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CHAPTER 9 BRIDGES, STRAIN GAGES AND SOME VARIABLE IMPEDANCE TRANSDUCERS Many transducers translate a change in the quantity you wish to measure into a change in impedance, i.e., resistance, capacitance or inductance. For example a strain gage, changes strain ( l / l ) into a change in relative resistance ( o R/R ). We usually measure voltages and so would wish to change this impedance change into a voltage. There are many ways of doing this, e.g., use a known current source and measure the voltage across the impedance. Unfortunately the change in impedance is often very small compared to the total impedance of the transducer and the resulting signal will have a large DC shift. You are interested in the small deviations of the signal away from this large DC shift. In extreme cases, when measuring the total voltage with a data-acquisition system (ADC+PC) the quantization errors may be of the same order as the deviations in voltage you wish to measure. BRIDGE CIRCUITS A bridge circuit solves this problem by creating a voltage output that is proportional to the change in impedance rather than the absolute value of the impedance. Such a bridge circuit is shown below. Sometimes 1 Z is part of the transducer and we say that it is an active arm of the bridge, and the other impedances 2 3 4 Z , Z , and Z are rest impedances. In other cases more of the impedances are active. For example, as we will see later, sets of two or four strain gages are used together in a bridge to increase sensitivity and also to compensate for temperature effects. U P Z 1 Z 2 V supply S Z 4 Q Z 3 V out T R Figure 1: A bridge circuit.
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9-2 The output voltage of this bridge may be calculated by using Kirchoff's laws on circuits UPSRT and UPQRT. We will assume that no current flows to the next instrument, i.e., it is disconnected, or the next instrument has a very high input impedance. This output is, of course, the Thevenin voltage. The current flowing in RTUP is I, and the currents flowing in PSR and PQR are 1 I and 2 I , respectively. supply 2 2 2 3 V I Z I Z (1) supply 1 1 1 4 V I Z I Z (2) out 1 4 2 3 V I Z I Z (3) 1 1 2 2 I Z I Z Substituting expressions for 12 I and I , derived from the first two of these equations, into the third equation yields 1 2 1 3 2 4 out supply supply 1 4 2 3 1 4 2 3 Z Z Z Z Z Z V V V Z Z Z Z (Z Z )(Z Z ) . (4) We will use this equation many times in this chapter. The output of this circuit is often a small voltage and so it is usually connected to some kind of amplifier. To examine what happens when the bridge is connected to the amplifier we need to generate the Thevenin equivalent circuit. The output voltage calculated above is the Thevenin voltage. We now need to calculate the Thevenin impedance or output impedance of the bridge. From this and knowledge of the input impedance, and transfer function of the amplifier we can generate an expression for the output of the amplifier.
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