Modeling amplifiers as analog filters improves SPICE simulation speed

IntroductionAmplifier
simulation models are usually implemented using resistors, capacitors, transistors, diodes, independent and dependent signal sources, and other analog components. An alternative approach is to use a second-order approximation of the amplifier behavior (Laplace transform), which speeds up the simulation and reduces the simulation code to three lines.

However, for high-bandwidth amplifiers, time-domain simulations using s-domain transfer functions can be very slow because the simulator must first calculate the inverse transform and then convolve it with the input signal. The higher the bandwidth, the higher the sampling frequency required to determine the time-domain function, which makes the convolution calculation more difficult and slows down the time-domain simulation.

This paper further improves the above method by synthesizing the second-order approximation as an analog filter instead of an s -domain transfer function, which greatly improves the time-domain simulation speed, especially for high-bandwidth amplifiers.

The second -order
transfer function of the amplifier simulation model can be implemented using the Sallen-Key filter topology, which requires two resistors, two capacitors, and a voltage-controlled current source, or the multiple feedback (MFB) filter topology, which requires three resistors, two capacitors, and a voltage-controlled current source. Both topologies should give the same results, but the Sallen-Key topology is easier to design, while the MFB topology has better high-frequency response and may be more suitable for programmable gain amplifiers because it is easier to switch to different resistor values.

First, the frequency and transient response of the amplifier are modeled using the standard form of the second-order approximation:

Figure 1 shows how to convert to the Sallen-Key and multiple feedback topologies.

Figure 1. Filter topology

The natural undamped frequency of the amplifier, ω _n, is equal to the corner frequency of the filter, ω _c , and the amplifier's damping ratio, ζ, is equal to ½ times the inverse of the filter's quality factor , Q. For a two-pole filter, Q represents the radial distance of the pole from the j ω axis; larger values ​​of Q indicate that the pole is closer to the j ω axis. For amplifiers, larger damping ratios result in lower peaking. These relationships provide a useful means of converting s- domain ( s = j ω ) transfer functions to analog filter circuits.

Design Example: Gain-of -5 Amplifier This design consists of three main steps: First, measure the amplifier's overshoot ( MP ) and settling time ( _ts ) . Second, use these measurements to calculate a second-order approximation of the amplifier's transfer function. Finally, convert this transfer function into an analog filter topology to generate a SPICE model of the amplifier _.

Figure 2. A gain-of-5 amplifier.

For example, a 5x gain amplifier is simulated using the Sallen-Key and MFB topologies. As shown in Figure 2, the overshoot ( M _p ) is about 22%, and the 2% settling time is about 2.18 μs. The damping ratio ζ is calculated as follows:

Rearrange the terms to solve for ζ:

Next, use the settling time to calculate the natural undamped frequency in radians per second.

^{For a step input, the s2} and s terms (radians per second) in the denominator of the transfer function are given by:

and

The unity-gain transfer function then becomes:

Multiplying the step function by 5 gives us the final transfer function for the gain-of-5 amplifier:

The netlist below simulates the Laplace transform of the transfer function of an amplifier with a gain of 5. Before converting to a filter topology, it is a good idea to run a simulation to verify the Laplace transform and to lengthen or shorten the settling time as needed to adjust the bandwidth.

***GAIN_OF_5 TRANSFER FUNCTION***

.SUBCKT SECOND_ORDER +IN –IN OUT

E1 OUT 0 LAPLACE {V(+IN) – V(–IN)} = {89.371E12 / (S^2 + 3.670E6*S + 17.874E12)}

.END

Figure 3 shows the simulation results in the time domain. Figure 4 shows the simulation results in the frequency domain.

Figure 3. Gain-of-5 amplifier: time domain simulation results.

Figure 4. Gain-of-5 amplifier: frequency domain simulation results

The peaking of the impulse response makes it easy to maintain a constant damping ratio while varying the settling time to adjust the bandwidth. This changes the angle of the complex conjugate pole pair relative to the real axis by an amount equal to the arccosine of the damping ratio, as shown in Figure 5. Reducing the settling time increases the bandwidth, while increasing the settling time decreases the bandwidth. As long as the damping ratio remains constant and only the settling time is adjusted, the peaking and gain are not affected, as shown in Figure 6.

Figure 5. Complex conjugate pole pairs for the gain-of-5 transfer function.

Figure 6. Settling time adjustment vs. bandwidth

Once the transfer function matches the characteristics of a real amplifier, it can be converted into a filter topology. Two topologies will be used in this example: Sallen-Key and MFB.

First, the transfer function is converted to resistor and capacitor values ​​using the canonical form of the unity-gain Sallen-Key topology.

Based on the s term, C ₁ can be calculated :

Choose readily available resistor values, such as 10 _{kΩ for both} R1 and R2 , _and calculate C1 _.

Use the relationship for the corner frequency to solve for C ₂ .

The corresponding netlist is shown below and the Sallen-Key circuit is shown in Figure 7. E1 multiplies the step function to obtain a gain of 5. Ro provides a 2 Ω output impedance. G1 is a VCCS with a gain of 120 dB. E2 is a differential input block. The simulation of frequency vs. gain is exactly the same as that using Laplace transform.

.SUBCKT SALLEN_KEY +IN –IN OUT

R1 1 4 10E3

R2 5 1 10E3

C2 5 0 10.27E–12

C1 2 1 54.5E–12

G1 0 2 5 2 1E6

E2 4 0 +IN –IN 1

E1 3 0 2 0 5

RO OUT 3 2

.END

Figure 7. Simulation circuit of a gain-of -5 amplifier using a Sallen-Key filter.

Next, the transfer function is converted to resistor and capacitor values ​​using the standard form of the MFB topology.

The conversion begins by calculating R _2. To do this, the transfer function can be rewritten in the following more general form:

Set C ₁ = 10 nF, and then choose C ₂ so that the quantity under the square root is positive. For convenience, choose C ₂ to be 10 pF. Substituting the known values ​​of C ₂ = 10 pF, a ₁ = 3.67E6, K = 5, and a ₀ = 17.86E12, calculate the value of R ₂ :

The value of R ₁ is easily calculated and is equal to R ₂ /K = R ₂ /5 = 33. The standard polynomial coefficients can be solved for R ₃ . Substituting the known values ​​of a ₀ , R ₂ , and C ₂ yields:

Finally, verify that the component ratios are correct; that is, when substituting the known values ​​for a ₀ , R ₂ , R ₃ , gain K , and C ₂ (from the s term), C ₁ should equal 10 nF.

Once you have the component values, substitute them into the equation to verify that the polynomial coefficients are mathematically correct. This can be easily done using a spreadsheet calculator. The component values ​​shown are realistic values ​​that can be used in the final SPICE model. In a practical application, ensure that the minimum capacitor value is not less than 10 pF.

The netlist for the gain-of-5 amplifier is shown below, and the model is shown in Figure 8. G1 is a VCCS (voltage-controlled current source) with an open-loop gain of 120 dB. Note that the number of components required would be much higher if resistors, capacitors, diodes, and dependent sources were used.

.SUBCKT MFB +IN –IN OUT

***VCCS – 120 dB OPEN_LOOP_GAIN***

G1 0 7 0 6 1E6

R1 4 3 330

R3 6 4 34K

C2 7 6 1P

C1 0 4 1N

R2 7 4 1.65K

E2 3 0 +IN –IN 1

E1 9 0 7 0 –1

***OUTPUT_IMPEDANCE RO = 2Ω***

RO OUT 9 2

.END

Figure 8. Simulation circuit of a 5x gain amplifier using an MFB filter.

Design Example: Gain-of -10 Amplifier In the second example, consider the pulse response of a gain-of-10 amplifier with no overshoot, as shown in Figure 9. The settling time is approximately 7 μs. Since there is no overshoot, the pulse response can be approximated as critically damped, ζ ≈ 0.935 ( M _p = 0.025%).

Figure 9. No-overshoot gain-of -10 amplifier.

In the absence of overshoot, it is easy to keep a constant settling time and adjust the damping ratio to simulate the correct bandwidth and peaking. Figure 10 shows how the poles change with the damping ratio while the settling time remains constant. Figure 11 shows how the frequency response changes.

Figure 10. Pole locations for different damping ratios, with settling time remaining constant

Figure 11. Frequency response for different damping ratios, with settling time remaining constant.

***AD8208 PREAMPLIFIER_TRANSFER_FUNCTION (GAIN = 20 dB)***

.SUBCKT PREAMPLIFIER_GAIN_10 +IN –IN OUT

E1 OUT 0 LAPLACE {V(+IN)–V(–IN)} = {3.734E12 / (S^2 + 1.143E6*S + 373.379E9)}

.END

_{To find the resistor and capacitor values ​​for the unity-gain topology, choose} R1 = R2 = ₁₀ kΩ as before . Calculate the capacitor value using the same method as in the gain-of-5 amplifier example:

The netlist is shown below and the Sallen-Key simulation circuit model is shown in Figure 12. E2 is a gain-of-10 block placed in the output stage along with a 2 Ω output impedance. E2 amplifies the unity-gain transfer function by a factor of 10. The Laplace transform and Sallen-Key netlists produce the same simulation, as shown in Figure 13.

***AD8208 PREAMPLIFIER_TRANSFER_FUNCTION (GAIN = 20 dB)***

.SUBCKT AMPLIFIER_GAIN_10_SALLEN_KEY +IN –IN OUT

R1 1 4 10E3

R2 5 1 10E3

C2 5 0 153E–12

C1 2 1 175E–12

G1 0 2 5 2 1E6

E2 4 0 +IN –IN 10

E1 3 0 2 0 1

RO OUT 3 2

.END

Figure 12. Simulation circuit of a 10x gain amplifier using a Sallen-Key filter.

Figure 13. Frequency domain simulation of a gain-of -10 amplifier using a Sallen-Key filter.

A similar derivation can be performed using the MFB topology. The netlist is shown below and the simulation model is shown in Figure 14.

***AD8208 PREAMPLIFIER_TRANSFER_FUNCTION (GAIN = 20 dB)***

.SUBCKT 8208_MFB +IN –IN OUT

***G1 = VCCS WITH 120 dB OPEN_LOOP_GAIN***

G1 0 7 0 6 1E6

R1 4 3 994.7

R2 7 4 9.95K

R3 6 4 26.93K

C1 0 4 1N

C2 7 6 10P

EIN_STAGE 3 0 +IN –IN 1

***E2 = OUTPUT BUFFER***

E2 9 0 7 0 1

***OUTPUT RESISTANCE = 2Ω***

RO OUT 9 2

.END

Figure 14. Simulation circuit of a 10x gain amplifier using an MFB filter.

Conclusion
For high bandwidth amplifiers, building SPICE models using analog components can provide much faster time domain simulations than using s- domain (Laplace transform) transfer functions. The Sallen-Key and MFB low-pass filter topologies provide a method to convert s- domain transfer functions to resistors, capacitors, and voltage-controlled current sources.

_The non-ideal operation of the MFB topology comes from the _{fact that C1} and C2 behave as an impedance short circuit relative to resistors R1, R2, and R3 at high frequencies. Similarly, the non-ideal operation of the Sallen-Key topology comes from the fact that C1 and C2 behave as an impedance short circuit _relative to resistors _R1 and _R2 at high frequencies . A comparison _of the two topologies is shown in Figure 15 _.

Existing commonly used circuits for CMRR, PSRR, offset voltage, supply current, spectral noise, input/output limiting, and other parameters can be incorporated into the model as shown in Figure 16.

Figure 15. Bode plots of Sallen-Key and MFB topologies

Figure 16. Complete SPICE amplifier model including error terms.

References
Karpaty, David. “Create Spice Amplifier Models Using Second-Order Approximations.” Electronic Design, September 22, 2010.

About the Author

David Karpaty [david.karpaty@analog.com] is a senior engineer in the Integrated Amplifier Products (IAP) Group at Analog Devices. He is responsible for product and test engineering support for precision signal processing devices, with a focus on automotive applications. David holds a bachelor’s degree in electrical engineering from Northeastern University and a bachelor’s degree in electrical engineering technology from Wentworth Institute of Technology.