Finding AC Steady State Values in Nodal Analysis

November 3, 2020 Cadence PCB Solutions

Key Takeaways

  • Nodal analysis is one way to calculate the voltage and current distribution in a circuit, including in AC circuits with harmonic/time-dependent sources.

  • Designers can find the steady state value in nodal analysis for an AC circuit using the same techniques as in DC nodal analysis.

  • These results are equivalent to the results from KCL and KVL calculations, but they bear closer resemblance to SPICE results.

Steady state value nodal analysis for circuit designs

Can you find the steady state values in nodal analysis for these circuits?

Any circuit can be difficult to analyze by hand, but some simple techniques can help you see exactly how voltage and current are distributed in different circuits as you create your designs. Ohm’s law, KVL, and KCL are the basic methods  engineers and scientists learn to use for circuit design and evaluation, but nodal analysis gives designers an alternative way to examine voltage and current distribution in  a complex circuit. No matter which way you choose to go, you’ll be able to get the same results.

When looking at AC systems with time-dependent sources, you may need to find the steady state values in nodal analysis, especially when you know there will be a transient response in the system. To get a steady-state value in nodal analysis, you can use the same procedure found in DC circuits. Here’s how this works in your circuits and how you can use nodal analysis as part of your design process.

What Is Nodal Analysis?

Nodal analysis involves determining the voltage and current at each point in a circuit in a manner similar to that used in Kirchoff’s laws. Just like in KCL, a circuit is composed of multiple nodes, and voltage and current are calculated at each node in the system. The process for setting up these problems is rather simple, where the goal is to derive a system of linear equations describing the voltage and current in the system.

  1. Select a reference node in the circuit (usually taken as ground)

  2. Use KCL to write out the currents entering/exiting each node

  3. Write the voltage measured at each node with respect to the reference node

  4. Solve the system of simultaneous equations to find all unknown voltages and currents

Defining Voltages

Step 3 is quite critical as it formulates voltage drops in KVL in terms of differences in voltage between a node and the reference node (i.e., the lowest ground potential point in the system). However, this makes nodal analysis more powerful than KVL as it can be used in systems with two different ground potentials. To see how voltages are defined in a system, take a look at the example below. In this circuit, there are two AC voltage sources with different frequencies and 2 impedances.

Steady state value nodal analysis in a simple circuit

Nodal analysis for an AC circuit with two sources.

Note that, in this circuit, we’ve specified the impedance values; for capacitive or inductive impedances, and when the system is being driven by sources with different frequencies, you’ll need to consider the problem in the frequency domain rather than the time domain. In the above problem, we’ve assumed the complex impedances are the same at both frequencies. Such a situation would be typical in an RF system with a wideband phase shifter, attenuator, or other passive structures.

This problem is one of finding the voltage and current at nodes B and C in the above circuit diagram. Note that, as all voltage measurements are differential measurements, the voltage at nodes B and C are defined with respect to the reference node below the left AC source. Assuming the reference node is defined as 0 V, we can construct the following equations for the voltage at nodes B and C:

Steady state value nodal analysis

Voltage at the two nodes shown in the above example.

If there were more than two nodes in the system with unknown values, we could apply the same process to the unknown voltages/currents at each node. The problem is now one of finding the voltage at each node in the above equation in the steady state.

Finding Steady State Values in Nodal Analysis

In the problem shown above, the steady state voltages at each node in a system driven with sinusoidal sources will also be periodic waveforms. In this case, where the two sources have different frequencies, the resulting waveforms may be rather complex, but they will still be periodic. To determine the steady state values in nodal analysis, simply solve the above equations for VB and VC. This is most easily done by writing the above system of equations in matrix form: 

Steady state value nodal analysis

Voltage at the two nodes shown in the above example.

To solve this equation, simply multiply both sides by the inverse of the coefficient matrix. You’ll now have the voltages at nodes B and C. You’ll find that the resulting waveforms measured at nodes B and C are a rather complex superposition of the source voltages measured at nodes A and D. You can now determine the current in each impedance by using Ohm’s law.

Using Nodal Analysis for Design

Oftentimes, you have a voltage/current specification at a particular node or between a set of nodes, and you need to find the right impedance value required to produce that specific current. This takes the above process with nodal analysis and runs it in reverse. You know the voltage and current, but you need to find an impedance that produces the right voltage/current with the right phase relationship with respect to the input signal.

This is a simple problem to solve with nodal analysis, KVL, or KVL; they will all produce the same results. This is often used in designing voltage divider legs in complex circuits, in feedback loops, and other non-trivial circuits. Instead of having an unknown voltage, you’ll have a known voltage at two nodes but an unknown impedance between the two nodes. This now becomes a problem of solving for the unknown impedance with the system of equations you construct for the system. An example for the above circuit is shown below.

Steady state values in nodal analysis unknown impedance

If VB and VC are known, then Z1 and Z2 can be determined by solving the above system of equations.

Another method for doing this is to use a parameter sweep in your circuit simulator. This involves sweeping through a range of values for a circuit element and calculating the voltage/current in the relevant circuit element. This is particularly useful in DC analysis as you will be working with a resistor and voltage/current values.

When you’re designing your circuits, you can find the steady state value in nodal analysis using the best front-end design features from Cadence. The PSpice Simulator application lets you create and simulate your circuits, and you can perform a range of analyses using built-in functionality. Once you’ve designed your circuits, you can capture your circuits and start creating a PCB layout.

If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.

 

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