Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using Zout · 2017. 10. 27. ·...

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Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT

Application ReportSLYA029–October 2017

Closed-Loop Analysis ofLoad-Induced Amplifier Stability Issues Using ZOUT

HarshaMunikoti

ABSTRACTThis application note discusses techniques using amplifier closed-loop output impedance (ZOUT) to solveload-dependent stability issues.

Contents1 Introduction ................................................................................................................... 22 Basic Properties of Electrical Impedance ................................................................................ 33 Properties of Closed-Loop Amplifier Output Impedance (ZOUT) ...................................................... 104 Stability Analysis of a Loaded Amplifier Using ZOUT and ZLOAD ....................................................... 145 Conclusions ................................................................................................................ 216 References .................................................................................................................. 21

List of Figures

1 Impedance Test Circuit ..................................................................................................... 32 Measuring ZCAPACITOR ......................................................................................................... 43 Bode Plots of Ideal Resistive and Reactive Impedances .............................................................. 44 Transient Response Based on Various Pole Locations in S-Plane ................................................... 55 Example Circuit With Both Series and Parallel Impedance Combinations........................................... 66 Unstable Transient Response of Circuit in .............................................................................. 77 Bode Plots of ZEQ for Circuit in ............................................................................................ 78 RLC Circuit of is Stable With R = 2 × √(L/C) ............................................................................ 99 Iterative Stability Improvement of Circuit ................................................................................ 910 ZOUT Test Circuit ............................................................................................................ 1011 Op Amp AOL Model With a Single Pole ................................................................................. 1212 ZOUT Simulation With Single-Pole AOL and Resistive ZO ............................................................... 1313 ZOUT Simulation With Single-Pole AOL and Capacitive ZO ............................................................. 1314 ZOUT Simulation With Single-Pole AOL and Inductive ZO ............................................................... 1315 Example of Circuit With Resonant LC Interaction Between ZOUT and ZLOAD ........................................ 1416 Unstable Load Transient Response of Circuit in ...................................................................... 1417 Adding Series Resistance Eliminates LC Resonant Peak in ZEQ .................................................... 1518 Load Transient Response of Circuit is Stable for 20 Ω < R < 30 Ω ................................................ 1519 Increasing ESR Also Eliminates Resonant LC Peak in ZEQ .......................................................... 1620 Load Transient Response With Increasing ZLOAD ESR ................................................................ 1621 Example Circuit Showing Double-Inductive ZOUT Driving Resistive ZLOAD ........................................... 1722 Unstable Load Transient Response of Circuit ......................................................................... 1723 Adding Series Inductance to ZOUT Stabilizes Load Step Response ................................................. 1824 Double-Inductive Region No Longer Dominates ZOUT With Higher ESL ............................................ 1825 Example Circuit With Double-Inductive ZOUT Driving Capacitive ZLOAD .............................................. 1926 Unstable Load Step Response of Circuit ........................................................................... 19

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27 Modified ZOUT is More Inductive Around fP = 16 kHz ................................................................... 2028 Adding Series Resistance Eliminates Peak in Modified ZEQ Magnitude............................................. 2029 Load Step Response of Stabilized Circuit .............................................................................. 20

List of Tables

1 Computation of Poles and Zeros of ...................................................................................... 62 Pole and Zero Signatures on Bode Plots................................................................................. 83 ZEQ Poles and Zeros With R = 2 × √(L/C) ................................................................................ 84 ZOUT for Resistive ZO and Single-Pole AOL ............................................................................... 115 ZOUT for Capacitive ZO and Single-Pole AOL ............................................................................. 116 ZOUT for Inductive ZO and Single-Pole AOL .............................................................................. 11

TrademarksAll trademarks are the property of their respective owners.

1 IntroductionAmplifier stability is load-dependent. An amplifier that is stable with a resistive load may ring or oscillatewith a reactive load.

The traditional method for evaluating the impact of loading on the stability of a closed-loop amplifierinvolves analyzing Bode plots of the amplifier loaded loop-gain function. The load element interacts withthe amplifier open-loop output impedance (ZO) to alter the frequency response of the amplifier loop gainfunction, and the available stability margins. Analyzing these load-induced changes in loop gain requiresopen-loop conditions or breaking the amplifier feedback loop. Breaking the loop is not possible in the caseof closed-loop amplifier devices such as current-sense amplifiers because the feedback loop is internal tothe device and cannot be manipulated.

Nevertheless, there is a closed-loop amplifier property that is affected by changes to loop gain, and can beused for stability analysis without breaking the loop. This property is the amplifier closed-loop outputimpedance (ZOUT), an increasingly common specification in the data sheets and SPICE macro-models ofTI current sensing and precision amplifier products. This application report demonstrates through TINA-TISPICE simulations a method of using ZOUT to analyze the stability of an amplifier load transient responseunder various load conditions.

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ZBLOCK(f)

ILOAD(f)

VDC (= 0VAC)

- VDROP(f) +

www.ti.com Basic Properties of Electrical Impedance




2 Basic Properties of Electrical Impedance

2.1 Measuring ImpedanceBefore delving into the details of closed-loop stability analysis, a review of a few basic concepts relating toelectrical impedance is helpful. Impedance is the current-to-voltage transfer gain over frequency, of anelectrical circuit or component. The stimulus is an ac small-signal current input of variable frequency, andthe response is the resulting change in voltage at the test frequency. Impedance is then calculated byapplying Ohm’s law to the recorded changes in current and voltage.

Figure 1 depicts the impedance test circuit. The unknown impedance (ZBLOCK) is excited by an ac small-signal current (ILOAD) of frequency f, with the other terminal driven to a fixed dc voltage (VDC). If VDROP(f) andILOAD(f) are both measured at time instants t1 and t2, then by Ohm’s law, the following two equations arederived:

(1)(2)

Subtracting Equation 2 from Equation 1 and solving for ZBLOCK(f) yields Equation 3.

(3)

∆VDROP(f) and ∆ILOAD(f) represent changes in VDROP and ILOAD between time instants t1 and t2. Therefore,there is no difference in the value of ∆VDROP whether VDROP is measured with respect to VDC or with respectto GND.

Figure 1. Impedance Test Circuit

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Basic Properties of Electrical Impedance www.ti.com




Figure 2 shows the test circuit of Figure 1 configured for measuring the impedance of a capacitor (C1). Inthis case, the response is measured directly relative to GND. The 1-GΩ resistor (R1) is for simulation onlyand is required for dc convergence. Capacitor C1 can be substituted with any two-terminal device tomeasure the impedance of the circuit.

Figure 2. Measuring ZCAPACITOR

2.2 Visualizing Impedance Using Bode PlotsImpedance, like any ac gain transfer function, is a complex function of frequency having magnitude andphase, and usually represented using Bode plots to simplify analysis. Figure 3 shows the Bode magnitudeand phase plots of the ideal resistive (purely real) and reactive (purely imaginary) impedance elements.

Figure 3. Bode Plots of Ideal Resistive and Reactive Impedances

Observe that the linear magnitude characteristic of the capacitor over frequency is the result of usinglogarithmic scales for the vertical and horizontal axes.

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Re(s)

j×Im(s)

×

×

× ×

×

×

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2.3 Poles of ZEQ Determine Load Transient StabilityImpedances combine in series or parallel configurations. The combination of resistive and reactiveelements in a circuit produces poles and zeros in the Thévenin equivalent impedance function (ZEQ). Thelocations of the poles of ZEQ in the complex s-plane determine the stability of the circuit response to loadtransients. For a detailed review of pole-zero analysis of system transfer functions, see UnderstandingPoles and Zeros .

For most circuits, a stable transient response is one that converges asymptotically to a finite, steady-statevalue without ringing. Based on Figure 4, the required transient response is obtained from a transferfunction (ZEQ) with poles that are not only all in the left half plane (LHP), but are also purely real (that is, lieon the negative real axis marked by the red line in Figure 4).

Figure 4. Transient Response Based on Various Pole Locations in S-Plane

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http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf


VDROP

C 1

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L 1mH R 10Ohm





2.4 Finding Poles and Zeros of ZEQ

The poles and zeros of ZEQ can be found by solving for the roots of the denominator and numerator,respectively. For example, the ZEQ transfer function of the circuit in Figure 5 is given by Equation 4.

Figure 5. Example Circuit With Both Series and Parallel Impedance Combinations

(4)

Table 1 summarizes the procedure to solve for the poles and zeros of ZEQ.

Table 1. Computation of Poles and Zeros of Equation 4

Poles of ZEQ Zeros of ZEQ

Substituting Values of R, C and L From Figure 5:

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The following observations can be made based on Table 1:• ZEQ has a single real LHP zero (that is, has negative real part) corresponding to f = 1.59 kHz.• ZEQ has a pair of LHP complex conjugate poles corresponding to f = 4.97 kHz, indicating an oscillatory

transient response with 4.97 kHz frequency, as shown in Figure 6.

Figure 6. Unstable Transient Response of Circuit in Figure 5

In many cases, the poles and zeros of ZEQ can also be identified graphically using Bode plots. The generalshape of the ZEQ magnitude plot can be obtained by superimposing the magnitude plots of the individualimpedances and tracing the path of the dominant impedance at each frequency. Impedances with highermagnitude dominate in a series combination, and impedances with lower magnitude dominate in a parallelcombination. Figure 7 shows an example.

Figure 7. Bode Plots of ZEQ for Circuit in Figure 5

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After the shape of the ZEQ magnitude plot has been constructed, standard rules (summarized in Table 2)can be applied to identify whether the break frequencies correspond to poles or zeros, as well as whetherthey are real or complex.

Table 2. Pole and Zero Signatures on Bode Plots

Change in Slope ofMagnitude Plot Around fBREAK

Change in Phase Around fBREAK fBREAK Corresponds to

20 dB/decade 45°/decade over 0.1fBREAK < f < 10fBREAK Single real LHP zeroN × (20 dB/decade) N x (45°/decade) over 0.1fBREAK < f < 10fBREAK N real repeated LHP zeros–20 dB/decade -45°/decade over 0.1fBREAK < f < 10fBREAK Single Real LHP poleN x (–20 dB/decade) N x (-45°/decade) over 0.1fBREAK < f < 10fBREAK N real repeated LHP polesPeak Sharp decrease Pair of complex conjugate LHP polesNotch Sharp increase Pair of complex conjugate LHP zeros20 dB/decade -45°/decade over 0.1fBREAK < f < 10fBREAK Single real RHP zero–20 dB/decade 45°/decade over 0.1fBREAK < f < 10fBREAK Single real RHP polePeak Sharp increase Pair of complex conjugate RHP polesNotch Sharp decrease Pair of complex conjugate RHP zeros

Applying the rules specified in Table 2 to Figure 7, ZEQ has a real LHP zero around the ZR – ZL matchingfrequency (1.59 kHz), and a pair of complex conjugate poles around the ZL – ZC matching frequency (4.97kHz). In this case, the complex conjugate poles physically represent resonance due to the ZL – ZCinteraction when ILOAD stimulates the circuit.

2.5 Stabilizing the Load Transient ResponseRinging can be eliminated by making sure that ZEQ has no complex poles. Referring to the expression forsp1,p2 in Table 1, the poles of ZEQ are real if the quantity under the square root is positive. Assuming thevalues of L and C are retained, and the value of R is varied, Equation 5 specifies the values of R thatstabilize the circuit.

(5)

Table 3 computes the effect of setting R = 2 × √(L/C) on the poles and zeros of ZEQ.

Table 3. ZEQ Poles and Zeros With R = 2 × √(L/C)

Poles of ZEQ Zeros of ZEQ

Substituting values of L, and C from Figure 5:

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ZEQ no longer contains complex poles, and Figure 8 confirms the circuit’s stable load transient response.

Figure 8. RLC Circuit of Figure 5 is Stable With R = 2 × √(L/C)

In most cases, Bode plots can also be used in an iterative fashion to improve stability. The goal is toeliminate any peaks in the ZEQ magnitude plot. Peaks caused by resonant LC interactions are eliminatedby increasing the value of the resistance. Figure 9 illustrates the effect of iteratively increasing the value ofR in the Figure 5 circuit on the magnitude peak and step response. The steady-state error is increasingbecause ILOAD is a step function and nonzero at steady-state, thus creating a bigger voltage drop as theseries resistance is increased.

Figure 9. Iterative Stability Improvement of Figure 5 Circuit

A value of R = 75 Ω (blue traces in Figure 9) is suitable because this value produces the most desirableresponse in terms of gain peaking and settling speed. The value is also compliant with Equation 5.

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ZO

Feedback Network (�)

+-

AOL(VDC ± �VOUT)

VOUT(f)

ILOAD(f)

VDC (= 0VAC)

+

-

Properties of Closed-Loop Amplifier Output Impedance (ZOUT) www.ti.com




3 Properties of Closed-Loop Amplifier Output Impedance (ZOUT)ZOUT is the frequency-dependent Thévenin equivalent impedance offered to a load by a closed-loopamplifier. ZOUT is measured using the test circuit depicted in Figure 10, and is derived by replacing ZBLOCKin Figure 1 with the closed-loop amplifier model, consisting of an op amp and a feedback network.

Figure 10. ZOUT Test Circuit

AOL is the amplifier open-loop voltage gain, and β is the voltage gain of the feedback network.

ZO is the op amp open-loop output impedance. Writing ac node equations (similar to Equation 3) based onFigure 10, and solving for the gain (∆VOUT / ∆ILOAD) produces Equation 6.

(6)

The following observations can be made based on Equation 6.• Any changes in amplifier stability affecting loop gain (AOLβ) also affect ZOUT. Therefore, ZOUT can be

used to assess amplifier stability.• The frequency response of ZOUT can be derived from the amplifier AOLβ and ZO transfer functions.

Familiarity with commonly occurring ZOUT regions over frequency is important for analyzing amplifierload transient behavior

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www.ti.com Properties of Closed-Loop Amplifier Output Impedance (ZOUT)




A simple amplifier model can be constructed to explore the effects of loop gain and ZO on ZOUT. For easeof analysis, β is assumed to be constant so that loop gain is determined by AOL alone. AOL can berepresented using the dominant pole approximation given by Equation 7.

where• AOL(0) is the amplifier dc open-loop gain.• s = –2πfP1_AOL is the dominant pole of the AOL transfer function.• (AOL(0) × 2πfP1_AOL) is the amplifier gain-bandwidth product. (7)

Substituting Equation 7 into Equation 6 and simplifying yields Equation 8.

(8)

Observe that ZOUT has:• a real zero at sZ_ZOUT = –2πfP1_AOL corresponding to fZ_ZOUT = fP1_AOL

• a real pole at sP_ZOUT = –2πf P1_AOL(1 + AOL(0)β corresponding to fP_ZOUT = fP1_AOL × (1 + AOL(0)β)• fZ_ZOUT < fP_ZOUT

Additional poles or zeros may be present based on the characteristics of ZO. As described in Modeling theoutput impedance of an op amp for stability analysis , ZO can be some combination of the fundamentalimpedances: resistive, capacitive or inductive. Therefore, Equation 8 can be examined under each ofthese three ZO conditions.1. If ZO is purely resistive then ZOUT contains no additional poles or zeros and has the general frequency

characteristics described in Table 4.

Table 4. ZOUT for Resistive ZO and Single-Pole AOL

ZOUT Frequency RangeResistive 0 < f < fZ_ZOUT

Inductive fZ_ZOUT < f < fP_ZOUT

Resistive f > fP_ZOUT

2. If ZO is purely capacitive (ZO = 1 / (sC)), then ZOUT contains an additional pole at f = 0, and has thegeneral frequency characteristics described in Table 5.

Table 5. ZOUT for Capacitive ZO and Single-Pole AOL

ZOUT Frequency RangeCapacitive 0 < f < fZ_ZOUT

Resistive fZ_ZOUT < f < fP_ZOUT

Capacitive f > fP_ZOUT

3. If ZO is purely inductive (ZO = sL), then ZOUT contains an additional zero at f = 0, and has the generalfrequency characteristics described in Table 6.

Table 6. ZOUT for Inductive ZO and Single-Pole AOL

ZOUT Frequency RangeInductive 0 < f < fZ_ZOUT

Double Inductive fZ_ZOUT < f < fP_ZOUT

Inductive f > fP_ZOUT

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http://www.ti.com/lit/pdf/SLYT677

http://www.ti.com/lit/pdf/SLYT677

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Properties of Closed-Loop Amplifier Output Impedance (ZOUT) www.ti.com




The term double inductive denotes a ZOUT region with a +40-dB/decade slope in magnitude. A double-inductive impedance can only occur in an active circuit using negative feedback, but modeling theimpedance as a passive element in a combination is useful for stability analysis. On a linear frequencyscale, the magnitude of ZOUT in the double-inductive region increases as (∆f)2 for a ∆f change in frequency.Thus, the impedance transfer function of a double-inductive element (LD) is ZLD = s2LD.

The effects of amplifier ZO and AOL on ZOUT can be validated through simulation. Figure 11 shows a single-pole op amp AOL model with a dc gain of 100,000 V/V (or 100 dB), and a single dominant pole at 15.9 Hz.ZO has been initialized to 0 Ω. The 1-GH inductor makes sure that the circuit is in closed-loopconfiguration for dc convergence, and in open-loop configuration for ac-gain measurement. The 1-GFcapacitor keeps the inverting input of the op amp from floating when the feedback loop is broken.

Figure 11. Op Amp AOL Model With a Single Pole

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www.ti.com Properties of Closed-Loop Amplifier Output Impedance (ZOUT)




For measuring ZOUT, the amplifier must be in closed-loop gain configuration with the appropriate ZOelement, and excited with an ac current source according to Figure 10. Using β = 1 V/V for ease ofanalysis, the circuits of Figure 12 through Figure 14 depict ZOUT versus frequency for the three basic ZOtypes. fZ_ZOUT = fP1_AOL = 15.9 Hz and fP_ZOUT = fP1_AOL (1 + AOL(0)β) = 1.59 MHz.

Figure 12. ZOUT Simulation With Single-Pole AOL and Resistive ZO

Figure 13. ZOUT Simulation With Single-Pole AOL and Capacitive ZO

Figure 14. ZOUT Simulation With Single-Pole AOL and Inductive ZO

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Stability Analysis of a Loaded Amplifier Using ZOUT and ZLOAD www.ti.com




4 Stability Analysis of a Loaded Amplifier Using ZOUT and ZLOAD

The stability of a loaded amplifier can be evaluated by examining the Thévenin equivalent impedance(ZEQ) for complex poles. ZEQ for a loaded amplifier is simply the parallel combination of ZOUT and ZLOAD, asshown in Equation 9:

where• ZOUT can be resistive, capacitive, inductive or double inductive over frequency.• ZLOAD can be resistive, inductive, or capacitive over frequency (9)

Therefore, the properties of ZEQ can be studied by considering the various combinations of ZOUT and ZLOAD.Obviously, a resistive ZOUT requires no special consideration because a resistive ZOUT is unconditionallystable with any resistive or reactive ZLOAD. From a stability perspective, only reactive ZOUT regions arerelevant when combined with specific ZLOAD elements, as discussed in subsequent sections.

4.1 Inductive ZOUT Driving a Capacitive ZLOAD or Capacitive ZOUT Driving an Inductive ZLOAD

This section analyzes stability issues caused by LC interactions between ZOUT and ZLOAD. The analysis isthe same regardless of which element is inductive and which is capacitive. In this case, however, theanalysis focuses on the more common scenario where ZOUT is inductive and ZLOAD is capacitive. Bothconditions produce complex conjugate poles in ZEQ. The circuit shown in Figure 15 is a typical example.

Figure 15. Example of Circuit With Resonant LC Interaction Between ZOUT and ZLOAD

With the individual impedances superimposed, tracing the path of the lower impedance reveals a peak inthe ZEQ magnitude plot around the ZOUT – ZLOAD matching frequency. The peak signifies complex conjugatepoles and the oscillatory load transient response is shown in Figure 16.

Figure 16. Unstable Load Transient Response of Circuit in Figure 15

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www.ti.com Stability Analysis of a Loaded Amplifier Using ZOUT and ZLOAD




As discussed in Section 2.5, an LC resonant peak in ZEQ can be eliminated by adding sufficiently largeseries resistance: R ≥ 2 × √(L/C) (see Equation 5). While the circuit diagram shows the value of C (=ZLOAD), the value of L must be calculated using the ZOUT magnitude plot by identifying a point in theinductive region. Observing that |ZOUT| ≈ 50 mΩ at f =75 Hz, the value of L can be calculated as L = |ZOUT| /(2πf) ≈ 106 μH. Thus, the required resistance value for stability is R ≥ 20.6 Ω. Stabilized responses areshown in Figure 17 and Figure 18. Clearly, a value of R in the 20-Ω to 30-Ω range is suitable.

Figure 17. Adding Series Resistance Eliminates LC Resonant Peak in ZEQ

Figure 18. Load Transient Response of Figure 17 Circuit is Stable for 20 Ω < R < 30 Ω

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The series resistance creates a steady-state dc error, and takes several milliseconds to settle, which maynot be acceptable in some applications. A simple workaround is to move the resistor in series with theload capacitor, as shown in Figure 19.

Figure 19. Increasing ESR Also Eliminates Resonant LC Peak in ZEQ

There is still a stability improvement because, as shown in Figure 19, the added ESR makes ZLOAD moreresistive around the ZOUT – ZLOAD matching frequency, thereby eliminating resonance. Figure 20 depicts theload transient response as ESR is increased.

Figure 20. Load Transient Response With Increasing ZLOAD ESR

Observe that the response becomes increasingly stable, and the dc steady-state error as well as settlingtime become negligible. However, the peak overshoot becomes progressively higher as the capacitorinitially sinks most of the load current (a positive step function, in this case), generating a bigger voltagedrop across the ESR. This overshoot may be acceptable in applications where dc accuracy and fastsettling are more important.

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T

Time (s)

0 15u 30u

ILOAD

0

10u

VDROP

-64m

100m

-

+

-

+

AOL_DC 100k RP1 1kOhmC

P1

10uF -

+

-

+

Output Stage 1 Zo 10mH

ILOAD

VDROP

RLO

AD

10k

Ohm

T

Impe

danc

e (O

hm)

100n

10u

1m

100m

10

1k

100k

10MEG

Frequency (Hz)

100m 1 10 100 1k 10k 100k 1MEG 10MEG 100MEG

Pha

se [d

eg]

90

135

180

ZEQ

ZLOADZOUT





4.2 Double-Inductive ZOUT Driving Resistive ZLOAD

Figure 21 shows the model of a circuit with a double-inductive ZOUT driving a resistive ZLOAD.

Figure 21. Example Circuit Showing Double-Inductive ZOUT Driving Resistive ZLOAD

There is no LC resonance in this case; however, ZEQ has an unexpected peak around the ZOUT – ZLOADmatching frequency, and the load step response is unstable, as shown in Figure 22.

Figure 22. Unstable Load Transient Response of Figure 21 Circuit

For stability analysis in this case, the ZEQ transfer function must be solved algebraically.

Equation 10 expresses ZEQ as the combination of inductive (L1, L2) and double-inductive (LD) portions ofZOUT with resistive ZLOAD = RLOAD = 10 kΩ. L2 can be eliminated from the expression for ZEQ because RLOADdominates ZEQ at high frequencies.

(10)

(11)

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T

Frequency (Hz)

100m 1 10 100 1k 10k 100k 1MEG 10MEG 100MEG

Impe

danc

e (O

hm)

100n

10u

1m

100m

10

1k

100k

ZOUT + 0[H] ZOUT + 3m[H] ZOUT + 6m[H] ZOUT + 9m[H] ZLOAD ZOUT

-

+

-

+


CP

1 10

uF -

+

-

+

Output Stage 1

ILOAD

VDROPL1 10mH

R1

10kO

hm

L2 10mH

T

Time (s)

10u 15u 20u

Cur

rent

(A

)

-64.14m

17.87m

99.87m ILOAD VDROP[1] 0[H] VDROP[2] 3m[H] VDROP[3] 6m[H] VDROP[4] 9m[H]





Substituting the values of L1 = 132 nH and LD = |ZOUT|/(2πf)2 ≈ 1 n units (from the ZOUT Bode plot) confirmsthat ZEQ has a pair of complex conjugate LHP poles near the ZOUT – ZLOAD matching frequency ofapproximately 500 kHz.

Stabilizing the load transient response requires the quantity under the square root in Equation 11 to bepositive

(12)

Equation 12 suggests that the complex poles can be eliminated by increasing the value of L1, which isequivalent to adding series inductance to ZOUT. According to Figure 23, the oscillatory responsedisappears when the value of the series inductance exceeds 6 mH. Figure 24 shows how adding seriesinductance eliminates the double-inductive characteristic of ZEQ.

Figure 23. Adding Series Inductance to ZOUT Stabilizes Load Step Response

Figure 24. Double-Inductive Region No Longer Dominates ZOUT With Higher ESL

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T

Time (s)

0 100u 200u

ILOAD

0

10u

VDROP

-2m

5m

-

+

-

+


CP

1 10

uF -

+

-

+

Output Stage 1 Zo 10mH

ILOAD

VDROP

C1

1uF

T

Impe

danc

e (O

hm)

100n

10u

1m

100m

10

1k

100k

10MEG

Frequency (Hz)

100m 1 10 100 1k 10k 100k 1MEG 10MEG 100MEG

Pha

se [d

eg]

90

180

270

ZEQ

ZLOAD

ZOUT





4.3 Double-Inductive ZOUT Driving Capacitive ZLOAD

The ZEQ Bode plots for this configuration show signs of gross instability. According to the magnitude plot ofFigure 25, there are three poles corresponding to the ZOUT – ZLOAD matching frequency, but phase leadincreases sharply by 90° in just one octave around the matching frequency. Table 2 shows that thissignature represents a pair of complex conjugate RHP poles that produce an exponentially increasingoscillatory response, as confirmed by Figure 26.

Figure 25. Example Circuit With Double-Inductive ZOUT Driving Capacitive ZLOAD

Figure 26. Unstable Load Step Response of Figure 25 Circuit

In this case, it is difficult to find the poles of ZEQ algebraically because this case involves solving a third-order polynomial equation, which is nontrivial. For reference, the ZEQ transfer function is given byEquation 13.

(13)

In lieu of an algebraic analysis, the strategies discussed in Section 4.1 and Section 4.2 can be applied.The basic idea is to add series inductance so that ZOUT transforms into being more inductive around theZOUT – ZLOAD matching frequency. Consequently, the circuit simplifies to an LC resonant circuit that canthen be stabilized relatively easily by adding series resistance.

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T

Time (s)

0 250u 500u 750u 1m

ILOAD

0.00

10.00u

VDROP

0.00

300.00u

-

+

-

+


CP

1 10

uF -

+

-

+

Output Stage 1

ILOAD

VDROPL1 10mH L2 200uH

C1

1uF

R1 30Ohm

T

Impe

danc

e

1m

10m

100m

1

10

100

Frequency (Hz)

100.00m 3.16k 100.00MEG

Pha

se [d

eg]

-92

-46

0

-

+

-

+


CP

1 10

uF -

+

-

+

Output Stage 1

ILOAD

VDROPL1 10mH L2 200uH

C1

1uF

T

Impe

danc

e (O

hm)

100n

10u

1m

100m

10

1k

100k

10MEG

Frequency (Hz)

100m 1 10 100 1k 10k 100k 1MEG 10MEG 100MEG

Pha

se [d

eg]

90

180

270

ZLOADZEQ

ZOUT + 200�H





From the Bode plot |ZOUT| ≈ 10 Ω at the ZOUT – ZLOAD matching frequency (fP) of approximately 16 kHz. ForZOUT to become more inductive, the impedance of the series inductor at 16 kHz must be higher than 10 Ω.Use Equation 14 to select a suitable value.

(14)

The value of the series resistance required to eliminate the resonant peak appearing in Figure 27 can nowbe calculated using Equation 5: R ≥ 2 × √(L1/CLOAD) ≈ 30 Ω. Figure 28 and Figure 29 depict the Bode plotand transient response of the stabilized circuit, respectively.

Figure 27. Modified ZOUT is More Inductive Around fP = 16 kHz

Figure 28. Adding Series Resistance Eliminates Peak in Modified ZEQ Magnitude

Figure 29. Load Step Response of Stabilized Circuit

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www.ti.com Conclusions




Finally, the compensation strategies presented in this article must be considered alongside other systemconstraints such as cost, component availability, performance, and so on. For example, introducing a low-cost op amp buffer between the amplifier and load capacitor in the circuit shown in Figure 25 could bemore cost-effective and enable faster settling across the load capacitor than using a 200-μH inductor forcompensation. However, an op amp buffer would likely also consume more board area and supplycurrent, and introduce additional errors in the signal path, thus requiring careful consideration againstdesign objectives.

5 ConclusionsThe closed-loop output impedance (ZOUT) of an amplifier makes it possible to evaluate the stability of theamplifier load transient response under closed-loop conditions.

A stable load transient response is characterized by exponential settling to steady state without ringing.This requires a Thévenin equivalent impedance function that has purely real poles in the left half of thecomplex s-plane (LHP).

Simple amplifier simulation models constructed using the dominant pole AOL approximation provide usefulinsights into the frequency response of ZOUT and the various impedance profiles offered to a load.

Compensation strategies were developed using algebraic and geometric methods to overcome loadtransient stability issues for various amplifier ZOUT and load configurations. These strategies were validatedthrough simulation.

6 References• Texas Instruments, Solving Op Amp Stability Issues Wiki• Texas Instruments, Modeling the Output Impedance of an Op Amp for Stability Analysis Analog

Applications Journal• Massachusetts Institute of Technology – Department of Mechanical Engineering, 2.14 Analysis and

Design of Feedback Control Systems, Understanding Poles and Zeros

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https://e2e.ti.com/support/amplifiers/precision_amplifiers/w/design_notes/2645.solving-op-amp-stability-issues

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Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using Zout · 2017. 10. 27. ·...

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