# How to Calculate the Operating Windows for a Remote Antenna Current-Sense Amplifier and Switch

Abstract: This application note helps designers choose the correct external components to ensure that automobile antenna-detection circuitry meets performance objectives. A calculator details how to specify the critical external components for the MAX16913/MAX16913A remote antenna current-sense amplifiers and switches. The calculator also determines the device's operational windows and analog output voltage accuracy. An example calculation is given.

## Introduction

The MAX16913/MAX16913A (**Figure 1**) are precision current-sense amplifiers (CSAs) and switches that provide phantom power to remote radio antennas in automotive applications. In addition, they provide short-circuit protection, current-limit protection, and open-load detection. To ensure that their antenna detection circuitry meets performance objectives, the design engineer must choose the correct external components for a design.

When working with CSAs and switches for antenna applications, the designer must often determine the operational windows for an open load, normal operation, a short circuit, and current limiting (

**Figure 2**). In addition, the accuracy of the CSA's analog output voltage must also be verified.

*Figure 1. Typical operating circuit of the MAX16913A remote antenna CSA and switch.*

*Figure 2. Operation ranges for these CSAs.*

This application note presents a calculator that shows how to determine the proper sense resistor and the resistor-divider for setting the open-load threshold (detection range) tolerance. It considers the tolerances of both the external components and the MAX16913/MAX16913A, and then calculates the appropriate tolerance window ranges for optimal performance. The calculator is available here, and an example calculation follows.

## Calculate the Required Sense Resistor

Ideally the maximum operating current develops the full-scale sense voltage across the current-sense resistor, R_{SENS}(Figure 1). Calculate the maximum value for R

_{SENS}so that the differential voltage across IN and SENS does not exceed the minimum full-scale sense voltage (87mV)*:

Where V

_{DIFF(MIN)}= V

_{IN}- V

_{SENSE}= 87mV (min) at the maximum guaranteed output current, I

_{LOAD(FULL-SCALE)}(Figure 2).

However, resistors always have tolerances, so the actual resistor value can be higher by its tolerance rating, thus causing the device to detect a short circuit too early. After considering the resistor's tolerance rating, the nominal maximum resistor value can be calculated:

Where R

_{SENS(MAX)}is the maximum sense resistor calculated above, and R

_{SENS-TOLERANCE}is the tolerance rating of the resistor. Remember that exact values for the calculated sense resistor may not be available. If that is the case, choose the closest smaller value for R

_{SENS(MAX)(NOM)}and use that to calculate R

_{SENS_P(NOM)}. Alternatively serial or parallel combinations of standard resistors can be used to attain the optimal sense resistor.

## Calculate the Short-Circuit Current-Detection Window

The nominal sense resistor has been chosen. Now the typical current through the sense resistor, when a short circuit is detected, can be calculated as follows:Where V

_{SC(TYP)}is the typical value of the short-circuit voltage threshold (100mV)* and R

_{SENS_P(NOM)}is the sense resistor selected above.

However, as V

_{SC}and R

_{SENS}have uncorrelated tolerances (i.e., have minimum and maximum values that vary independently of each other), an additional error has to be considered. So the worst-case short-circuit, current-detection window is:

And

Where V

_{SC(MIN)}is the minimum value of the short-circuit voltage threshold (87mV)* and V

_{SC(MAX)}is the maximum value (110mV).* Therefore:

R_{SENS_P(MAX)} |
= R_{SENS_P(NOM)} + its tolerance rating + R_{SENS_P(MIN)} |

= R_{SENS_P(NOM)} - its tolerance rating |

The short-circuit flag (active-low SC) will thus go low when the current is in the range between I

_{SC(MIN)}and I

_{SC(MAX)}.

## Calculate the Current-Limit Range

Analogous to the short-circuit current-detection window, the current-limit range is typically:Where V

_{LIM(TYP)}is the typical value of the current-limit threshold voltage between IN and SENS (200mV),* and R

_{SENS_P(NOM)}is the sense resistor selected.

Considering that V

_{LIM}and R

_{SENS}have uncorrelated tolerances, the worst-case current-limit range through the sense resistor can be calculated:

And

Where V

_{LIM(MIN)}is the minimum value of the voltage between IN and SENS (173mV),* and V

_{LIM(MAX)}is the maximum value (225mV).*

## Calculate the Open-Load Detection Window

This procedure differs for the MAX16913 and MAX16913A.### For the MAX16913

The open-load detection threshold (active-low OL) for the MAX16913 is set internally to V_{OLT}= 0.66V.* The associated current range using a 1Ω resistor is specified in the data sheet as 10mA (min), 20mA (typ), and 30mA (max). These values include the tolerance of the open-load comparator, the gain amplifier, and the external sense resistor (1Ω).

To determine the open-load detection window using a different sense resistor, first the given current levels must be converted to a voltage:

Then using the values calculated above, the typical value of the open-load current detection threshold calculates to:

Where V

_{OLT(TYP)}is the typical value of the open-load detection-threshold voltage calculated above, and R

_{SENS_P(NOM)}is the sense resistor selected.

Considering also the tolerances of the open-load current threshold and the tolerance of the sense resistor, then the current range for open-load detection calculates to:

And

Where V

_{OLT(MIN)}and V

_{OLT(MAX)}are the minimum and maximum values of the open-load detection-threshold voltage; R

_{SENS_P(MIN)}and R

_{SENS_P(MAX)}are the minimum and maximum values of the sense resistor calculated above.

The worst-case open-load detection window lies between I

_{OL(MIN)}and I

_{OL(MAX)}.

### For the MAX16913A

The open-load threshold for the MAX16913A can be adjusted externally with a resistor-divider between REF, OLT, and GND. Therefore, the first task is to specify the external resistor-divider.**Specify the External Resistor-Divider**

To begin, choose the voltage needed on the OLT pin to set the desired nominal OL threshold (at the OLT pin, Figure 1) using the following formula:

VWhere AV is the (V_{OLT}(V) = I_{OLT}(A) × R_{SENS}(Ω) × A_{V}(V/V) + 0.133 × V_{REF}

_{IN}- V

_{SENS}) to V

_{AOUT}gain (13V/V) and V

_{REF}is the REF pin voltage (3V).* The ratio of the external resistors on the OLT pin can then be calculated using the following equation:

RWhere V_{2}/R_{1}= V_{OLT}/(V_{REF}× (1 - V_{OLT}/V_{REF}))

_{REF}is the voltage on the REF pin (3V). An arbitrary standard value can now be chosen for R

_{1}or R

_{2}, and the other resistor value can then be calculated (Figure 1). However, ensure that the impedance of the resistor-divider does not load the internal reference voltage excessively.

**Determine the Open-Load Threshold-Voltage Range**

The standard resistor values for R1 and R2 have now been defined. Next, considering the uncorrelated tolerances of V

_{REF}and the resistors R1 and R2, the worst-case voltage range for the open-load pin, V

_{OLTw}, can be calculated:

And

Where R

_{2(MIN)}is the nominal value of R

_{2}minus its tolerance value. This can be restated as R

_{2(MIN)}= R

_{2}- (R

_{2}× (R

_{2TOL}[%]/100%)) and R

_{1(MAX)}is the nominal value of R

_{1}plus the tolerance.

**Determine the Worst-Case, Open-Load Current-Detection Window**

At this point V

_{OLTw(MIN)}and V

_{OLTw(MAX)}have been calculated. Now taking into consideration the tolerances of the REF output voltage, V

_{REF}, the sense resistor, R

_{SENS_P(NOM)}, and the gain, AV, the worst-case current window when open load is detected (active-low OL) can be calculated:

And

Where V

_{OLTw(MIN)}, V

_{OLTw(MAX)}, R

_{SENS_P(MIN)}, and R

_{SENS_P(MAX)}have been calculated above; A

_{V}is the (V

_{IN}- V

_{SENS}) to V

_{AOUT}gain which has minimum and maximum values of 12.87 and 13.13, respectively*; and V

_{REF(MIN)}and V

_{REF(MAX)}are the minimum and maximum values of the REF pin voltage (2.7V and 3.3V).*

## Evaluate the Current Through R_{SENS} by Measuring Voltage on A_{OUT}

### A_{OUT} Accuracy

With a given sense resistor, R_{SENS}, and a defined current through it, I

_{SENS}, then the worst-case range of voltage values measured at the current-sense amplifier's output, A

_{OUT}(e.g., a microcontroller's analog-to-digital converter (ADC)), can now be calculated. Consider also the uncorrelated tolerances of A

_{OUT_Z}and the sense resistor, R

_{SENS}. Therefore:

VAnd_{AOUT(MIN)}(V) = A_{OUT_Z(MIN)}(V) + A_{V(MIN)}(V/V) × R_{SENS(MIN)}(Ω) × I_{SENS}(A)

VWhere A_{AOUT(MAX)}(V) = A_{OUT_Z(MAX)}(V) + A_{V(MAX)}(V/V) × R_{SENS(MAX)}(Ω) × I_{SENS}(A)

_{V(MIN)}is 12.87V and A

_{V(MAX)}is 13.13V;* and A

_{OUT_Z(MIN)}and A

_{OUT_Z(MAX)}are the minimum and maximum values of the A

_{OUT}zero-current output voltage (340mV)* (460mV);* and R

_{SENS(MIN)}and R

_{SENS(MAX)}are the sense resistor plus/minus its tolerance.

Stated in other words, the sensed current produces a worst-case A

_{OUT}voltage variation between V

_{AOUT(MIN)}and V

_{AOUT(MAX)}.

Taking the above worst-case voltage levels and using a microcontroller's software to calculate those voltages back to a current, one can calculate:

And

Where V

_{AOUT(MIN)}and V

_{AOUT(MAX)}have been calculated above; AV is the (V

_{IN}- V

_{SENS}) to V

_{AOUT}gain (13V/V);* A

_{OUT_Z(TYP)}is the typical value of the A

_{OUT}zero-current output voltage (400mV);* and R

_{SENS}is the nominal value of the sense resistor.

Thus when the analog output voltage is used to measure a certain current through the sense resistor, the microcontroller's ADC gives a current value between I

_{EVALUATED(MIN)}and I

_{EVALUATED(MAX)}.

The current measurement tolerance, I

_{TOL}, is:

### Example Calculations

For these example calculations we assume an antenna phantom supply application where the upper end of the normal operation window (I_{LOAD(FULL-SCALE)}) is at 100mA. Then the maximum value of the sense resistor required is:

When using a resistor with a 1% tolerance, the maximum sense resistor that can be selected is:

As a 0.861Ω resistor is not available as a standard value, we select the next smaller value from the E96 series for R

_{SENS-P(NOM)}= 0.845Ω. We use this value for our subsequent calculations.

Next, the typical current value for short-circuit detection can be calculated:

As previously shown, the minimum and maximum values for the short-circuit current-detection window lie between I

_{SC(MIN)}and I

_{SC(MAX)}. To calculate these values, we first need the minimum and maximum values of the selected sense resistor.

This allows us to derive the limits of the short-circuit current-detection window:

And

Analogous to the short-circuit current-detection window, the typical value of the current-limit range is:

Considering the tolerances, the minimum and maximum values for the current-limit range lie between I

_{LIM(MIN)}and I

_{LIM(MAX)}:

And

Now for the MAX16913, the typical value for the open-load detection threshold is:

Including the tolerances, the minimum and maximum values are:

And

Turning now to the MAX16913A, we assume an application where the maximum current value of the open-load detection window is at 30mA. Therefore, the maximum voltage value for the center point of the resistor-divider is:

Next we pick a standard resistor for R2 from the E96 series, 90.9kΩ (1%), and calculate its maximum value:

The minimum resistor value for the upper resistor of the divider is then:

The nominal value, assuming also a 1% tolerance, is:

The closest higher standard value to be selected with the same tolerance is R1 = 392kΩ. Considering also its tolerance, we calculate:

And the minimum value for R2 is:

Continuing with these values, the open-load threshold-voltage range is:

And

Then the worst-case current window for the open-load detection of the MAX16913A is:

And

To evaluate the analog output, A

_{OUT}, accuracy, we assume the same sense resistor selected above (0.845Ω) and evaluate the accuracy at a load current of 100mA. At this current, the minimum and maximum values of the A

_{OUT}voltage are between:

VAnd_{AOUT(MIN)}(V) = A_{OUT_Z(MIN)}(V) + A_{V(MIN)}(V/V) × R_{SENS(MIN)}(Ω) × I_{SENS}(A) = 340mV + 12.87(V/V) × 0.837Ω × 100mA = 1.417V

VTaking these voltages and calculating back as the microcontroller's software would do (i.e., taking the typical values from the data sheet), we derive an evaluated current between:_{AOUT(MAX)}(V) = A_{OUT_Z(MAX)}(V) + A_{V(MAX)}(V/V) × R_{SENS(MAX)}(Ω) × I_{SENS}(A) = 460mV + 13.13(V/V) × 0.853Ω × 100mA = 1.58V

And

The worst-case tolerance of the measured current can then be up to:

*For more details on these calculations, see the data sheet for the MAX16913/MAX16913A.