# Circuit Analysis Using a Two-Port Transformation

Quote of the Day

The power of instruction is seldom of much efficacy except in those happy dispositions where it is almost superfluous.

— Gibbon. I think this quote is a bit harsh, but not off too much. I recently have been taking some online classes where I work problems and can ask the instructor questions if I have issues. In this case, there really is no instruction – I am just reading the book and getting the opportunity to ask an expert questions. It works.

## Introduction

Figure 1: Simple Voltage Regulator with Current Limit.

I was doing some reading on the Planet Analog web site when I encountered an interesting blog post by Dennis Feucht on a simple BJT-based voltage regulator with an output current limit.

I thought Dennis' post was good because it (1) provided a very clean demonstration of the use of Thevenin and beta transform equivalent circuits for analysis, and (2) it also provide me a good demonstration for how to use a computer-algebra system to help you design a circuit.

I am always looking for good Mathcad reference applications for my staff. In this post, I illustrate the the basic circuit transformation and then use Mathcad to determine component values and predict circuit performance. I also simulate the circuit using LTSpice (Appendix A).

I should point out that even though this circuit is simple, the algebra can get overwhelming. The gods of electronics work by simple rules, but they have no fear of algebra.

## Background

### Motivation

I am always looking for simple power conversion circuits to use for my home projects. I like to see current-limited power sources for safety reasons. The performance of this circuit is not great, but there are ways to improve it – I will cover these later.

### General Operation

This circuit really operates in two modes (see Appendix A for simulation details):

• Q1 Saturated

When not limiting the output current, Q1 is saturated. As such, Q1 dissipates relatively little power.

• Q1 Active

When limiting the output current, Q1 is in the active region and is dissipating significant power.

## Analysis

### Simple Model

Figure 2 shows the circuit of Figure 1 with a simple DC PNP transistor model and the base circuit transformed to a Thevenin equivalent.

Figure 2: Equivalent Circuit.

### Circuit with Beta Transformation

Figure 3 shows the circuit of Figure 2 using a beta transformation.

Figure 3: Equivalent Circuit with Impedance Transformation.

### Derivations

#### Derivation of Formulas for the RI and RB Values

Figure 4 shows how to analyze the circuit in Figure 3 for RE and RB. Note how I grabbed an intermediate term to determine the constraint for a positive RE value.

Figure 4: Derivation of RI and RB.

#### Derivation of Constraint on RE

Figure 5 shows how to derive the constraint on RB that ensures positive RE.

Figure 5: Derive Constraint on RB.

#### Derivation of RE Equation

Figure 6 shows to derive the expression for RE. I start with the expression shown in Figure 4 with the bubble numbered 1.

Figure 6: Derivation of RE.

#### Example

Figure 7 shows the example worked on the blog post.

Figure 7: Example from the Blog Post.

## Conclusion

This was a good illustration of the capabilities of a computer algebra system for a simple electronic circuit. This current-limited voltage source served that purpose well.

## Appendix A: LTSpice Simulation

I captured the circuit in LTSpice (Figure 8).

Figure 8: LTSpice Version of This Circuit.

Figure 9 shows my simulation result. The values are in the range I would expect for this circuit.

Figure 9: LTSpice Output.

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### 2 Responses to Circuit Analysis Using a Two-Port Transformation

1. Gene Mirro says:

In your figure 1, shouldn't R1 be connected to Vs+ ?

• mathscinotes says:

You are correct! Thank you very much. Correction has been made.

mathscinotes