Series Resistance
6.5 The Thevenin Model of a Voltage Source
Is this resistance a lot or a little? How much resistance is too much?
There are two criteria to judge the relative importance of the series resistance of a trace:
✓ How much voltage drop there is in the signal trace due to the IR drop
✓ How large the trace resistance is compared to the resistance of the voltage source.
For a signal trace and a maximum resistance of 0.40 ohms, the voltage drop with a current of 100 mA would be 0.040 V or 40 mV. In most applications, this voltage drop in a signal path carrying 100 mA may not be significant. When connecting an analog signal, in which case small voltage drops may be significant, the current might be on the order of 1 mA and the voltage drop would be as low as 0.4 mV.
For a power path with a trace series resistance of 0.125 ohms, the maximum current we might see, limited by potential trace heating, is 3 A. This would result in a voltage drop of as much as 3 A x 0.125 ohms = 0.375 V. This can be a significant voltage drop in some situations. For a 5 V rail, a 0.375 V drop to the device being powered is a 7.5% drop.
In very high current applications, and when small voltage drops are important, it may be necessary to use a trace wider than 20 mils for delivering power. This is why it is so important to get in the habit of putting in the numbers to estimate performance and doing your own analysis.
The second criteria to use to evaluate how much resistance is too much is in comparison to the resistance of the source that drives the current.
6.5 The Thevenin Model of a Voltage Source
The absolute simplest ideal equivalent circuit model for a power source is an ideal voltage source. The fundamental property of an ideal voltage source is that its output voltage is constant, no matter
the load attached. This model matches the behavior of many real- world power sources when they supply very low currents.
A far more accurate equivalent circuit model that takes into account the behavior of a real power supply to higher current loads is a Thevenin circuit model. This is a powerful model and should always be the starting place to describe a real power supply.
Every voltage source, either from a microcontroller’s digital output pin or from a voltage regulator module (VRM) such as a switch mode power supply (SMPS), a low drop out (LDO) regulator, and even a battery, can all be described to first order as an ideal Thevenin voltage source.
This means the voltage source can be described by an equivalent circuit model composed of an ideal voltage source and an ideal series resistor. This equivalent circuit is shown in Figure 6.7.
Figure 6.7 Basic Thevenin model of a voltage supply.
An ideal voltage source will always keep its output voltage constant, no matter the current load. This means that the impedance looking into an ideal voltage source is 0 ohms: Z = V/I = 0/anything = 0.
The series resistor in the circuit means that as the current from the source increases, the voltage at the output of the power source will decrease due to the internal voltage drop across the Thevenin resistor.
Only two terms completely characterize a Thevenin voltage source:
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✓ The Thevenin voltage, Vth
✓ The Thevenin resistance, Rth
These two terms are the important figures of merit of any power supply. Knowing their values will tell us about the performance of the power supply under many typical applications.
Unfortunately, however valuable this model and these two figures of merit are in describing a voltage source and using it effectively, rarely are these terms directly specified in the datasheet of a component. This is one reason it is important to be able to reverse engineer these figures of merit by measuring them.
Reverse engineering is a very valuable skill we will use over and over again. It is a process that extracts the behavior and properties of a component or system based on an assumed model.
Using our best guess, intuition, or other insight, we create an ideal model of the device. We perform measurements on the device and compare the measurements to the simulations of the model. We adjust the parameters of the model until the simulated voltages or currents match the measurement.
When we get good agreement, we have confidence the model is a good approximation of the real component, and the parameter values we used become the figures of merit for the device.
To apply reverse engineering principles to a power source, we assume the equivalent circuit model of the actual VRM DUT is the ideal Thevenin voltage source model. The voltage on the output of the voltage supply, Vout, with no load is a measure of the ideal Thevenin voltage parameter. This is the easy part.
In principle, the way to measure the Thevenin resistance parameter is to short the end of the voltage source and measure the current through the short. The Thevenin resistance is the ratio of the
Thevenin voltage to the short circuit current. In practice this is never a good idea.
The Thevenin model is only a first-order model to describe the real voltage supply. It may apply well when the current draw is small, but this simple model may not match the device behavior when the current load is large and approaches the short circuit current. Many voltage sources have a current clamp or have a nonlinear transistor output stage. This means when the output of the voltage source is shorted, the device may be operating in a completely different mode than when the output is not shorted, and our first-order model may not be a good approximation.
A practical approach to extract the equivalent ideal Thevenin resistance is to add a resistive load to the source that is comparable to or a little larger than the internal Thevenin resistance. This forms a voltage divider. We should choose a load resistance so that the voltage drop with the load is not larger than half the Thevenin voltage of the source.
The output voltage across the load is a direct measure of the Thevenin resistance. Figure 6.8 shows the equivalent circuit.
Figure 6.8 Circuit used to measure the loaded voltage of a Thevenin circuit.
The extracted Thevenin resistance is:
th load
th load
load
V V
R R
V
−
=
The typical Thevenin resistance of a simple AC to DC converter or a battery or a signal source is on the order of 0.2-10 ohms. A 50 ohm
6.5 The Thevenin Model of a Voltage Source 161
load is a convenient value resistive load to add to reverse engineer the Thevenin resistance.
Many AC to DC power supplies, for example, show a current rating along with their DC voltage output. This current rating is very misleading. What limits the current? If the current rating is 1 A, does this mean that if the external load is low enough to draw 1.1 A, the power supply will not work or will turn off, or will explode? Will the 1 A-rated power supply never provide 1.1 A of current?
The current rating says nothing about the output Thevenin
resistance. It is usually based on the maximum power the supply can provide with acceptable temperature increase due to the power dissipation handling of the packaging. This is the maximum current the power supply can handle without any long-term thermal issues.
It is not about the maximum current the supply can deliver under shorted load or related to the Thevenin resistance.
A simple way of reverse engineering the Thevenin resistance of a voltage source is to measure the output voltage of the source open circuit and then again after a resistive load is connected across the output of the power source. A scope or DMM can be used to measure the output voltage of the power source with and without the external resistor attached.
The resistor attached as the load should be low enough to cause an easily measured voltage drop when loaded, but not a voltage drop larger than half the unloaded voltage. A good initial resistance to try is a value on the order of 50-100 ohms.
As a simple alternative, an oscilloscope can be used to perform this measurement, taking advantage of its built-in 50 ohm input
resistance. To use the internal 50 ohm resistance of most scopes, it is important to use a direct coax cable connection to the scope and NOT a 10x probe. The voltage measured on the output of the source under this no-load condition, with a 1 Meg input to the scope, is the
Thevenin voltage.
Then the scope is set for a 50 ohm input resistance. Just be sure the voltage on the output of the source is not larger than 5 V. There is a limit to 5 V as the largest RMS voltage that should be connected to
any scope set for 50 ohms due to the power dissipation ability of the scope’s internal 50 ohm resistor. It can only consume 0.5 watts before it may be damaged. If an input rms voltage of more than 5 V is applied to the 50 ohm input resistance of the scope, the internal resistor and circuit board to which it is mounted may be thermally damaged.
The measured voltage drop on the supply output when the 50 ohm scope resistance is applied to the device can be used to extract the Thevenin resistance.
This technique was used to measure the 5 V rail from an Arduino Uno powered by an external USB hub. The Thevenin voltage, measured with a 1 Meg termination in the scope, was 5.15 V. When the scope was set for a 50 ohm load, the voltage measured on the 5 V rail was 4.93 V. The Thevenin resistance is
th load
th load
load
V V 5.15V 4.93V
R R 50 2.23
V 4.93V
− −
= = = There is no law that says a real voltage source must match the behavior of this first-order ideal circuit model. However, it is remarkable that many real voltage sources do show an output voltage drop with current that matches the behavior predicted by this simple ideal circuit model.
The fact that real voltage sources really do behave as this first-order Thevenin model predicts is what makes this model so valuable in describing real sources. Using this model, the extracted figures of merit of the Thevenin voltage and resistance become valuable figures of merit to describe any real voltage source.
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Watch this video and I will walk you through this simple process of reverse engineering the Thevenin voltage and
resistance of two power sources.
A simple circuit can be used to routinely measure the equivalent Thevenin voltage and resistance for any voltage supply using an Arduino microcontroller board as an automated measurement tool.
An example of this circuit is shown in Figure 6.9.
Figure 6.9 Circuit used to automatically measure the output voltage on chan2 of the Arduino, and the current load on chan1 of an Arduino. This will automatically extract the
figures of merit of a Thevenin circuit model.
A digital output pin of an Arduino generates a slowly increasing voltage using an RC low-pass filter. This voltage ramp drives a transistor current source. As the current load to the power supply increases, the voltage across the source and the current through it are measured by two of the analog to digital converter (ADC) input channels of an Arduino used as a measurement instrument.
From the measured voltage drop on the rail and the measured current from the source, the open circuit voltage and the Thevenin resistance can be extracted. These are the parameters or figures of merit of the equivalent ideal circuit model of the voltage source.
For example, the output voltage of a 100 mA rated, 12 V AC to DC convertor was measured as the DC current load changed. The slope
of the V vs I curve is a direct measure of the output resistance.
Figure 6.10 shows the measured data with a slope that is very constant with a value of 7.2 ohms. This is the measured Thevenin output resistance of the power supply.
Figure 6.10 Measured output voltage as the current load increases on a 12 V AC to DC converter.
In this example, the slope is very constant, but above 0.2 A current load, the output voltage drops faster than just a simple 7.2 ohm resistor would predict. This measurement shows when the simple first-order model begins to break down and a more complicated behavior begins to appear. However, even up to a current of 300 mA, the Thevenin model is still a very good approximation.
Not all voltage sources match this simple Thevenin model, especially as the current load increases. Many devices, especially signal
sources, have a current clamp in their outputs making the output resistance very nonlinear. The Thevenin model is only a rough approximation within a limited current range.
For example, a digital output pin of an Atmega 328 microcontroller, the heart of an Arduino Uno, shows this nonlinear behavior. Figure 6.11 is the measured V-I curve and the resulting extracted Thevenin resistance as the current load to an Arduino digital I/O pin was