11.1.1.6 The Ideal Voltage Source The ideal constant-voltage source Figure 11.1g will maintain a constant voltage drop across its terminals, no matter what.. 11.1.1.7 The Ideal Current S
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Notice that an instantaneous change in current requires an infinite voltage drop across the inductor If an attempt is made to switch an inductive load (a circuit that is being electrically powered and contains an inductive element) off, the current has to go somewhere In the absence of any provision made for this,
it will break down the insulation of the switch, causing a spark (arc)
Also note that the inductor is a short circuit to DC and an open circuit at high frequencies
11.1.1.6 The Ideal Voltage Source
The ideal constant-voltage source (Figure 11.1g) will maintain a constant voltage drop across its terminals, no matter what This means current may flow into or out of either terminal, and it can even supply the practically impossible infinite current necessary to charge a capacitor instantaneously (This makes it somewhat similar to an ideal wire with a constant voltage drop placed across it.)
11.1.1.7 The Ideal Current Source
The ideal constant-current source (Figure 11.1h and Figure 11.1i) will maintain
a constant current flow in the direction indicated by the arrow, no matter what voltage drop appears across it (This makes it somewhat similar to an ideal open circuit with a constant current flowing through it.)
The previous descriptions may be a little counterintuitive (current flowing through an open circuit?), but they are useful Deeper consideration of the con-cepts introduced in this section will reveal other useful qualities or analogies, some of which will become apparent later (see the book by Bogart for a complete introduction to the subject [1])
11.1.1.8 Controlled Sources
Often, the voltage or current in one part of a circuit is mathematically related to
a voltage or current in another This is represented in the circuit diagram by the
More Units
• Resistance is measured in ohms — Ω
• Capacitance is measured in farads — F
• Inductance is measured in henries — H
Note on Strange-Seeming Analogies
In the case of the ideal current source, it can be found in Subsection 11.1.1.2 that no current will flow through an open circuit irrespective of the voltage dropped across it Turning the analogy around, one could argue that the open circuit is a special case of the ideal current source, one with a 0-A current rating DK3182_C011.fm Page 243 Friday, January 13, 2006 11:01 AM
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controlled voltage and current sources (Figure 11.1j and Figure 11.1k) In this case, the voltage appearing across the terminals (or current flowing through the component) is mathematically related to some other parameter in the circuit — the relationship normally being noted next to the source Otherwise, these ideal components exhibit the same properties as the constant sources mentioned before Controlled sources appear in models for amplifying components, where the output
is some multiple of the input, for example
11.1.1.9 Power Calculations
Power can be computed by multiplying the voltage drop across a circuit element
by the current passing though it:
(11.6)
In the case of a resistor (or resistive load), this equation can be readily employed to yield the power dissipated by the component Ohm’s law can be employed to transform Equation 11.6, in which case it will be seen that the power dissipated is equal to the square of the current multiplied by the value of the resistor; one may conclude that reducing the current flowing through a resistive element is most effective in reducing the power dissipated In the case of reactive loads (capacitive or inductive loads), the same equation may be employed, but the power will depend on the waveform being applied
If Equation 11.6 is applied to voltage or current sources, it will normally yield a negative result — the voltage drop across the circuit element being in opposition to current flow This indicates that the source is supplying energy
to the circuit, not dissipating it As capacitors and inductors both store energy, they extract energy from the circuit (positive value) during one part of the cycle and deliver it (negative value) during another part of the cycle (the signs noted in parentheses being correct if normal sign conventions have been followed)
11.1.1.9.1 Switching Losses
The ideal switch acts either as a short circuit (on state) or open circuit (off state)
In either state, no power would be dissipated by the switch The nonideal switch (in this case the transistor can be considered when employed in a switching capacity) would appear as a very-low-value resistor when in the on state and a very high value resistor when in the off state It would therefore dissipate some power when in either (static) state During the act of switching, however, the current rises (or falls) as the voltage dropped across the switch falls (or rises)
At high switching frequencies, this dynamic power dissipation (power dissipated during a change of state) is typically greater than the static power dissipation This is particularly true when highly capacitive loads are involved because the time taken to charge and discharge the capacitance through the (ideal and non-ideal) resistances in the circuit and, hence, the time taken for the switch to open
or close will be relatively long
p=iv
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11.1.1.10 Components in Series and Parallel
Component elements may be placed in series (Figure 11.4a, Figure 11.4c, and Figure 11.4e) or parallel (Figure 11.4b, Figure 11.4d, and Figure 11.4f) Consid-eration of Equation 11.1 to Equation 11.5 and Figure 11.4 will show that several instances of the same component connected together in such a network will have the same combined effect as a single instance of that component of a value computed as follows:
resistors in parallel (11.8)
capacitors in parallel (11.10)
inductors in parallel (11.12)
FIGURE 11.4 (a) Resistors in series, (b) resistors in parallel, (c) capacitors in series, (d) capacitors in parallel, (e) inductors in series, (f) inductors in parallel.
L1 L2
L3
C1 C2 C3
(c)
C1 C2 C3
(d)
L1 L2 L3 (e)
(f )
R1 R2 R3
(b) R3
R2 R1
(a)
R TOTAL =R1+R2 + +Rn
R
TOTAL =
1 1 1
1 2
1
C
TOTAL=
1 1 1
1 2
1
C TOTAL =C1+C2 + +Cn
L TOTAL =L1+L2 + +Ln
L
TOTAL =
1 1 1
1 2
1
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11.1.1.11 Kirchoff’s Laws
The consideration alluded to in the previous section gives rise to Kirchoff’s current and voltage laws Simply stated, the current law states that current cannot simply appear from or vanish into nothing The voltage law does the same for voltages
Kirchoff’s current law, more formally, states that the sum of currents flowing into a node (a junction between components) must be zero In Figure 11.5a:
(11.13)
It should be recalled that when analyzing a circuit and computing the actual value of a current, a positive result means that the current actually flows in the direction indicated, i.e., into the node, and a negative result will indicate that the current in fact flows in the direction opposite to that indicated, i.e., out of the node Figure 11.5b shows the junction between just two components, two resistors, marked up for analysis such that:
(11.14) Clearly, at any instant that the circuit is in operation the current will flow into the node through the one resistor and the same current will flow out through the other (unless, of course, no current at all flows) This is the same as saying that the same current flows through both resistors, and Kirchoff’s current law formal-izes this for multiple currents (components)
Kirchoff’s voltage law states that the sum of voltage drops around a closed path in a circuit must be equal to zero This is explained in Figure 11.6 Consid-ering Figure 11.6a, it is apparent that the voltage dropped across the circuit enclosed by the box, V AB, must be equal to the source voltage (V SOURCE) Starting
at node A in the circuit and working clockwise, applying Kirchoff’s voltage law,
it can be seen that:
(11.15)
Notice that the source voltage appears with a negative sign in the equation This is because in working clockwise around the circuit the voltage dropped
FIGURE 11.5 (a) Three current nodes, (b) two current nodes.
I3
I1+I2+I3=0
I1+I2=0
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connections to allow certain (internal) parameters of the amplifier itself to be
adjusted — an offset null being common
The difference between the voltage applied to the noninverting input (VIN+)
and the inverting input (VIN) is commonly termed the error voltage, marked as
e in Figure 11.7 The op-amp has a very high open-loop gain, AOL, and the output
voltage is related to the input error voltage by the formula:
(11.18) Open-loop gains in the region of 200,000 or greater are not unusual
11.2.1 T HE I DEAL O P -A MP
The ideal op-amp is principally characterized as having:
• infinite open-loop gain
• infinite input impedance
• zero output impedance
Additional points can be appended to this list Suffice it to say that the two
inputs do nothing more than sense the voltages appearing at them, and the output
acts as a perfect voltage controlled voltage source
FIGURE 11.7 Operational amplifier.
Gain, Transconductance, and Transimpedance
The gain of a circuit is the ratio of the output parameter of interest to the input
parameter of interest Normally this will be either the voltage gain (output
voltage/input voltage) or current gain (output current/input current); as a
con-sequence it is represented by a dimensionless number Sometimes it will be
the ratio of the output current to the input voltage or the output voltage to the
input current In the former of these two cases, it is commonly referred to as
the transconductance and normally measured in siemens, the unit of
conduc-tance (the reciprocal of resisconduc-tance) In the latter, it is normally referred to as
transimpedance and measured in ohms Examination of Equation 11.1 will
make it apparent as to why these terms and units arerelevant
+
−
V in−
Vin+
+
e
V OUT = ×e A OL
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The issue of infinite open-loop gain appears nonsensical, but, in practice, it
means that for a noninfinite VOUT:
This enables an amplifier with an arbitrary closed-loop gain to be constructed
using an op-amp and two external resistors Figure 11.8 shows an op-amp in
inverting configuration with negative feedback (resistor RF being the feedback
resistor)
If an ideal amplifier is assumed, no current flows into or out of the inputs
(infinite input impedance) Thus, by applying Kirchoff’s current law it can be
seen that the current flowing through RF also flows through R1 Furthermore,
given that the ideal op-amp has infinite open-loop gain, from Equation 11.19 it
can be seen that the voltages at both input terminals of the amplifier are the same
As one of these (the noninverting input) is connected to 0 V (or ground,
symbol-ized by the three diminishing lines; the symbol is not normally labeled), it follows
that the voltage at the inverting input must also be 0 V Applying Kirchoff’s
current law then gives:
(11.20)
This can be rearranged to give the classic equation for the closed-loop voltage
gain (ACL) of an ideal op-amp in inverting configuration (Figure 11.8):
(11.21)
The gain of the resulting amplifier is set by the ratio of the two resistors, and
it is called an inverting amplifier because of the negative gain.
FIGURE 11.8 Inverting amplifier.
RF
Vout
Vin R1
− +
V R
V R
OUT F IN
−
V
R R
CL OUT IN
F
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A similar analysis can be applied to the noninverting configuration of Figure 11.10, yielding:
(11.22)
Because the input signal is connected directly to the very-high-impedance noninverting input of the op-amp in the noninverting configuration and the gain will be at least unity, it is common to see it used as a unity-gain buffer amplifier, the inverting input being connected directly to the output with no resistors involved
FIGURE 11.9 Potential divider.
The Potential Divider
Although gains of less than unity can be achieved with an inverting amplifier,
a signal that is too large can be more economically attenuated (reduced in amplitude) by a potential (or voltage) divider This consists of two resistors
in series (Figure 11.9), characterized by the equation:
FIGURE 11.10 Noninverting amplifier.
Vin
R2
R1
Vout
V OUT =V IN R R+2R
1 2
V V
R R
OUT IN
F
= +1 1
RF
Vout
Vin
R1
− +
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Consideration of the concepts presented in the previous sections of this chapter will show that if the object is to measure a voltage that arises from a source that is not an ideal voltage source (i.e., it could be modeled as an ideal voltage source — the signal that is to be measured — within a network incorpo-rating at least one series resistor and possibly capacitive and inductive elements
as well), then connecting this source to an input with a finite impedance will degrade the signal; exactly what will happen will depend on the nature of the source and the input impedance of the measuring circuit
11.2.1.1 Nonideal Sources, Inverting, and Noninverting
Op-Amp Configurations
Op-amps are used to condition signals from sensor elements For the most part, these are not going to be ideal voltage or current sources Most nonideal voltage sources can be modeled as an ideal voltage source, Vs, in series with a source
Decibels
Gains are often expressed in decibel units These are formally defined in terms
of power:
gain in dB
It is, however, more common to find voltage gains expressed in dB:
gain
Filter cutoff points (Section 11.3 of this chapter) are taken to be the frequencies at which attenuation is by a factor of 3 dB Beyond this, attenuation may well increase by a factor that is a multiple of –20 dB per decade (i.e., the signal is attenuated by an additional 20 dB, 40 dB, 60 dB, etc., for every factor-of-ten increase in frequency, depending on the design of the filter)
Power Transfer
In many cases, it is required to transfer maximum power from the source to the input of the measuring circuit In this case, the input impedance of the measuring circuit should be the same as the output impedance of the source
10log10 P
P
out in
20log10 V
V
out in
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impedance, Zs, as illustrated in Figure 11.11a The nonideal current source is illustrated in Figure 11.11b For simplicity, only a nonideal voltage source with
a resistive source impedance (source resistance, Rs) will be considered, as illus-trated in Figure 11.12 This is a very common model for a signal source Returning to Figure 11.10, it can be seen that the input signal is connected directly to the high-impedance input of the op-amp When the nonideal source
of Figure 11.12 is connected to this configuration (Figure 11.13a), it is apparent that no current, ideally, flows through Rs It follows that there is no voltage drop across Rs, and therefore VIN+ = Vs In other words, the ideal circuit amplifies the signal of interest, Vs, according to Equation 11.22
Turning to Figure 11.8, it is apparent in the inverting amplifier configuration, the source “sees” one end of R1 It has already been noted that in this case, VIN will be the same as VIN+, which in this case is tied to 0 V The source sees the resistor, R1, connected to ground In Figure 11.13b, it can be seen that Rs and R1 form a voltage divider (see box titled “The Potential Divider”) that attenuates the signal of interest before it can be amplified This has noise implications, as will be discussed later, as well as design implications
The overall gain of the circuit will, therefore, be dependent on Rs As this is beyond the designer’s control, the noninverting configuration is preferred The noninverting configuration is also used to buffer voltage dividers and filters constructed using passive components (capacitors, inductors, and resistors) before the signals enter the next stage of a circuit
FIGURE 11.11 (a) Nonideal voltage source, (b) nonideal current source.
FIGURE 11.12 Nonideal voltage source with the source impedance, Zs, replaced by
a resistance, Rs.
+ Vs
Zs
(b) (a)
+ Vs Rs
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minimize offsets, and still others have provision for an external potentiometer (variable resistor) to be attached so that the offset can be adjusted once the circuit has been built Even without this provision, it is always possible to design an external circuit that will accommodate offsets (For production versions, the design should avoid potentiometers, as the time taken to set them up adds con-siderably to the final costs)
11.2.2.1 Bandwidth Limitations and Slew Rate
The incredibly high gain of the op-amp leaves it susceptible to instability, unless something is done to prevent this
When discussing the capacitor (Subsection 11.1.1.4), it was noticed that at high frequencies the capacitor appeared as a short circuit In an electrical circuit there are lots of parasitic (unwanted) capacitors where two conductors run near to one another There may also be inductive loops that allow one signal to couple mag-netically to another If the op-amp exhibited its incredibly high gain at all frequen-cies, it is quite likely that high-frequency signals at the output would couple back
to the input through these routes and cause the whole system to oscillate Therefore, op-amps normally only operate at their maximum open-loop gain
up to a few hertz (1 Hz to 10 Hz is typical) Thereafter, the open-loop gain of the amplifier drops off at, usually, about 20 dB per decade The frequency at which the gain becomes 1 (i.e., unity) can be found from the gain–bandwidth product listed
in the amplifier’s data sheet Typically, this will be in the 1 MHz to 10 MHz range The op-amps chosen must have sufficient open-loop gain at the frequency of interest for it to operate As the open-loop gain approaches the closed-loop gain, the op-amp will cease to approximate an ideal one and the design equations discussed previously will fail Most modern op-amps will operate at moderate closed-loop gains up to frequencies in the 100-kHz range or above, and amplifiers with hundreds of megahertz bandwidth are available, although special care has
to be taken when designing circuits that involve such high frequencies
It may sometimes be found that although an op-amp with sufficient gain–band-width product has been selected, the circuit still fails to function as expected, with the output waveform being distorted in shape — particularly for square and other non-sinusoidal waves This is due to another limiting parameter of the nonideal op-amp, the slew rate This is the maximum rate at which the output of the amplifier can change and is usually expressed in volts per microsecond It can be thought of
as the op-amp’s ability to track a fast-moving input signal
Gain–Bandwidth Product and Slew Rate
• The gain–bandwidth product (in Hz) gives the frequency at which the open-loop gain of the op-amp drops to unity
• The slew rate (in V/µsec) is the maximum rate of change that the
op-amp output is capable of