Series vs Parallel Networks - First Order Comparison

Despite many of the myths that surround series networks and their acclaimed superiority over conventional parallel networks for loudspeaker design, both networks can be designed with identical transfer functions if the load impedance remains constant. Most of the claims regarding series networks are either grossly overstated or blatantly wrong and may cause deleterious effects on system performance. As with all aspects of design, there are compromises that must be made, and it is impossible to make an informed decision if you are unaware of the facts.

This article is intended to show that there are no greatly enhanced features in a series or parallel network - if properly designed their performance is essentially identical in terms of response, phase and (by extension) transient response. It is unwise to claim that one type of network is superior to the other, when simple logic dictates that if amplitude and phase response are the same, then all of the filter's other characteristics are also the same.

There are other factors than just the response, and this is where the differences between the network topologies exist. Each has good and bad points that must be considered.

Schematic Representation of an Equivalent First Order Series and Parallel Network

Illustrated above are equivalent series parallel first order crossovers with 1kHz crossover points for a fixed load. Note, resistive loads were used in order to minimize analysis variables. Complex load impedances, typical of a loudspeaker, will be covered in latter parts of this article.

Input Impedance Comparison between First Order Series & Parallel Networks

Summed Output Impedances of First Order Series and Parallel Networks

Summary

As we can clearly see, both first order series and parallel networks have identical input impedances and very similar summed output impedances. The Output impedance of the parallel network has a bump at the crossover frequency of a mere 0.8mdB which can mostly be considered irrelevant.

Frequency Response Comparison (Constant Woofer and Tweeter Loads)

Summary

As we can see both series and parallel networks have identical frequency response and 1kHz 3dB points for the HP and LP filters when the loads are held constant.

Frequency Response of Parallel Network (Variable Woofer Load; Constant Tweeter Load)

Summary

By varying the woofer impedance ± 2 ohm from the nominal (8 ohms), we observe the nominal 1 kHz -3dB point of the LPF change from 759 Hz to 1.25 kHz, respectively. Since this is a parallel network, the changing woofer impedance has no affect on the tweeters crossover frequency as the HPF is electrically isolated from the woofers load impedance. However, for the case where Rpwoof = 6 ohms, we see the overlapping bandwidth between the HPF and LPF decrease while we see it increase when Rpwoof = 10 ohms.

Frequency Response of Series Network (Variable Woofer Load; Constant Tweeter Load)

Summary

For the series case, when varying the woofer impedance ± 2ohms from nominal (8 ohms), we see a slightly less 3dB point variance in the LP filter response, with a convergence at about ½ a decade past its corner frequency. We now also have variance of the HP filter response. This in turn has less affect on overlapping bandwidth changes between the two drivers since they somewhat track each other (an advantage to the series crossover).

Summary of Series vs Parallel Networks

In the parallel network, only the driver whose characteristics have changed is affected. In the series network, however, the change is somewhat complimentary.

Series vs Parallel Networks - First Order Comparison - page 2

Phase Response (deg) Comparison between First Order Series & Parallel Networks

Summary

As we can see, both series and parallel implementations of a first order network exhibit identical phase response with 90 degrees phase difference between LPF's and HPF's and a 90 degrees phase change between start and stop band roll off of each filter.

First Order Series and Parallel Networks Comparison with Reactive Woofer Load

For the next phase of our comparison between first order series and parallel networks, we expand our models to include a reactive load such as those typically found in a woofer as illustrated below.

Frequency Response Comparison ­of First Order Series & Parallel Networks with Reactive Loading

Summary

We can see that a reactive woofer load has had a and minor impact on the HPF 3dB point, as seen by the shift to 1.08kHz. More importantly, we note the profound affect the reactive woofer load had on the LPF of the parallel network as evident by the 3dB point shift from 1kHz nominal to 836 Hz, and its degenerative stop band attenuation rolloff. The series network also experiences 3dB point shift to a much lesser degree, with no ill effects on stop band rolloff. Note, Zobel compensation (series R & C) in shunt with the LPF of the parallel network is mandatory to restore comparable filter characteristics and is also recommended for the series network to restore optimal crossover points.

To figure out the correct R and C values for the Zobel network, the following equations are usually used:

Rz = 1.25 * Rs where Rs is the series resistance of the driver

Cz = 1*10^6 / (2*pi*Rs*fd) where fd is the frequency where the impedance of the driver doubles

However, by tweaking these values using trial and error, a more damped response was achieved.

Calculated values: Rz = 10 ohms , Cz = 2.97uF

Frequency Response Comparison ­of First Order Series & Parallel Networks with Reactive Loading (Zobel Compensated)

Summary

As evident in the graph above, both Series and Parallel networks exhibit identical frequency responses under reactive loading when Zobel compensated.

Summed Output Impedances of First Order Series and Parallel Networks with Reactive Loading (Zobel Compenstated)

Summary

The series network exhibits a perfectly flat summed output impedance while the parallel network has minor variations, which are mostly inconsequential given their magnitude (44mdB).

Driver Induced EMF
By changing the driver impedances, two things happen. The filter Q changes, and the reflected change affects the behavior of the other filter section. Although the individual response, Q and phase varies, the net result is that the effective crossover frequency is changed, but nothing more. This is a remarkable property, and the series first order is the only crossover filter circuit that has this ability.

Remarkable though it may be, it is still advisable to design the series network correctly, and maintain everything as close as possible to the design values. Should the woofer impedance increase (with voice coil temperature, for example), the crossover frequency will move upwards, thus providing a small measure of added protection for the tweeter at sustained high power levels.

However, all is not completely rosy. Everything in electronics is a compromise, and the selection of a crossover is no different. There is one final test that needs to be applied, and that is to examine the amount of woofer back EMF that reaches the tweeter. This is an area where the series network is inferior to the parallel.

Summary

With a parallel network, only the amplifier's output impedance plus the impedance of the cable allows any cross coupling between high and low pass sections. With a zero ohm source, attenuation is infinite, and is not shown above.

A series network relies solely on the isolation of the crossover filters, and as a result, the back EMF from the woofer is not attenuated as well. This may not be a major problem, since the attenuation of back EMF is the same as for amplifier power (actually, it is 3dB better), and the latter is at a far greater amplitude. It is a consideration nevertheless, so be aware that it may increase tweeter intermodulation.

Transient Response Comparison between First Order Series & Parallel Networks

Although not illustrated to save space, both first order series and parallel networks exhibit identical transient response under a fixed resistive load. However, when the load becomes complex, such as a real world loudspeaker load, the results are quite different as illustrated below.

Summary

Injecting a 1kHz square wave into each network loaded with a reactive load on the woofer, and looking at their summed response, we see the series network electrically passes the signal unadulterated while the parallel network exhibits overshoot on the rising and falling edges of the square wave. However, again by simply applying a Zobel network (series R and C) in shunt with the woofer load, we see the compensated parallel network can now pass the square wave just like the series network.

Conclusions

These simulations have revealed that first order series and parallel networks can be designed to exhibit very similar transfer functions as evident by their similar input impedance, frequency response, transient analysis, summed output impedance, and phase response. However, under a reactive load such as a loudspeaker, both parallel and series networks must be Zobel compensated to restore equivalent filter responses to their original responses during purely resistive loading. The series network is probably a better choice than parallel for a number of reasons. It retains a flat response even when the driver characteristics change, and is to an extent "self correcting". Implementation is no more difficult than for an equivalent parallel network, and the same component values are used. On the negative side, woofer back EMF suppression is significantly worse than with a parallel network - it is up to the designer to determine if this is likely to cause a problem.

Finally, it must be remembered that any first order network dictates that the drivers will have significant power applied at frequencies where their performance will be rapidly deteriorating, however for a system that will never be operated at high power, the performance can be very satisfying.