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Download this article in PDF format. (270 KB) Modeling Amplifiers as Analog Filters Increases SPICE Simulation Speed
With high-bandwidth amplifiers, however, time-domain simulations using This article presents a further refinement, synthesizing the second-order approximation as an analog filter rather than as an
We can begin the process by modeling the frequency and transient response of an amplifier with the following standard form for the second-order approximation:
Conversions to Sallen-Key and multiple feedback topologies are shown in Figure 1.
The natural undamped frequency of the amplifier, ω and the damping ratio of the amplifier, ζ, is equal to ½ times the reciprocal the quality factor of the filter, _{c},Q. For a two-pole filter, Q indicates the radial distance of the poles from the jω-axis, with higher values of Q indicating that the poles are closer to the jω-axis. With amplifiers, larger damping ratios result in lower peaking. These relationships serve as useful equivalencies between the s-domain (s = jω) transfer function and the analog filter circuit.
M) and settling time (_{p}t). Second, using these measurements, calculate the second-order approximation of the amplifier’s transfer function. Third, convert the transfer function to the analog filter topology to produce the amplifier’s SPICE model._{s}
As an example, a gain-of-5 amplifier will be simulated using both Sallen-Key and MFB topologies. From Figure 2, the overshoot (
Rearranging terms to solve for ζ gives
Next, calculate the natural undamped frequency in radians per second using the settling time.
For a step input, the
and
The unity-gain transfer function then becomes
The final transfer function for a gain-of-5 amplifier is obtained by multiplying the step function by 5:
The following netlist simulates the Laplace transform for the transfer function of the gain-of-5 amplifier. Before converting to a filter topology, it’s good to run simulations to verify the Laplace transform, adjusting the bandwidth as needed by making the settling time larger or smaller. ***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 results in the frequency domain.
The peaking in the pulse response makes it easy to maintain a constant damping ratio while varying the settling time to modify the bandwidth. This changes the angle of the complex-conjugate pole pair with respect to the real axis in an amount equal to the arccosine of the damping ratio, as shown in Figure 5. Decreasing the settling time increases the bandwidth; and increasing the settling time decreases the bandwidth. Peaking and gain will not be affected as long as the damping ratio is kept constant and adjustments are made only to the settling time, as shown in Figure 6.
Once the transfer function matches the characteristics of the actual amplifier, it is ready to be converted to a filter topology. This example will use both Sallen-Key and MFB topologies. First, use the canonical form for the unity-gain Sallen-Key topology to convert the transfer function into resistor and capacitor values.
From the
Choose convenient resistor values, such as 10 kΩ, for
Use the relationship for the corner frequency to solve for
The resulting netlist follows, and the Sallen-Key circuit is illustrated in Figure 7. E1 multiplies the step function to obtain a gain of 5. .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
Next, use the standard form for the MFB topology to convert the transfer function into resistor and capacitor values.
Begin the transformation by calculating
Set
Finally, to verify that the component ratios are correct,
Now that the component values are solved, substitute back into the equations to verify that the polynomial coefficients are mathematically correct. A spreadsheet calculator is an easy way to do this. The component values shown provide practical values for use in the final SPICE model. In practice, ensure that the minimum capacitor value does not fall below 10 pF. The netlist for the gain-of-5 amplifier follows 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 component count is much lower than would otherwise be required with transistors, capacitors, diodes, and dependent sources. .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 = 2 Ω*** RO OUT 9 2 .END
M = 0.025%)._{p}
With no overshoot, it is convenient to maintain a constant settling time and adjust the damping ratio to simulate the correct bandwidth and peaking. Figure 10 shows how the poles move as the damping ratio is varied while maintaining a constant settling time. Figure 11 shows the change in frequency response.
***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 resistor and capacitor values for the unity gain Sallen-Key topology, choose
The netlist follows and the Sallen-Key simulation circuit model is shown in Figure 12. E2, a gain-of-10 block, is placed at the output stage along with a 2-Ω output impedance. E2 multiplies the unity gain transfer function by 10. Both Laplace and Sallen-Key netlists produced identical simulations, 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
A similar derivation can be done using the MFB topology. The netlist follows, 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 = 2 Ω*** RO OUT 9 2 .END
s-domain (Laplace transform) transfer functions. The Sallen-Key and MFB low-pass filter topologies provide a method for converting s-domain transfer functions into resistors, capacitors, and voltage-controlled-current-sources. Non-ideal operation of the MFB topology results from Existing circuits commonly used for CMRR, PSRR, offset voltage, supply current, spectral noise, input/output limiting, and other parameters can be combined with the model, as shown in Figure 16.
Electronic Design, September 22, 2010.
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