature measurement
• Photo-resistors for light measurement
• Strain gages for measurement of mechanical strain
• Sensors which produce a voltage change for a change in a physical variable.
• Thermocouples for temperature
• Solar cells for light.

• Sensors which produce some other sort of electrical change.  Some examples might include the following.
• A tachometer that produces a frequency proportional to rpm.
• Sensors which produce a set of signals in binary code proportional to pressure.
• Sensors which produce a voltage signal with a frequency proportional to flow rate.

What we found in the voltage divider lesson is the following.

• At some nominal value of the temperature (if we have a temperature sensor) the voltage divider will have a nominal output voltage.
• As the temperature changes the voltage output of the voltage divider changes.
• The voltage change will probably be a nonlinear function of temperature as the temperature deviates from the nominal value of the temperature.

• You might want an output signal that is zero at the nominal conditions.
• You could then have an output signal that was positive when the temperature deviated in one direction and a negative output signal when the temperature deviated in the other direction.
• If the outut is something other than temperature that becomes even more clear.
• If the sensor is a strain gage, the strain could be positive or negative, and you would want the output voltage to be similarly positive or negative, and ideally it would be proportional to strain.
• If the sensor is a pressure sensor, you could want a positive signal for a presssure above atmospheric (or any other reference) and negative for a pressure below.

        What is a bridge circuit?  It's easier to look at one than to try to describe it.  Here is a bridge circuit.

The bridge circuit has two arms (R_a and R_b constitute one arm here, and R_c and R_s constitute the other arm).  Each arm is composed of two resistors in series, and you may want to think of each arm as a voltage divider.  The output is the difference between the outputs of the two voltage dividers.  In the bridge circuit above we have also included some source resistance for the source which drives the bridge circuit.  This is the circuit we want to understand.

        What are you trying to do in this lesson?

• Given a sensor that changes resistance as some physical variable changes,
• Be able to use the sensor in a bridge circuit.
• Be able to choose components for the bridge circuit that will produce good performance.

        We have noted that it might be possible to get a bridge output of zero volts.  That's true, but it only happens under certain conditions.  When the output of a bridge is zero, the bridge is said to be balanced.  The first thing we will do is to determine the conditions for a bridge circuit to be balanced.

        If the output voltage of a bridge circuit is zero, that will happen when the outputs of both dividers is the same.  Here's the bridge circuit again.  We'll probably have to be looking at it as we make this argument.

        The first thing that we notice is that both voltage dividers have the same voltage at the "top" of the bridge.  Call that voltage V_top.  Then, the voltage at the left terminal (labelled "+") is given by:

Similarly, the voltage at the right terminal (labelled "-") is given by:

The difference between these two voltages - the output voltage - is given by:

Setting the output voltage to zero (the condition for a balanced bridge), we get:

Since V_top is a common factor it can be removed.  Then, we get:

Now, cross-multiply the denominators.

Note that the term R_sR_b appears on both sides of the equation and can be taken out on both sides.  That gives us:

This is the condition for balance that we were looking for.  It is a very simple relationship that must be obeyed by the resistors in the bridge portion of the circuit.

Q1  For these values of resistance would the bridge be balanced?

• R_a = 1000 W
• R_b = 2000 W
• R_c = 3000 W
• R_s = 6000 W

P1   For these values of resistance determine the value of R_s for which the bridge is balanced.

• R_a = 1000 W
• R_b = 2000 W
• R_c = 5000 W

• R_b = 1000 W
• R_c = 2000 W
• R_s = 5000 W

• R_a = 10 W
• R_b = 10000 W
• R_c = 10 W

P4   For these values of resistance you should have found that the bridge would be balanced.  (These are the values from Question 1 above.)  For these values, what are the values of V₊ and V_-?  (And remember, both values of voltage are the same when the bridge is balanced.)

• R_a = 1000 W
• R_b = 2000 W
• R_c = 3000 W
• R_s = 6000 W

• V_c = 10v
• R_o = 0 W (Ideal voltage source)

P5   For these values of resistance bridge would be unbalanced.

• R_a = 1000 W
• R_b = 2000 W
• R_c = 3000 W
• R_s = 6600 W

• V_c = 10v
• R_o = 0 W (Ideal voltage source)

        Now, we want you to check another situation.  Here's the problem.

• R_a = 10 W
• R_b = 10000 W
• R_c = 10 W
• R_s = 10000 W

        There is an issue buried in here.  By now you should realize that it is possible to have the bridge balanced, but it might be unusable because the voltages out of the voltage dividers in the two different arms fall into one of these two categories.

• The voltage out of the voltage dividers can be very small (close to zero) if R_s and R_b are both small compared to the other resistors.
• The voltage out of the voltage dividers can be very close to the supply voltage if R_s and R_b are both large compared to the other resistors - the situation we contrived above.

        Earlier, we found that the output voltage from the bridge was given by:

We want to determine the conditions for which this output voltage changes the most for a given change in a resistor - and we will choose the sensor resistance.  We start by computing the rate of change of the output voltage as a function of R_s and R_c.  The derivative of output voltage (dV_out/R_s.)with respect to the sensor resistance is:

Now, if you want the maximum slope, you would choose R_s = 0. However, you don't have control over the sensor resistance, but you can choose R_c to get the maximum slope.  And, to determine the maximum slop, we differentiate the expression for the slope and set the result to zero.  Doing that, we have the expression below.

And, that tells us that the maximum slope occurs when you pick R_c = R_s.

        Summing that up, if you have a resistive sensor and you want to use it in a bridge circuit, the resistor in series with the sensor should be chosen so that the resistance in series has the same value as the nominal resistance of the sensor.

E1   You have a strain gage with a nominal resistance of 350W.  The resistor in series with the strain gage should have the same value, i.e. 350W, if you want the bridge to have maximum sensitivity.

        There may be times when you want the output voltage to change linearly with resistance change in the sensor.  In that case, recall the result above.

The only way that the output voltage will change linearly with the sensor resistance is if you have:

Then, if you have that situation, the output from the right leg of the bridge will be almost zero, and you will have sacrificed sensitivity for linearity.  It will be a tradeoff, and you will have to make the call.

• Problem Resist4P01 - Calculating a resistance value to balance a bridge
• Problem Resist4P02 - Bridge with thermistor

• Resistors
• RCombinations
• VoltageDividers
• Bridge Circuits
• Resistance Experiments