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Designing for Low Distortion with High-Speed Op Amps
By James L. Karki, High Performance Linear, Texas Instruments

0   Introduction

Total distortion in an amplifier circuit is a combination of intrinsic non-linearity and reduction due to negative feedback. Distortion in the forward amplification path of an op amp is referred to as intrinsic non-linearity. Loop gain in negative feedback reduces the op ampýs intrinsic non-linearity. The goals in designing for low distortion are to minimize the intrinsic non-linearity and maximize the curative effects of feedback.

Many high-speed op amps use bipolar transistors as the basic active element to amplify the signal. In bipolar transistors, junction capacitances are a function of voltage; current gain is a function of collector current; collector current is influenced by collector to emitter voltage; transconductances are typically exponential functions; and so on. These all cause non-linear transfer functions in the circuits for which they are used to build. When a transistor is driven into saturation or cut-off, it exhibits strong non-linearity. In this discussion, it is assumed the devices are being operated well within their saturation limits in what is typically referred to as linear operation.

Designing for low distortion is essential to ensure top performance of high-speed op amps in a variety of applications ranging from communications to medical imaging to high definition TV. This article explores a typical architecture of a voltage feedback (VFB) high-speed op amp. Distortion mechanisms are identified and quantified. The effectiveness of the negative feedback in reducing distortion is examined, and design recommendations for achieving low distortion are given. As a practical example test results of a high-speed low-distortion op amp circuit are shown.

1   Power Series Expansion
Referring to the formula of a straight line: y = b + mx, non-linearity refers to any deviation the output (y) may have from a constant multiple (m) of the input (x) plus any constant offset (b).

Expanding the non-linear transfer functions of basic transistor circuits into a power series is a typical way to quantify distortion products . For example: assume a circuit has an exponential transfer function y = e^x, where x is the input and y is the output. Expanding e^x into a power series around x = 0 results in: . Figure 1 shows the function y = e^x along with estimates that use progressively more terms of the power series. The farther x is from 0, the more terms are required to properly estimate the value of ex. If x < 0.25, the linear term, 1 + x, provides a close estimate of the actual function, and the circuit is linear. As x becomes larger, progressively more terms are required to properly estimate e^x, and the output now contains quadratic, cubic and higher order distortion terms.

Figure1

If the input to this circuit is a sinusoid: , trigonometric identities1 show the quadratic, cubic and higher order terms give rise to 2^nd, 3^rd and higher order harmonic distortion. In a similar manner² , if the input is comprised of two tones, trigonometric identities show that the quadratic and cubic terms give rise to 2^nd, 3^rd and higher order intermodulation distortion.

2   Simplified Op Amp Schematic

A simplified schematic of a high-speed op amp is shown in Figure 2. A simple feedback network and load resistor are included for completeness. A brief overview of its operation is given first for those unaccustomed to op amp circuit design.

Vcc + is the positive power supply input, and Vcc - is the negative power supply input. Vin + and Vin - are the signal input pins, and Vout is the signal output. The op amp amplifies the differential voltage across its input pins to generate the output. By convention, the input voltage is the difference voltage, Vid = (Vin +) - (Vin -). It is amplified by the open loop gain of the amplifier to produce the output voltage, Vout = a(f)Vid, where a(f) is the frequency dependent open loop gain of the amplifier. The input differential pair, Q1 and Q2, is balanced so the collector currents, Ic1 and Ic2, are equal when the input differential voltage is zero. Applying a voltage across the input pins causes Ic1 Ic2.

Q3 and Q4 fold the current from the input differential pair into the Wilson current mirror formed by Q5, Q6, and Q7. The mirror tries to maintain equal collector currents. Any difference current, Ic1 - Ic2, develops a voltage at Vmid.

The capacitor, Cc, shown from Vmid to ground is really multiple capacitors to the supply rails. It is used to stabilize the amplifier via dominant pole compensation. Vmid is then buffered to the output via Q8 and Q10, or via Q9 and Q11, depending on polarity. In the following sections, we will look at distortion mechanisms in each of these sub-circuits more closely.

Figure2

3   Intrinsic Linearity

Intrinsic linearity refers to the linearity of each sub-circuit without the effect of feedback.

3.1   The Input Stage

The transfer function of the input differential pair is given in many texts3, 4. The input is the differential voltage, Vid = (Vin+) - (Vin-), and the output is the difference current, I_C1 - I_C2, which is proportional to , where Vt is the thermal voltage . Expanding tanh(x) around x = 0 in a power series, we find . The power series contains no even order terms, and a well-matched differential pair will generate very low even order harmonics. Figure 3 shows three estimates of y(x) = tanh(x) each using progressively more terms in the power series.

Figure 3

If x < 0.25, which corresponds to Vid < 0.5 Vt, the function is approximately linear, i.e. y1(x) = x. Under that condition, the input stage is very linear. With larger excursions, 3rd and 5th order terms are required for an adequate approximation, and the input stage will distort the signal. A typical high-speed op amp, without degeneration*, has a very large open loop gain - on the order of 100 dB. Very small input voltages are required to keep the output from saturating. Vid is normally much less than Vt, and the input stage is normally considered linear. For example: assume the output voltage swing is 10 Vpk, with 100 dB of gain Vid = 0.1 mVpk, which is much less than .

*Often emitter degeneration is used as part of the overall compensation scheme, where resistors are placed in the emitter leads of Q1 and Q2. This increases the linear input voltage range by reducing the gain.

3.2   The Intermediate Stage

The difference current, I_C1 - I_C2, from the input stage is the input to the intermediate stage and the output is the voltage at Vmid. The voltage swing at Vmid in conjunction with the early voltages, VA, and collector capacitances, Cj, of Q4 and Q7 cause non-linear behavior in the stage. The other transistors in the intermediate stage see very small voltage and current variations, and do not play a major role in distortion.

3.3   Early Voltage Effects

V_A appears in the formula for a transistorýs collector current5 : , where Vbe is the base emitter voltage, Vt is the thermal voltage, Vce is the collector to emitter voltage, and Is is the saturation current.

Ic comprises the input to the intermediate stage and Vce the output. Assuming is linear should be valid considering the previous discussion about the linearity of the input stage. As such, we can write a generalized formula for the intermediate stage: , where y is Vmid, z is the impedance of the stage, x is the input current, and depends on V_A. Rearranging: . Using conservative numbers, z = 10⁶ and = 100, the power series expansion around x = 0 is: 10⁶x + 10¹⁰x² + 10¹⁴x³ + 10¹⁸x⁴ + 10²²x⁵ .ý Figure 4 shows three estimates each using progressively more terms in the power series - linear, quadratic, and cubic.

It can be seen that with voltage swings up to 4 V at Vmid, the linear term, 10⁶x, is a close approximation, and it is expected that the transfer function is linear within these limits. At higher excursions, between 4V and 12 V, the quadratic terms are required, resulting in 2^nd order distortion products. Above 12V, the cubic terms are required, giving rise to 3^rd order distortion products. Limiting voltage swings at Vmid reduces distortion due to the non-linear effects of V_A.

3.4   Junction Capacitance Effects

A simple model of junction capacitance6 is given by the formula: , where Co is the zero-bias capacitance, Vj is the junction voltage, is the built in potential, and MJE is the grading coefficient. The transfer function of the intermediate stage is influenced by this non-linear capacitance at Vmid.

Looking at the capacitive reactance and taking y as the independent variable, we can write:

Typical numbers for TIýs high-speed BiCom 1 process include: , and Cc = 5 pF. Using these values and setting = 106, the plot shown in Figure 5 is drawn. Again, power series estimates are shown that use progressively more terms of the series - linear, quadratic, and cubic.

With voltage swings of about 3 Vp-p, the linear approximation is valid. At higher voltage swings, the quadratic and cubic terms are required, resulting in 2^nd and 3^rd order distortion products.

The effect of V_A is not frequency dependent, but the effect of Cj is. Below the dominant pole of the amplifier, V_A dominates the non-linearity of the intermediate stage. Above the dominant pole, the non-linearity is a combination of the two. In either case, limiting the voltage swing at Vmid (and thus Vout) is the key to linear operation.

3.5   The Output Stage - Q8 through Q11

Q8 through Q11 form a double buffered class AB output stage. The voltage at Vmid is buffered via these transistors to produce Vout. Depending on polarity, the signal path is either through Q8 and Q10, or through Q9 and Q11. The analysis is the same in either case. Assuming the polarity is positive, Vout = Vmid + Vbe_Q8 - Vbe_Q10 =

Q10 will see variations in collector current as it delivers power to the output load. Variations in Q8ýs collector current are reduced by the beta of Q10 resulting in VbeQ10 being the dominant non-linearity in the signal path from Vmid to Vout.

Again, using a power series expansion helps to highlight the nonlinear terms responsible for distortion. Expanding the natural log function around the point x = a in a power series:

. Using typical numbers for Vbe = 0.6 V, Figure 6 shows Vbe and three estimates each using progressively more terms in the power series - linear, quadratic, and cubic.

Variations of 20% or more in collector current cause the output stage to become non-linear. Increasing the amplifierýs load impedance reduces the current variations in the output transistors and helps to reduce distortion in the output stage.

A typical scenario where the output stage is designed for a quiescent bias current of 5 mA, and can deliver up to 100 mA to the load, the output current varies by a factor of 20. Under these circumstances, distortion in the output stage will dominate the intrinsic distortion of the amplifier.

4   Power Supply Bypass Capactitors

Figure 7 illustrates power supply bypassing for a high-speed op amp. The current supplied to the op amp from the power supply must pass through the distributed impedance between the power supply and the op amp. When the amplifier outputs fast rising (or falling) waveforms, the voltage drop across this impedance can cause significant voltage drop and reduce the voltage across the op amp. The transistor operation moves toward saturation, causing significant distortion. Bypass capacitors are required for high-speed operation.

The strategy shown uses bulk capacitors, typically 6.8 ýF to 10 ýF tantalum, within an inch (or two) of the power pins, along with high frequency capacitors, 0.01 ýF to 0.1 ýF ceramic, within 0.1 inch of the power pins.

The bulk capacitors store more energy but tend to have larger parasitic inductance and equivalent series resistance. For this reason, they can only provide energy at low frequencies. Due to the relaxed requirements for placement on the board, they are typically shared between several devices.

The high frequency capacitors have lower inductance and equivalent series resistance and are capable of supplying fast transients. They are located as close as physically possible to the op amp power pins.

Figure 7

5   Using Differential Amplification to Reduce Even Order Distortion

Differential signaling has been commonly used in audio, data transmission, and telephone systems for many years because of its inherent resistance to external noise sources. Today, differential signaling is becoming popular in high-speed data acquisition, where the analog -to-digital converterýs (ADC) inputs are differential and a differential amplifier is needed to properly drive them.

Another attractive advantage to differential signaling is that it significantly reduces even order harmonics. This is easy to see by using a generic power series expansion of the output voltage.

A differential amplifier has two outputs that are ideally 180ý out of phase with one another: (Vout +) = - (Vout -). The output is the differential voltage: Vout = (Vout +) - (Vout -). Assuming the amplifier is perfectly balanced, the generic expansion of each output is:
(Vout +) = K₁(Vin) + K₂(Vin)² + K₃(Vin)³ + K₄(Vin)⁴ + K₅(Vin)⁵ . . .,
(Vout -) = K₁(-Vin) + K₂(-Vin)² + K₃(-Vin)³ + K₄(-Vin)⁴ + K₅(-Vin)⁵ . . .,
where K₁ through K₅ are constants that depend on the characteristics of the amplifier. The odd order terms retain their original polarity, but the even order terms are always positive. Since Vout is the difference, the even order terms cancel, but the odd order terms increase by a factor of two:
Vout = 2K₁Vin + 2K₃Vin³ + 2K₅Vin⁵ . . .

Balance is very important. Any imbalance in the two amplification paths will compromise the cancellation of even order terms. Symmetrical layout and matched amplifiers are required.

There are many ways to convert single ended signals to differential signals as shown in Figure 8. Some employ the use of transformers, multiple single ended op amps, and various passive components. Integrated fully differential op amps are available that are better suited to wide-band applications and can provide a more elegant solution.

Figure8

6   Loop Gain in Feedback Reduces Distortion

Figure 9 shows a block diagram of an op amp using negative feedback. The open loop gain or forward gain of the amplifier is A1A2. The loop gain is A1A2ý. e1, e2, and e3 are generalized error sources. The following discussion analyzes the output response due to the individual error sources. e1 is amplified by the full open loop gain of the amplifier. Setting all other sources to zero:
If there were no feedback, Vout = e1A1A2, but with feedback, if
A1A2 >> 1. Thus, e1 will be amplified by the closed loop gain of the amplifier.

e2 is amplified only by A2. Setting all other sources to zero:
If there were no feedback, Vout = e2A2, but with feedback, if
A2 >> 1. e2 is attenuated by A1ý.

e3 is buffered by a gain of +1 to the output. Setting all other sources to zero:
If there were no feedback, Vout = e3, but with feedback, if
A1A2ý >> 1. e3 is attenuated by the loop gain, A1A2ý .

Figure 9

The loop gain of the op amp is effective in reducing distortion in the intermediate and output stages where the distortion is highest.

7   Design Guidelines for Low Distortion

So how does one go about designing for low distortion with high-speed op amps? The following are important design considerations:
Op amp selection:

1. Look for an amplifier with low intrinsic distortion, high open-loop gain at the frequencies of operation, and high slew rate.
2. For VFB op amps, gain bandwidth products (GBW) in the GHz range may be required to have enough loop gain to significantly reduce distortion in the 10 MHz to 100 MHz range.
3. Current feedback (CFB) op amps have much higher slew rates than VFB op amps. If the output cannot track the input due to slew rate limitations, the effectiveness of negative feedback is null and void. For this reason, CFB op amps can provide lower distortion in high frequency applications.
4. For high gain applications, use de-compensated op amps. De-compensated op amps sacrifice stability at lower gain for higher GBW, higher slew rate, and lower noise. They are easily spotted in data books and selection guides by their minimum gain requirements.
5. CFB op amps allow you to optimize the loop gain based on the closed loop gain of the amplifier by selection of the feedback resistor value. At higher gains, lower feedback resistors can be used without sacrificing stability (see OPA685 data sheet).