Loading Reduction

An adverse effect of improper impedance conditions is the loading effects, which distort signals. The resulting error can far exceed other types of error such as measurement error, sensor error, noise, and input disturbances. Loading can occur in any physical domain such as electrical and mechanical.

Electrical loading errors result from connecting an output unit such as a measuring device or signal acquisition hardware that has low input impedance to an input device such as a signal source or a sensor with low to moderate impedance. Mechanical loading errors can result in an input device (e.g., an actua-tor) because of inertia, friction, and other resistive forces generated by an output component connected to it (e.g., a gear transmission, a mechanical load).

In engineering systems, loading errors can appear as phase distortions as well. Digital hardware also can produce loading errors. For example, ADC hardware in a DAQ board can load the amplifier output from astrain gauge bridge circuit, thereby affecting the digitized data.

2.3.4.1 Cascade Connection of Devices

To obtain a model for loading distortion and a method for reducing loading effects, we now consider cascade connection of two-port electrical devices. A model for a two-port electrical device is shown in Figure 2.6a. It shows in particular the input impedance Zi and the output impedance Zo of the device.

They are defined in the following.

Input impedance: Input impedance Zi is defined as the ratio of the rated input voltage to the corre-sponding current through the input terminals while the output terminals are maintained in open circuit.

v_o1

v_i Z_i1

+

– –

(b)

+

G₁ G₂

+ – +

v_o v_i2

Z_i2 Zo2

Z_o1

Input v_i Z_i Output

Z_o +

– –

(a)

+ G

v_o v_o= Gv_i

FIGURE 2.6 (a) Schematic representation of input impedance and output impedance and (b) cascade connection of two two-port devices.

Output impedance: The output impedance Zo is defined as the ratio of the open-circuit (i.e., no-load) voltage at the output port to the short-circuit current at the output port. Open-circuit voltage at output is the output voltage present when there is no current flowing at the output port. This is the case if the output port is not connected to a load (impedance). As soon as a load is connected at the output of the device, a current will flow through it, and the output voltage will drop to a value less than that of the open-circuit voltage. To measure the open-circuit voltage, the rated input voltage is applied at the input port and maintained constant, and the output voltage is measured using a voltmeter that has a high (input) impedance. To measure the short-circuit current, a very low-impedance ammeter is connected at the output port.

These definitions are given with reference to an electrical device. However, a generalization to mechanical devices is possible by interpreting voltage and velocity as across variables, and current and force as through variables, as noted earlier. Then, mechanical mobility should be used in place of electrical impedance, in the associated analysis. Similar generalization is possible for other physical domains as well.

It is seen that the input impedance Zi and the output impedance Zo represented in Figure 2.6a agree with their definitions as given earlier. Note that vo is the open-circuit output voltage. When a load is connected at the output port, the voltage across the load will be different from vo. This is caused by the presence of a current through Zo. In the frequency domain, vi and vo are represented by their respec-tive Fourier spectra (or in the complex form with a real part and an imaginary part or a magnitude and a phase). The corresponding transfer relation can be expressed in terms of the complex frequency response (transfer) function G(jω) under open-circuit (no-load) conditions:

vo=Gvi (2.11)

Next consider two devices connected in cascade, as shown in Figure 2.6b. It can be easily verified that the following relations apply:

These relations can be combined to give the overall input/output (I/O) relation:

v Z

We observe from this result that the overall frequency transfer function differs from the ideally expected product (G2G₁) by the factor

Note from Equation 2.12 that cascading has distorted the frequency response characteristics of the two devices, and this represents loading error. The loading error becomes insignificant when Zo1/Zi2≪ 1.

From this observation, it can be concluded that when two components are interconnected (cascaded), in order to reduce the loading error, the input impedance of the second device (output device) should be much larger than the output impedance of the first device (input device).

Example 2.2

A lag network used as the compensation element of a control system is shown in Figure 2.7a.

Show that its transfer function is given by vo/vi = Z2/(R1 + Z2) where Z2 = R2 + (1/Cs).

What are the input and output impedances of this circuit?

Moreover, if two such lag circuits are cascaded as shown in Figure 2.7b, what is the overall trans-fer function? How would you bring this transtrans-fer function close to the ideal result, {Z2/(R1 + Z2)} ²? Solution

To solve this problem, first note that in Figure 2.7a, voltage drop across the element R2 + 1/(Cs) is

v R Cs

R R Cs v

o= + i

+ +

( ( / ))

{ ( / )}

2

1 2

1 1

C –

(b) –

v_i R₂

R₁

+

Z₂

R₂

C Z₂

v΄_o R₁

+

(a) v_i+

–

R2

R₁

+

Z₂ v_o

C

–

Z R₁

v_i+ –

v΄_o +

(c) –

FIGURE 2.7 (a) A single circuit module, (b) cascade connection of two modules, and (c) an equivalent circuit for (b).

Hence,

Next, consider the equivalent circuit shown in Figure 2.7c. Since Z is formed by connecting Z2

and (R1 + Z2) in parallel, we have

1 1 1

2 1 2

Z =Z +R Z

+ (2.2.4)

Voltage drop across Z is

¢ = +

Now apply the single-circuit module result 2.2.1 to the second circuit stage in Figure 2.7b. We get vo=(Z2/(R1+Z2))vo¢. Substituting this into Equation 2.2.5, we get

The overall transfer function for the cascaded circuit is

G v

Now, substituting Equation 2.2.4 for 1/Z we get

G Z

® = +

2.3.4.2 Impedance Matching for Loading Reduction

From the analysis given in the preceding section, it is clear that the signal-conditioning circuitry should have a considerably large input impedance in comparison with the output impedance of the sensor–transducer unit to reduce loading errors. The problem is quite serious in measuring devices such as piezoelectric sensors, which have very high output impedances. In such cases, the input impedance of the signal-conditioning unit might be inadequate to reduce loading effects; also, the output signal level of these high-impedance sensors is quite low for signal transmission, processing, actuation, and control. The solution for this problem is to introduce several stages of amplifier cir-cuitry between the output of the first hardware unit (e.g., sensor) and the input of the second hardware unit (e.g., DAQ unit). The first stage of such an interfacing device is typically an impedance-matching amplifier (or impedance transformer) that has high input impedance, low output impedance, and almost unity gain. This is known as impedance bridging. The last stage is typically a stable high-gain amplifier stage to step up the signal level. Impedance-matching amplifiers are, in fact, op-amps with feedback.

In conclusion, we make the following comments:

1. When connecting a device to a signal source, loading problems can be reduced by making sure that the device has a high input impedance. Unfortunately, this will also reduce the level ( amplitude, power) of the signal received by the device. A stage of signal amplification may be needed then, while maintaining the required impedance levels at the output.

2. A high-impedance device may reflect back some harmonics of the source signal, as we noted under signal reflection. As presented there, a termination resistance (impedance pad) might be connected in parallel with the device to reduce this problem of signal reflection. In many DAQ systems, output impedance of the output amplifier is made equal to the transmission line imped-ance (characteristic impedimped-ance).

3. When maximum power amplification is desired, conjugate matching is recommended. In this case, input and output impedances of the matching amplifier are made equal to the complex conjugates of the source and load impedances, respectively.