Understanding the Characteristic Impedance Formula

October 13, 2021 Cadence System Analysis

Key Takeaways

  • Every circuit is different, and will have a characteristic impedance that depends on several factors.

  • A characteristic impedance formula will only be specific to a given circuit, and it can be calculated from that circuit’s various elements and components.

  • If you have voltage and current measurements from a circuit network, you can calculate the network’s characteristic impedance using Ohm’s law.

Characteristic impedance formula

The term “characteristic impedance” can simply refer to a circuit’s impedance as calculated from equivalent circuit rules or Ohm’s law. With real circuits that are used as networks, the delineation between a network’s characteristic impedance and its input impedance becomes less clear, and the two terms are often misunderstood or poorly communicated. Then, when you have a circuit with a common-mode or differential-mode input, the characteristic impedance of a circuit becomes more complicated to describe. Nonlinearity in a system also makes characteristic impedance difficult to describe from a circuit perspective.

When you want to predict how an input signal, whether harmonic or an arbitrary input, will behave as it encounters a circuit network, then you need to know the circuit’s characteristic impedance. There is no characteristic impedance formula for every circuit, but there are some simple ways you can think about how a circuit is affected by a load component and how this relates to the circuit’s characteristic impedance.

Is There a Characteristic Impedance Formula?

The simple answer is “no,” there is no single characteristic impedance formula. This term is normally used to refer to the characteristic impedance of a transmission line, whose characteristic impedance depends on its dimensions, so there are many different formulas for characteristic impedance. The term is sometimes used to refer generally to the impedance of a circuit, which could be called a characteristic impedance, but they are not the same thing.

For a transmission line with known R, L, C, and G values, you have the classic characteristic impedance formula from transmission line theory:

Transmission line characteristic impedance formula

A transmission line structure in an integrated circuit, on a PCB, or in any other structure that supports wave propagation, will always have R, L, C, and G values that depend on the geometry of the structure. In other cases, the value of a load component/network connected to your intended circuit will cause the input impedance of that network to be much different from the characteristic impedance. The input impedance is an important point to distinguish from characteristic impedance, as the two values are not always the same.

An Example With 3 Series Impedances

To see how the input impedance can differ from the characteristic impedance, it helps to look at a simple example of a circuit with 3 elements in series. In the circuit shown below, we have 3 impedances in series; the terminals on the left side of the circuit are the location where we would input some signal (a voltage or current source, or the output from some other circuit). If we measure the total impedance of these three elements (by replacing the voltage source with an Ohmmeter), we would see that the equivalent impedance is the sum of the 3 impedances.

Impedance of a circuit network

The impedance of this circuit network is the sum of the individual impedances

The impedance of this circuit is just the impedance measured across the inputs when the output is not connected to anything. In this case, the input impedance is equal to the equivalent impedance when the output is connected to an open circuit. This is only true for this particular circuit; it is not generally true for every circuit.

Adding a Load Impedance

Now, let’s suppose that the output is connected to a load impedance Z4. If we measure the impedance across the input, we would have a different result, as Z2 is in parallel with Z4. In other words, the presence of the load component modifies the equivalent impedance of the combined circuit and the input impedance is no longer going to be equal to the equivalent impedance.

Input impedance in a circuit

The load component impedance and the characteristic impedance combine to produce an input impedance, seen at the input terminals on the left side of this circuit

This should illustrate what the definition of a network’s equivalent impedance is: it is the impedance of the network when it is isolated from all other circuit networks, meaning the network is not connected to anything. The same idea applies to a transmission line’s characteristic impedance; the real impedance that matters is the input impedance, which may or may not be equal to the equivalent impedance (or the characteristic impedance for a transmission line). You can’t actually measure the characteristic impedance except in specific situations.

This should illustrate why there is no single characteristic impedance formula for every circuit network. If we combined the original 3 impedances differently (such as in parallel) and connected them to the same load impedance, we would get a different characteristic impedance and a different input impedance. In contrast, if we use the same circuit shown above and just connect the load impedance across one of the other two elements, such as across Z1 or Z3, the input impedance would be different but the characteristic impedance would not be.

Different Connections Give Different Impedances

This should illustrate why a SPICE simulation for the voltage and current in a circuit network will be different if the load impedance is varied. This might sound like it should be obvious, but it’s common to be surprised when the load impedance connected to a circuit causes the simulated voltage and current to appear to be related by something other than the equivalent/characteristic impedance. In reality, and specifically in a SPICE simulation, it is the input impedance that determines the circuit’s electrical behavior, not the equivalent impedance or characteristic impedance.

Matching network

The matching network (in blue) can be designed such that the antenna impedance appears to match to 50 Ohms

The above principles might seem odd when taken in isolation, but they are essential to understanding impedance matching and designing an impedance matching network for an interconnect. For example, if you want to prevent a reflection on a 50 Ohm feedline that is connected to a (7 + i6) Ohm antenna, you need to add a circuit that causes the input impedance of the (circuit + antenna) network to be equal to 50 Ohms.

In a way, Ohm’s law is a tool that can be used as a characteristic impedance formula. To use it, take the circuit in isolation, source a harmonic voltage, measure the current, and divide these at each frequency to get the characteristic impedance. A more sophisticated (and accurate) method is to use an S-parameter measurement, which is captured with a VNA and with the DUT matched to some desired reference impedance (normally 50 Ohms). In this case, the input impedance can be calculated directly from an S11 measurement, and the characteristic impedance can then be extracted.

With a complete set of system analysis tools from Cadence, designers can create and simulate their circuit and network designs as well as use these tools to capture the input and characteristic impedance of different circuit networks. If you like, you can use the PSpice modeling application to determine a characteristic impedance formula using regression, or you can calculate it directly from simulated electrical measurements. The complete set of simulation features in Cadence’s powerful field solvers also integrate with your PCB layout software, creating a complete systems design package for any application and level of complexity. 

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