Conceptualising impedances and filters

[Sifunkle]

Wondering if someone can help me find the right angle to wrap my head around how impedances and filters relate. I have no formal education, just what I’ve managed to glean from the internet over the past few months. At the moment I’m going in circles.

My current situation arose from calculating some input and output impedances bookending an amplifier (small signal audio if it matters). Taking the output pathway (a single coupling capacitor in series with a complex of various resistances leading to the output)… I simplified the resistances down to a single equivalent value that I’m considering in series with the coupling cap, and then can calculate impedance from that resistance and capacitative reactance. Given capacitative reactance is frequency dependent, it seems fair to say there’s filtering going on.

So simplifying things to the one capacitor followed by a single equivalent resistance made me think ‘Seems awfully similar to an RC HPF’ (other than the resistance being in series rather than to ground before the output)… How do I integrate these two similar ideas conceptually? Can I calculate a single corner frequency for this output section like I could for a simple RC HPF? If so, would that have the same meaning as for a one-pole RC HPF (i.e. describing the same curve)? Or is it erroneous because I’m losing info as part of the simplification?

(Some other quick ones:
- Why does a simple passive HPF actually need the resistance to ground? Is a single cap not already filtering all on its lonesome? Is it a reference frame thing: that sooner or later there’s always a load that will act as the R in the RC?
- As someone who never learnt imaginary numbers and cannot understand the mathematical explanations, does the number of poles a passive filter has more-or-less equate to how many series RC pairs there are?)

All replies appreciated!

[wasedadoc]

The series C and R to ground act in a simiiar way to a voltage divider of series R and another R to ground.  In both cases the actual value of the R to ground affects the division ratio.  So that the ratio is properly defined it is customary to have a dedicated known value resistor.

Although the ratio is (almost) constant for all frequencies in the resistive divider, in the high pass filter the impedance of the C depends on frequency.

[gcewing]

And so the answer to your question about the R and C in series is that its behaviour as a filter depends on the input impedance of the load it's connected to.

If the load is a pure resistance, I think the whole thing will behave like a single-stage high-pass RC filter with the output reduced in amplitude by the voltage divider action of the series resistance and the load.

[PGPG]

Why does a simple passive HPF actually need the resistance to ground? Is a single cap not already filtering all on its lonesome? Is it a reference frame thing: that sooner or later there’s always a load that will act as the R in the RC?

If you intentionally design HPF you want to decide what is corner frequency and not let it be random. So in paralel to "sooner or later there’s always a load that will act as the R" you connect much smaller R and it is the R you ask about.

Nowadays you need not to know imaginary numbers to find circuit characteristics. Just learn to use one of free available Spice simulators and you will be able to find characteristics of any complicated circuit that practically no one analyses them mathematically, but just let computer to calculate it.
KiCad (Open Source PCB design project) allows for Spice simulation. It uses the external to it Spice project (I don't remember the name) that I don't know if is easy to use without KiCad. I just don't know if that project contains schematic drawing tool or is text only Spice.
LTSpice is the other (I have tried it only once - may be 10+ years ago).

[Geoff-AU]

...an amplifier (small signal audio if it matters). Taking the output pathway (a single coupling capacitor in series with a complex of various resistances leading to the output)…

...one capacitor followed by a single equivalent resistance made me think ‘Seems awfully similar to an RC HPF’ (other than the resistance being in series rather than to ground before the output)

My first comment is that "awfully similar" is incorrect.  A series element and a shunt element behave completely differently in most cases.  It's the same as having a wheel on its edge vs face-down on the ground.  It's still a wheel, but it behaves very differently in those two positions.

- Why does a simple passive HPF actually need the resistance to ground? Is a single cap not already filtering all on its lonesome? Is it a reference frame thing: that sooner or later there’s always a load that will act as the R in the RC?

If you don't have the shunt R then the filter's cutoff frequency depends on the load. 

If the load is a very low impedance then the capacitor will get charged very quickly by current flow (until it opposes the input voltage), and therefore the signal will get blocked.  So basically all frequencies will be blocked. 

If the load is a very high impedance then the capacitor stays uncharged, and the electric field easily crosses to the other side of the capacitor and stays there.  So basically all frequencies will be passed.

One key point is that for your highpass, adding the load resistance shouldn't change your filter's shunt resistance much.  Choose suitable component values.

- As someone who never learnt imaginary numbers and cannot understand the mathematical explanations, does the number of poles a passive filter has more-or-less equate to how many series RC pairs there are?)

Yep, exactly.  Also it's described as "order"s.  A 1st order filter has 1 set of element pairs, a 2nd order has 2 sets, etc.

Don't get bogged down into formulas.  Try to have an intuitive understanding for how the circuit works based on first principles.  Even when you do start working with numbers and formulas, you can get a fairly close estimate for the expected result in your head.


To get back to the original scenario, your amplifier has a series C and a series R for the output.  If your amplifier has only 1 positive power rail then the output can only swing between 0V and the rail voltage.  To amplify an AC signal this means you set the "resting" (bias) point of the amplifier to half of the rail voltage.  Therefore you have constant DC on the output, with AC superimposed as necessary.  The series output capacitor blocks this DC while allowing AC to pass through.  DC will kill headphones/speakers quickly by overheating them.  The series resistor I'm guessing is for amplifier short-circuit protection, it will limit the AC current the amplifier tries to produce into a faulty load.

[radiolistener]

Wondering if someone can help me find the right angle to wrap my head around how impedances and filters relate.

Impedance changes in a transmission line lead to wave reflections. Filters and matching circuits utilize the effect of wave reflections to accumulate the energy of RF oscillations in reactive components, such as inductors or capacitors. This allows them to convert impedance or block specific frequencies.

Don't confuse impedance with resistance, they are different concepts. When designing a filter, you need to know the required input and output impedances as well as the desired frequency response.

If you use a filter designed for an input impedance of Z1 and an output impedance of Z2 in a circuit expecting an input impedance of Z3 and an output impedance of Z4, where Z1 ≠ Z3 and Z2 ≠ Z4, the filter's response will change. This effect can be leveraged when designing a filter with specific impedances is challenging. In such cases, you can design the filter for different, easier-to-achieve impedances and then use impedance matching circuits at the filter's input and output.

- Why does a simple passive HPF actually need the resistance to ground? Is a single cap not already filtering all on its lonesome? Is it a reference frame thing: that sooner or later there’s always a load that will act as the R in the RC?

A simple passive high-pass filter (HPF) needs the resistance to ground because the resistor, together with the capacitor, forms the RC network that determines the filter's frequency response. A capacitor alone does block DC and attenuate low frequencies, but without a resistor to define the cutoff frequency and provide a reference for the output voltage, it doesn't form a complete filter. The capacitor by itself would not provide the expected frequency-dependent attenuation across a load.

- As someone who never learnt imaginary numbers and cannot understand the mathematical explanations, does the number of poles a passive filter has more-or-less equate to how many series RC pairs there are?)

Yes, the number of poles in a passive filter generally corresponds to the number of reactive components (inductors or capacitors) in the circuit. For an RC filter, each series RC pair typically adds one pole to the filter's frequency response. But in more complex filters (like Butterworth or Chebyshev), the poles are determined by the overall design, which might involve combinations of multiple RC pairs or other components arranged in a specific way to achieve a desired response.