1SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
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
Introduction www.ti.com
2 SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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.
ZBLOCK(f)
ILOAD(f)
VDC (= 0VAC)
- VDROP(f) +
www.ti.com Basic Properties of Electrical Impedance
3SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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
T
Impe
danc
e (O
hm)
10m
100m
1
10
100
1k
10k
100k
Frequency (Hz)
10 100 1k 10k 100k 1MEG 10MEG
Pha
se [d
eg]
-180
-90
0
90
180arg(ZL)
arg(ZR)
arg(ZC)
abs(ZR)
abs(ZL)
abs(ZC)
VDC 5V
C1 1uF
R1 1GOhm
ILOAD V+
VDROP
Basic Properties of Electrical Impedance www.ti.com
4 SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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.
Re(s)
j×Im(s)
×
×
× ×
×
×
(0, j0)
www.ti.com Basic Properties of Electrical Impedance
5SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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
VDROP
C 1
uF
VDC ILOAD
L 1mH R 10Ohm
Basic Properties of Electrical Impedance www.ti.com
6 SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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:
T
Impe
danc
e [o
hm]
10m
100m
1
10
100
1k
10k
100k
Frequency (Hz)
10 100 1k 10k 100k 1MEG 10MEG
Pha
se [d
eg]
-90
-45
0
45
90
ZC
ZEQ = (ZL + ZR) // ZC
ZR
ZL
T
Time (s)
0 100u 200u 300u 400u 500u 600u 700u 800u 900u 1m
ILOAD
0.00
100.00u
VDROP
-469.39u
3.38m
www.ti.com Basic Properties of Electrical Impedance
7SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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
Basic Properties of Electrical Impedance www.ti.com
8 SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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:
T
Frequency (Hz)
10 100 1k 10k 100k 1MEG 10MEG
Impe
danc
e (O
hm)
2m
20m
200m
2
20
200
VDROP[1] 10[Ohm] VDROP[2] 25[Ohm] VDROP[3] 50[Ohm] VDROP[4] 75[Ohm] VDROP[5] 100[Ohm]
T
Time (s)
0 1m 2m 3m 4m 5m
Vol
tage
(V
)
0
5m
10m
VDROP[1] 10[Ohm] VDROP[2] 25[Ohm] VDROP[3] 50[Ohm] VDROP[4] 75[Ohm] VDROP[5] 100[Ohm]
T
Time (s)
0 250u 500u 750u 1m
ILOAD
0.00
100.00u
VDROP
0.00
6.32m
T
Frequency (Hz)
10 100 1k 10k 100k 1MEG 10MEG
Impe
danc
e (O
hm)
10m
100m
1
10
100
1k
10k
100k
ZC
ZEQ = (ZL + ZR) // ZC
ZR = 63.25
ZL
www.ti.com Basic Properties of Electrical Impedance
9SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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.
ZO
Feedback Network (�)
+-
AOL(VDC ± �VOUT)
VOUT(f)
ILOAD(f)
VDC (= 0VAC)
+
-
Properties of Closed-Loop Amplifier Output Impedance (ZOUT) www.ti.com
10 SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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
www.ti.com Properties of Closed-Loop Amplifier Output Impedance (ZOUT)
11SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using 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
T
Gai
n (d
B)
-40
-20
0
20
40
60
80
100
Frequency (Hz)
100m 1 10 100 1k 10k 100k 1MEG 10MEG 100MEG
Pha
se [d
eg]
-90
-45
0
fP1_AOL = 15.9Hz
RP1 1kOhm
CP
1 10
uF-
+
-
+
AOL_DC 100k VOUT
+
VIN -
+
-
+
Output Stage 1
L1 1GH
C1 1GF
Zo 0Ohm
Dominant pole
fP_AOL = 15.9 Hz
Properties of Closed-Loop Amplifier Output Impedance (ZOUT) www.ti.com
12 SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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
RP1 1kOhm
CP
1 10
uF-
+
-
+
AOL_DC 100k VOUT
-
+
-
+
Output Stage 1
ILOAD
Zo 1mH
T
Impe
danc
e (O
hm)
6n
600n
60u
6m
600m
60
6k
600k
Frequency (Hz)
100m 1 10 100 1k 10k 100k 1MEG 10MEG 100MEG
Pha
se [d
eg]
90
135
180
Double Inductive InductiveInductive
fZ2_ZOUT fP_ZOUTfZ1_ZOUT
T
Impe
danc
e (O
hm)
1m
10m
100m
1
10
Frequency (Hz)
100m 1 10 100 1k 10k 100k 1MEG 10MEG 100MEG
Pha
se [d
eg]
-90
-45
0
Capacitive Capacitive
RP1 1kOhm
CP
1 10
uF-
+
-
+
AOL_DC 100k VOUT
-
+
-
+
Output Stage 1
ILOAD
Zo 1uF
R1 1GOhm
fZ_ZOUT fP2_ZOUTfP1_ZOUT
Resistive
T
Impe
danc
e (O
hm)
10m
100m
1
10
100
1k
Frequency (Hz)
100m 1 10 100 1k 10k 100k 1MEG 10MEG 100MEG
Pha
se [d
eg]
0
45
90
RP1 1kOhm
CP
1 10
uF-
+
-
+
AOL_DC 100k VOUT
-
+
-
+
Output Stage 1 Zo 1kOhm
ILOAD
Resistive Inductive Resistive
fZ_ZOUT fP_ZOUT
www.ti.com Properties of Closed-Loop Amplifier Output Impedance (ZOUT)
13SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using 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
T
Time (s)
0 250u 500u 750u 1m
ILOAD
0.00
10.00u
VDROP
-85.27u
97.80u
-
+
-
+
AOL_DC 100k RP1 1kOhm
CP
1 10
uF -
+
-
+
Output Stage 1
ILOAD
VDROP
ZLOAD 1uF
ZO 1kOhm
TIm
peda
nce(
Ohm
)
1m
10m
100m
1
10
100
1k
10k
Frequency (Hz)
100m 1 10 100 1k 10k 100k 1MEG 10MEG 100MEG
Pha
se [d
eg]
-90
-199m
90
ZLOAD ZOUT
ZEQ
Stability Analysis of a Loaded Amplifier Using ZOUT and ZLOAD www.ti.com
14 SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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
T
Time (s)
0.00 1.00m 2.00m
Vol
tage
(V
)
-84.88u
107.91u
300.71u
VDROP[1] 0[Ohm] VDROP[2] 10[Ohm] VDROP[3] 20[Ohm] VDROP[4] 30[Ohm]
-
+
-
+
AOL_DC 100k RP1 1kOhm
CP
1 10
uF -
+
-
+
Output Stage 1
ILOAD
VDROPR1 1kOhm
C1
1uF
R2 21Ohm
T
Impe
danc
e (O
hm)
1m
10m
100m
1
10
100
Frequency (Hz)
100.00m 3.16k 100.00MEG
Pha
se [d
eg]
-90
-45
0
www.ti.com Stability Analysis of a Loaded Amplifier Using ZOUT and ZLOAD
15SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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
T
Time (s)
0.00 50.00u 100.00u
Vol
tage
(V
)
-85.27u
0.10m
290.76u
VDROP[1] 0[Ohm] VDROP[2] 10[Ohm] VDROP[3] 20[Ohm] VDROP[4] 30[Ohm]
-
+
-
+
AOL_DC 100k RP1 1kOhmC
P1
10uF -
+
-
+
Output Stage 1
ILOAD
VDROP
C1
1uF
R1 1kOhm
R2
0Ohm
T
Frequency (Hz)
100m 1 10 100 1k 10k 100k 1MEG 10MEG 100MEG
Impe
danc
e (O
hm)
1m
10m
100m
1
10
100
1k
ZLOAD[i]
ZOUT
ZLOAD[1] = 1�F + 0 ZLOAD[2] = 1�F + 10 ZLOAD[3] = 1�F + 20 ZLOAD[4] = 1�F + 30 ZOUT
Stability Analysis of a Loaded Amplifier Using ZOUT and ZLOAD www.ti.com
16 SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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.
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
www.ti.com Stability Analysis of a Loaded Amplifier Using ZOUT and ZLOAD
17SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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)
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
-
+
-
+
AOL_DC 100k RP1 1kOhm
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]
Stability Analysis of a Loaded Amplifier Using ZOUT and ZLOAD www.ti.com
18 SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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
T
Time (s)
0 100u 200u
ILOAD
0
10u
VDROP
-2m
5m
-
+
-
+
AOL_DC 100k RP1 1kOhm
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
www.ti.com Stability Analysis of a Loaded Amplifier Using ZOUT and ZLOAD
19SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using 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.
T
Time (s)
0 250u 500u 750u 1m
ILOAD
0.00
10.00u
VDROP
0.00
300.00u
-
+
-
+
AOL_DC 100k RP1 1kOhm
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
-
+
-
+
AOL_DC 100k RP1 1kOhm
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
Stability Analysis of a Loaded Amplifier Using ZOUT and ZLOAD www.ti.com
20 SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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
www.ti.com Conclusions
21SLYA029–October 2017Submit Documentation Feedback
Copyright © 2017, Texas Instruments Incorporated
Closed-Loop Analysis of Load-Induced Amplifier Stability Issues Using ZOUT
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
IMPORTANT NOTICE FOR TI DESIGN INFORMATION AND RESOURCES
Texas Instruments Incorporated (‘TI”) technical, application or other design advice, services or information, including, but not limited to,reference designs and materials relating to evaluation modules, (collectively, “TI Resources”) are intended to assist designers who aredeveloping applications that incorporate TI products; by downloading, accessing or using any particular TI Resource in any way, you(individually or, if you are acting on behalf of a company, your company) agree to use it solely for this purpose and subject to the terms ofthis Notice.TI’s provision of TI Resources does not expand or otherwise alter TI’s applicable published warranties or warranty disclaimers for TIproducts, and no additional obligations or liabilities arise from TI providing such TI Resources. TI reserves the right to make corrections,enhancements, improvements and other changes to its TI Resources.You understand and agree that you remain responsible for using your independent analysis, evaluation and judgment in designing yourapplications and that you have full and exclusive responsibility to assure the safety of your applications and compliance of your applications(and of all TI products used in or for your applications) with all applicable regulations, laws and other applicable requirements. Yourepresent that, with respect to your applications, you have all the necessary expertise to create and implement safeguards that (1)anticipate dangerous consequences of failures, (2) monitor failures and their consequences, and (3) lessen the likelihood of failures thatmight cause harm and take appropriate actions. You agree that prior to using or distributing any applications that include TI products, youwill thoroughly test such applications and the functionality of such TI products as used in such applications. TI has not conducted anytesting other than that specifically described in the published documentation for a particular TI Resource.You are authorized to use, copy and modify any individual TI Resource only in connection with the development of applications that includethe TI product(s) identified in such TI Resource. NO OTHER LICENSE, EXPRESS OR IMPLIED, BY ESTOPPEL OR OTHERWISE TOANY OTHER TI INTELLECTUAL PROPERTY RIGHT, AND NO LICENSE TO ANY TECHNOLOGY OR INTELLECTUAL PROPERTYRIGHT OF TI OR ANY THIRD PARTY IS GRANTED HEREIN, including but not limited to any patent right, copyright, mask work right, orother intellectual property right relating to any combination, machine, or process in which TI products or services are used. Informationregarding or referencing third-party products or services does not constitute a license to use such products or services, or a warranty orendorsement thereof. Use of TI Resources may require a license from a third party under the patents or other intellectual property of thethird party, or a license from TI under the patents or other intellectual property of TI.TI RESOURCES ARE PROVIDED “AS IS” AND WITH ALL FAULTS. TI DISCLAIMS ALL OTHER WARRANTIES ORREPRESENTATIONS, EXPRESS OR IMPLIED, REGARDING TI RESOURCES OR USE THEREOF, INCLUDING BUT NOT LIMITED TOACCURACY OR COMPLETENESS, TITLE, ANY EPIDEMIC FAILURE WARRANTY AND ANY IMPLIED WARRANTIES OFMERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, AND NON-INFRINGEMENT OF ANY THIRD PARTY INTELLECTUALPROPERTY RIGHTS.TI SHALL NOT BE LIABLE FOR AND SHALL NOT DEFEND OR INDEMNIFY YOU AGAINST ANY CLAIM, INCLUDING BUT NOTLIMITED TO ANY INFRINGEMENT CLAIM THAT RELATES TO OR IS BASED ON ANY COMBINATION OF PRODUCTS EVEN IFDESCRIBED IN TI RESOURCES OR OTHERWISE. IN NO EVENT SHALL TI BE LIABLE FOR ANY ACTUAL, DIRECT, SPECIAL,COLLATERAL, INDIRECT, PUNITIVE, INCIDENTAL, CONSEQUENTIAL OR EXEMPLARY DAMAGES IN CONNECTION WITH ORARISING OUT OF TI RESOURCES OR USE THEREOF, AND REGARDLESS OF WHETHER TI HAS BEEN ADVISED OF THEPOSSIBILITY OF SUCH DAMAGES.You agree to fully indemnify TI and its representatives against any damages, costs, losses, and/or liabilities arising out of your non-compliance with the terms and provisions of this Notice.This Notice applies to TI Resources. Additional terms apply to the use and purchase of certain types of materials, TI products and services.These include; without limitation, TI’s standard terms for semiconductor products http://www.ti.com/sc/docs/stdterms.htm), evaluationmodules, and samples (http://www.ti.com/sc/docs/sampterms.htm).
Mailing Address: Texas Instruments, Post Office Box 655303, Dallas, Texas 75265Copyright © 2017, Texas Instruments Incorporated