Determining the Impedance of a Circuit and How It Affects Your PCB
Impedance affects how signals propagate through the board, how power is transferred between components, and how signals bleed into unwanted areas of your PCB. There are a number of analyses you can use to determine the impedance of a circuit, but these do not always produce realistic results unless you include the right parasitic elements in your models. If you are testing a prototype or designing a circuit for use with high-speed or high-frequency signals, you will need to have an understanding of the parasitic elements that affect the impedance of a circuit.
It is also important to understand how the impedance of a nonlinear circuit is affected by the input signal level, as the impedance of a nonlinear circuit element is really a function of the input. Analyzing the impedance of a circuit with nonlinear components becomes more complicated and requires a different class of analysis techniques.
What Is the Impedance of a Circuit?
The term impedance describes the behavior of electronic components in resisting AC current. A circuit is made up of various components that behave differently in the presence of an AC source.
The resistor behaves consistently regardless of the frequency of the AC source, as it is a pure DC component. Its impedance is a measurement of its resistance:
Z = R
Meanwhile, the capacitor’s impedance equals its capacitive reactance, which is given by:
Xc = 1/2 πfC
Capacitive reactance is an inverse function of the angular frequency. As the frequency increases, the capacitive reactance decreases.
As for inductors, they behave in a similar way with capacitors, except that inductive reactance is directly proportional to the angular frequency, with the formula:
While these are basic components often associated with impedance analysis, they are not the only ones present in a circuit. In a design, you are likely to have ICs, diodes, transistors, and other passive components, and they may introduce impedance to the circuit.
At a deeper level, your traces have some impedance, as they cannot be considered as long inductors. The board itself will affect the impedance of a circuit and your traces. The insulating PCB substrate creates parasitic capacitance, while the arrangement of traces and planes in interior layers creates parasitic capacitance and inductance. These parasitic effects contribute to capacitive crosstalk, and they determine the impedance of transmission lines and your power delivery network.
How to Calculate the Impedance of a Circuit
Linear elements are the simplest circuit elements in that their impedance is not a function of the input voltage. Basic electronics principles such as Kirchoff’s Law and Ohm’s Law are applicable in AC analysis to determine a circuit’s impedance, voltage, and current.
For a series RLC circuit, the total impedance can be calculated by the following formula.
Z = √(R2 + (XL - XC)2)
As both capacitive and inductive reactance are a function of angular frequency, the total impedance is expressed in a complex number with real and imaginary components.
Here is a common expression for circuit impedance:
Z = R + jX
Besides manual calculation, you can use the Gauss-Jordan method in SPICE simulators, which will give you the total impedance of the circuit and the respective components. In the time domain, the arrangement of circuit elements will affect the transition to steady-state behavior, which can be analyzed with transient analysis or pole-zero analysis.
Impedance calculations get trickier when the circuit contains elements like diodes, transistors, amplifiers, and other elements where the output is a nonlinear function of input signal strength. The impedance is actually defined in terms of the transimpedance at a particular input signal strength. In other words, if the input signal strength changes, so will the transimpedance of each nonlinear circuit element as well as the equivalent impedance of the circuit.
For more complex circuits and circuits with nonlinear elements, more advanced simulation and analysis techniques are required to determine the impedance seen by a digital or analog signal in a circuit.
The right analysis tools and component electrical models can help you analyze the impedance and behavior of circuits in this schematic
How PCB Layout Affects Circuit Impedance
In a real PCB layout, the impedance seen by signals can be very different from the ideal value you determined from a schematic. This is due to the presence of the substrate and the arrangement of traces on the board. This gives rise to important effects like crosstalk, which changes the impedance from ideal values. Circuits can experience power integrity problems such as ringing when circuits switch at high frequencies. The impedance of your power delivery network will also deviate from the ideal capacitive behavior at higher frequencies, which contributes to potential signal integrity and power integrity problems.
Transmission Line Impedance
The impedance of transmission lines can be characterized using a number of impedance values. The most important of these is the characteristic impedance, which is simply the impedance of a transmission line on a PCB in total isolation from any other transmission line. This value is normally 50 Ohms, although it may take a different value depending on the signaling standard used in your device. As an example, LVDS specifies that the differential impedance of a differential pair should be 85 Ohms.
The other metrics used to describe transmission line impedance depend on the relative arrangement of two transmission lines. Due to the parasitic capacitance provided by the PCB substrate and the mutual inductance between two nearby transmission lines, a transmission line can be characterized using even and odd mode impedance, which accounts for coupling between two nearby transmission lines and how the two lines are driven (i.e., in common mode or differential mode). Common and differential impedance are related to these other values, bringing the total number of impedance values used to describe transmission lines to five.
These traces may act like transmission lines if they are long enough.
Power Delivery Network Impedance
Your power delivery network will exhibit capacitive impedance at lower frequencies and reduces to the resistance of your power bus in series with your load components and ground return path at DC. This impedance is dominated by the physical separation between your power rails, traces, and internal planes in your board. As the driving frequency increases, mutual inductance between circuits in your board will cause the impedance of your power delivery network to increase. Eventually, the impedance of your power delivery network will exhibit many peaks at high frequencies.
Ideally, the impedance of your power delivery network should be flat within the band you want to work in. For digital signals, the relevant bandwidth is all frequencies between the clock rate and the knee frequency (0.35 divided by the signal rise time). If all harmonics that comprise a digital signal see the same frequency, then the transfer function for a return signal in a ground plane will be flat. The same idea applies to analog signals traveling throughout your board and ground planes.
While the impedance spectrum is important for identifying a bandwidth that has minimum power delivery network impedance, the spatial distribution of impedance in your ground plane is much more important, particularly in mixed-signal devices. Signals will follow the path of least reactance back to the ground return when traveling in the ground plane. Ideally, the path of least reactance in a star, point-to-point, or multipoint topology should lie directly beneath conductors in your board. This will ensure your circuits have minimized loop inductance and will have the least susceptibility to EMI.
Example power delivery network impedance spectrum
PCB Substrate Material Selection and Stackup Design
Due to the parasitic effects mentioned earlier, you will need to carefully select a substrate material and design your stackup. The dielectric constant of your substrate will affect the geometry required to produce a transmission line with specific impedance and will affect the impedance of the power delivery network. The presence of planes beneath conductors also determines the loop impedance in a circuit, which affects a circuit’s EMI susceptibility.
Impedance Matching Networks
The stackup will affect plenty of other aspects of the board design, such as the thermal resistance and routing strategy. When combined with the right substrate material, you can reduce the losses seen by signals while maintaining consistent impedance throughout your circuits. Maintaining the impedance of a circuit to a specific value during routing is important for ensuring impedance matching throughout a net. As signals transition to transmission line behavior, you will need to ensure that the transmission lines, drivers, and receivers have consistent impedance to prevent signal reflections.
You’ll need to determine the impedance of transmission lines and vias in this layout
Just like a PCB substrate will have some parasitics between neighboring conductive elements, so will vias in a multilayer board. Vias are essentially small inductors with an air-filled or conductive epoxy-filled core. The inductance of a via is on the order of nanohenries and depends primarily on its aspect ratio. Vias also have self-capacitance, and groups of vias have some mutual capacitance and mutual inductance. This leads to noise coupling between vias and causes vias to act as impedance discontinuities when placed on a transmission line. In general, the use of vias is generally kept to a minimum in high-speed and high-frequency circuits.
Measuring and Analyzing Impedance in a Circuit
Circuit impedance can be measured by a setup that includes a signal generator, oscilloscope, and the load with unknown impedance. It involves analyzing the output amplitude and phase shift against the input signal for a particular frequency. Equipment like impedance meters and impedance analyzers are also great options for impedance measurements.
However, the above measures are only applicable when you have a finished product. If you are in the midst of designing a circuit, you will need simulation and analytical tools to determine the impedance of a circuit.
If you have access to a SPICE-based simulator, you can easily determine the impedance of a linear circuit using an AC frequency sweep and you can visualize the transfer function for a circuit in a Bode plot. These tools can be used to show you the total impedance of a circuit block in your design as well as how a circuit block affects the magnitude and phase of an input signal.
These vias and conductors will affect the impedance seen by signals throughout your board.
With nonlinear circuits, you will need to use more advanced analyses. The most prominent tool for working with nonlinear circuits is to use a DC sweep, as this tells you how a DC input voltage propagates an output voltage and current. When working with AC signals or arbitrary oscillating waveforms, you can use small-signal analysis to examine change around some operating point. This analysis tells you the transimpedance of a circuit around a specific operating voltage in the circuit. A more powerful technique is harmonic balance analysis, which tells you how a sinusoidal signal and its harmonics behave in a nonlinear circuit.
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