DI deep dive
"Use a passive DI with an active source and vice versa."
In the next 10 minutes you'll learn why that common rule of thumb makes sense but also why it isn't the whole story. We'll also go through other confusing and often misunderstood concepts while delving into some audio and electrical engineering, so strap in. Here are the basic selling points of any DI:
- balancing an audio source
- (not) affecting signal level
- impedance bridging
- sound/tone coloring
Many DIs come with additional features, which may be important for you, including but not limited to stereo signals, ground lift, signal splitting, phase inversion and more. But I'll only cover the main features, why they matter and when they don't. Also note that I'm comparing the technical differences between well-made active and passive DIs, not taking into account shortcomings of specific units.
Balancing audio
First, creating balanced audio from an unbalanced source. This is done to reduce noise over long cable runs. Think live stages or big studios. Balancing takes advantage of the fact that any noise introduced in the cable is likely introduced equally to all conductors. By also sending an inverted signal, the noise can be canceled at the receiving end. In the illustration below, the original mono signal (two-connector, labeled with signal and ground) passes through a balancer (B) that copies and inverts the signal which is then fed through an XLR cable. On the other end, the signal is combined, canceling out the noise and producing the original noise-free mono signal.

Interestingly, carrying the inverted signal isn't strictly required as even a quiet signal would still pick up the noise to cancel it at the other end. Balancing circuits differ in implementation. Passive circuits use transformers while active ones use operational amplifiers (op-amps), but the result is identical. Microphones output balanced signals by default which is why they don't require DI boxes. Guitar pickups can also be wired to output a balanced signal, which could be useful in a noisy environment. For more details about balancing, see Neatcircuits.
Affecting signal level
DI boxes can be used to control signal levels. Here, passive and active boxes differ. Passive DIs use a transformer to step the voltage down according to the reported turns ratio (usually ~10:1 or 20 dB). Active DIs use a buffer amplifier which does not change signal voltage - an advantage with low-output sources. Using this circuit means however, that active DIs have an absolute maximum ceiling for their input voltage. It is often reported in dBu, typically 10 dBu (2.4 V) for active units. Passive ones also report a maximum input level, often around 20 dBu (7.7 V), but surpassing that will not lead to hard distortion but a more gentle saturation as the transformer is pushed. Attenuation or "pad" switches on DIs may be used to further reduce the voltage. Typically then, with a pad switch passive DIs make the signal 20-40 dB quieter, while active DIs often provide significant attenuators to compensate for the absolute voltage ceiling and lack of inherent attenuation, often up to 40 dB as well.
DIs must therefore be paired with the source and destination in terms of level to some degree. If the source is extremely quiet and the console does not have enough gain, using a passive DI will make the problem worse. On the other hand, using very hot sources with active units might distort them or a console's converters. In both cases, using a passive box or padding the input helps. Below is an example signal path with typical gain and voltage values for each stage. 20 dB of gain corresponds to a tenfold increase in voltage.

Let's present more examples to drive the point home. Microphone-level signals are usually around 1 mV, line-level signals in the order of 1 V (60 dB higher) and speaker-level signals far surpass that. Mixing consoles operate at line level. Suppose we have a console with 0-40 dB of gain (very little) and converters that can handle a 4 V signal:
- with a microphone that outputs 1 mV, we would boost the signal by 40 dB to get to a quiet 0.1 V
- with a loud active acoustic guitar that peaks at 7 V, we could use a DI box to attenuate the signal say 20 dB to a safe 0.7 V with no gain applied
- with a passive bass guitar that outputs 0.2 V, if we need to use a passive DI with a turns ratio of 10:1, our 40 dB of gain will still land us at a healthy 2 V
- with a rather quiet instrument source of 10 mV, if we were to use the same passive DI, our preamp would take us to 0.1 V, while an active DI with no pad engaged lifts us to 1 V
Impedance bridging
DI boxes help in achieving a bridging impedance. To understand what it is and why it matters requires a short detour into electronics. Sound in a circuit is most accurately characterised as changes in voltage, and those changes can be amplified and recorded. To do that we use a voltage divider that completes the circuit to measure the electric potential. Its response is dependent on the ratio of two impedances: load and source impedance. Source refers to the instrument being played, and load is the divider or the device capturing the audio. With a high ratio, meaning when the load impedance is much higher, voltage is transferred efficiently while minimally affecting the source.
If the load impedance is too low, the voltage transfer efficiency drops. This would be fine if the impedance of the source stayed constant, but crucially for audio, the impedance of most sources varies with frequency. So the measured frequency response of the source changes when the load impedance is too low, which is why we'd like to have as high of an impedance ratio as possible. However, there's a tradeoff between the efficiency and how much noise is introduced to the signal especially in long cables, because the high "sensitivity" of the load will also allow noise to drive the divider. And the returns of extreme impedance ratios definitely diminish.
Sources must therefore be paired with loads in terms of impedance. Ordinary microphone preamplifiers that typically have impedances around 1-10 kΩ cannot handle electric instruments (20-200 kΩ), much less piezo pickups (1-10 MΩ). But line inputs generally have a higher impedance, and may even come with a switch for a high-impedance (Hi-Z) mode that could raise it towards 1 MΩ. See this impedance bridging calculator for calculating how much signal loss occurs for a given setup. Again, passive and active DIs differ in how they are implemented, but the real-world performance of both is very comparable and the differences are subtle. The usage of transformers makes passive DIs amplify the impedance of the load with the square of the turns ratio:

With active units, their op-amps have a naturally high input impedance that is presented directly to the source while a low output impedance is presented to the load:

As a rule of thumb, an impedance ratio of 10 is considered bridging. But it is not a hard rule. If noise isn't an issue you could go way beyond, and having less is okay too. Let's go through more examples again:
- a microphone with a peak impedance of 200 Ω connects to a preamplifier with 2 kΩ and a bridging impedance is achieved
- a guitar with a peak impedance of 50 kΩ connects to a line input with a Hi-Z switch that brings the impedance to a bridging 1 MΩ
- a bass with a peak impedance of 20 kΩ connects into a preamplifier with 2 kΩ, so either a passive DI with a turns ratio of 10:1 or more (most of them), or an active DI with an input impedance of >200 kΩ and an output impedance of <200 Ω is needed to achieve a bridging impedance
- a keyboard with an output impedance of 200 Ω connects to a line input with 10 kΩ and a bridging impedance is achieved
Also, product pages for passive DIs often report static input and output impedances. This is misleading because the the transformer only scales the load impedance. And what's reported as the output impedance often describes the inherent impedance of the transformer itself, which should be negligible because the source impedance is much higher. So take note of the load impedance that the transformer was tested with, and ask manufacturers if using a drastically different load changes the characteristics of the transformer and the final sound. In this sense, an active circuit is a safer bet because the behavior is more consistent. For a detailed discussion, see here.
Sound and tone
Finally, a brief note on tone. Proper gain staging and impedance bridging make most sources sound at their best and avoid huge blunders in sound quality. Some poorly-constructed or old DI units may have issues with frequency response, often in the bass region. If that's not the case, and if the major issues in the signal chain are fixed, then components and circuit design of the DI affect tone in minor ways. Listen to audio examples or try various units out if possible to hear if the differences are significant enough for you to care.
In closing, DI boxes are designed to solve problems. If you don't have problems, you don't need one. Most modern instruments output a healthy signal level, most DI boxes have a reasonable design, and mixing consoles have plenty of gain. So as long as there is no fundamental mismatch between your DI and other equipment, you'll be fine with either type of DI, passive or active, especially if you are just starting out or playing live. If you can reliably hear a big difference, obviously go for what sounds best. Do also consider if you need the other features. But most importantly, I'm just a guy on the internet. Don't take my word for anything. But please do let me know if I made any significant factual errors!