General Properties of Electronic Single-Loop Feedback Systems

183

where x is a parameter in the NFB circuit. From Equation [4.39] we can write:

ln[Av (jω)]= ln		Av (jω)		+ jθ(ω)	(4.41)

Substitution of Equation [4.41] into Equation [4.40] yields:

SAv =	d(ln		Av (jω)		)	+ j	dθ(ω)	(4.42)
x	dx x						dx x

The real part of SAx v is called the magnitude sensitivity; the imaginary part of SAx v is the phase sensitivity. Generally, x will be a real number, which simplifies the calculation of Equation [4.42]. Sensitivity analysis using Equation [4.42] can be utilized to evaluate active filter designs.

4.4Effects of Negative Current Feedback

The purpose of negative current feedback (NCF) is to create an approximation to a voltage-controlled current source (VCCS) so that the output current, IL, is proportional to Vs, regardless of the load, which can be nonlinear. In NCF amplifiers, the signal fed back is proportional to the current through the load. NCF, like NVF, extends the bandwidth of the amplifier within the feedback loop. Unlike NVF, NCF raises the output resistance of the amplifier within the loop. A single op amp circuit with NCF, illustrated in Figure 4.5, will be investigated; frequency dependence will not be treated in this example. A VCCS is characterized by a transconductance, GM, and a Norton shunt conductance, Go.

First, it is necessary to find the circuit’s GM = IL/Vs. Note that the current

	is given by Ohm’s law:
	IL	=			Vi′Kvo			(4.43)
				Ro	+ RL + Rc
	The summing junction node voltage is found:
	Vi′[G1 + GF ]− (−ILRcGF ) = G1Vs							(4.44)
					¬
	V′=		G1Vs − ILRcGF					(4.45)
	i			G1 + GF

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184	Analysis and Application of Analog Electronic Circuits
R1	RF
		Ro	Vo
	Vi’
+	−		‘RL’
Vs	KvoVi’
	+			VF
				0
			IL	Rc

FIGURE 4.5

Schematic of a simple Thevenin VCVS with negative current feedback (NCFB).

Substituting Equation 4.45 into Equation 4.43, it is possible to solve for

IL(Vs):
	IL	= GM =	KvoG1	(4.46)
			(Ro + RL + Rc )(G1 + GF )+ KvoRcGF
	Vs
In general, Kvo RcGF (Ro + RL + Rc)(G1 + GF), so
		GM RF/Rc R1 siemens		(4.47)

How does NCF affect the VCCS’s Norton output conductance? First, calculate the circuit’s OCV, given RL •. This condition also makes IL = VF = 0. Now the summing junction voltage is

Vi′= Vs	RF	(4.48)
	R + R
	F 1

and the OCV is:

OCV = Vs	RF	Kvo volts	(4.49)
	R + R
	F 1

Next, find the short-circuit current, i.e., the ILsc when RL = 0. This condition also makes VF = Vo. The SCC is:

	ILsc	=	Vi′Kvo		(4.50)
			Ro	+ Rc

© 2004 by CRC Press LLC

General Properties of Electronic Single-Loop Feedback Systems

where Vi′ is found from:

Vi′[G1 + GF ]− (−ILsoRc )GF = VsG1

	¬
V′=	VsG1 − ILscGF
i	G1	+ GF

Equation 4.52 is substituted into Equation 4.50 and ILsc is found:

I = VsG1Kvo

Lsc (Ro + Rc )(G1 + GF )+ KvoRcGF

185

(4.51)

(4.52)

(4.53)

Now the Norton Gout of the amplifier with NCF is simply the ratio of ILsc to OCV:

			VsG1Kvo					(4.54)
G =	(Ro + Rc )(G1 + GV )+ KvoRcGF					=	1
								siemens
							(Ro + Rc )+ KvoRcR1 (RF + R1)
out			VsRFKvo
			(RF + R1)

The second term in the denominator of Equation 4.54 makes Gout Go, which is desirable in making an effective VCCS.

In order to see how the closed-loop VCCS’s frequency response is affected by NCF, replace the op amp’s open-circuit, VCVS with Kvo/(jωτa + 1). The transconductance is now found to be:

KvoRF

(4.55)

GM (jω) =

(R1

+ RF )(Ro + Rc + RL )+ KvoRcR1

RF (R1Rc )

jω

τa

+ 1

jω

τa(R1 + RF )(Ro + Rc + RL )

+ 1

(R1

+ RF )(Ro + Rc + RL ) + 1

KvoRcR1

KvoRcR1

Now the approximate GM(0) RF/(R1Rc) and the −3-dB frequency is:

fb				KvoRcR1		hertz	(4.56)
			πτa (R1	+ RF )(Ro + Rc	+ RL )
	2

In the preceding relations, it was assumed that KvoRcR1 (R1 + RF)(Ro + Rc + RL). Note that the closed-loop system’s bandwidth is extended by the

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186		Analysis and Application of Analog Electronic Circuits
	Vs R		R
		R
+	−VL	(0)	RF	VL IL
Vs		V2	POA
			Vo	0
				RL
			(VCCS)
		R
			R VL

FIGURE 4.6

A three-op amp VCCS in which the load is grounded. Analysis is in the text.

NCF. The NVF circuit of Figure 4.5 is simple, but suffers the disadvantage of having the RL “floating” between Vo and Rc. Also, the peak current is limited by the op amp used.

In Section 2.6.7 on laser diodes in Chapter 2, it was determined that an effective VCCS using NCF can be made from three op amps. The op amp that drives RL can be made a power op amp and the load can be grounded.

Figure 4.6 illustrates this NCF circuit. As in the earlier example, we will find the VCCS’s GM(jω) and Zout(jω). Because the two feedback amplifiers are

unity gain, they have −3-dB frequencies approaching their op amp’s fT and thus can be treated as pure gains (+1 and −1, respectively). The node equation for the summing junction is written:

Vi′ [3G] − VoG − (−VL)G = VsG						(4.57)
Two auxiliary equations are used:
Vo = − IL (RF + “RL”)						(4.58)
	V	jωτ	a	+ 1
V′= −	o (			)		(4.59)
i		Kvo

Substituting the preceding equations into Equation 4.57 and rearranging terms yields:

IL

Vs

		Kvo
= GM (jω) =		3(RF + RL )+ KvoRF		(4.60)
		τa 3(RF + RL )
	jω		+ 1
		3(RF + RL )+ KvoRF

© 2004 by CRC Press LLC