1. Passive Filter Transformations For every passive filter design, there are two ways of laying out the LC network. In many cases, one of these may be more appropriate for component simulation than the other. Eg. Three pole low-pass filter  network, best passive layout (fewest inductors) T network, best for component simulation (fewest FDNRs)

2. Layout Transformation Meshes become nodes Impedances between meshes become their reciprocal impedance between nodes. Resistors, R  Resistors, 1/R  Inductors, L H  Capacitors, L F Capacitors, L F  Inductors, L H Voltage sources become current sources

3. Example Three pole Butterworth low pass filter

4. Example 2 Five pole elliptic low pass filter. Pass band ripple = 1 dB Stop band attenuation = 60 dB

5. Component Simulation Summary Component simulation can be used to simulate relatively complex passive networks. Either inductances or FDNRs are simulated using GICs. Passive networks are proven to be robust against component tolerances (unlike cascade synthesis). But… The op-amp output voltages required within the GIC can be larger than the input signal. The maximum input range is restricted.

6. Operational Simulation Operational simulation proceeds as follows: A passive LC network is designed. A set of differential equations relating voltages and currents in the network is written. An analogue computer that solves these equations is designed. The analogue computer solves the set of differential equations is real time and hence simulates the LC network.

7. Two-Pole Passive Filter

8. Block Diagram Equivalent

9. Analogue Computer Elements (I) Inverting Summing Integrator Applying Kirchoff’s current law at the inverting input node:

10. Analogue Computer Elements (II) Inverting Amplifier Required to convert the inverting integrator into a non-inverting integrator.

11. Analogue Computer Design The voltage levels at the outputs of the op-amps are numerically related to variables in the equations These variables can be either voltages or currents To simplify matters: Signals representing voltages will be kept equal to the real voltage Signals representing currents will be kept numerically equal to the current in Amps (i.e. a scale factor of 1 Volt per Amp) In general, any scale factors can be used for voltages and currents

12. Building the Filter The passive filter analysis boils down to just two equations: Two integrators will, therefore, be required to simulate this circuit

13. Design 1

14. Design 2

15. Design 3 The ‘Ring of Three’ or ‘Two Integrator Loop’

16. Higher Order Filters The ring of three could be used as the second order sections in a cascaded filter. A more robust method is to simply extend the concepts of operational simulation to higher order passive networks. Such filters are sometimes called Leapfrog Filters. Leapfrog filters are built up stage by stage, simulating the V-I characteristics of each passive component. 3N/2 op-amps required.

17. Summary Operational simulation can be thought of as real-time circuit analysis performed by an analogue computer. All passive networks can be represented using summing integrators and invertors. The ring of three is one of the simplest examples, simulating a second order passive network. Next time: Design example for ring of three and module summary