# Operational Transconductance Amplifier

Model behavioral representation of operational transconductance amplifier

## Library

Integrated Circuits

## Description

The Operational Transconductance Amplifier block provides a behavioral representation of an operational transconductance amplifier. A transconductance amplifier converts an input voltage into an output current. Applications include variable frequency oscillators, variable gain amplifiers and current-controlled filters. These applications exploit the fact that the transconductance gain is a function of current flowing into the control current pin.

To support faster simulation, the behavioral representation does not model the detailed transistor implementation. Therefore, the model is only valid when operating in the linear region, that is, where the device input resistance, output resistance, and transconductance gain all depend linearly on the control current, and are independent of input signal amplitude. The dynamics are approximated by a first-order lag, based on the value you specify for the block parameter Bandwidth.

### Control Current

The control current pin C is maintained at the voltage that you specify for the Minimum output voltage. In practice, the Minimum output voltage equals the negative supply voltage plus the transistor collector-emitter voltage drop. For example, if the Minimum output voltage for a supply voltage of +-15V is -14.5, then to achieve a control current of 500μA, a resistor connected between the +15V rail and the control current pin must have a value of (15 - (-14.5)) / 500e-6 = 59kOhm.

### Transconductance

The relationship between input voltage, v, and transconductance current, igm, is:

$\begin{array}{l}v={v}_{+}-{v}_{-}\\ {i}_{gm}={g}_{m}\cdot v\\ {g}_{m}=\frac{{g}_{m0}\cdot {i}_{c}}{{i}_{c0}}\end{array}$

where:

• v+ is the voltage presented at the block + pin.

• v is the voltage presented at the block - pin.

• gm is the transconductance.

• ic is the control current flowing into the control current pin C.

• ic0 is the reference control current, that is, the control current at which transconductance is quoted on the datasheet.

• gm0 is the transconductance measured at the reference control current ic0.

Therefore, increasing control current increases the transconductance.

### Output Resistance and Determining Output Current

The output resistance, Rout, is defined by:

$\begin{array}{l}{i}_{gm}+{i}_{o}=\frac{{v}_{o}}{{R}_{out}}\\ {R}_{out}=\frac{{R}_{out0}\cdot {i}_{c0}}{{i}_{c}}\end{array}$

where:

• igm is the transconductance current.

• io is the output current, defined as positive if flowing into the transconductance amplifier output pin.

• ic is the control current flowing into the control current pin C.

• ic0 is the reference control current, that is, the control current at which output resistance is quoted on the datasheet.

• Rout0 is the output resistance measured at the reference control current ic0.

Therefore, increasing control current reduces output resistance.

### Input Resistance

The relationship between input voltage, v, across the + and - pins and the current flowing, i, is:

$\begin{array}{l}\frac{v}{i}={R}_{in}\\ {R}_{in}=\frac{{R}_{in0}\cdot {i}_{c0}}{{i}_{c}}\end{array}$

where:

• ic is the control current flowing into the control current pin C.

• Rin is the input resistance for the current control current value, ic.

• ic0 is the reference control current, that is, the control current at which input resistance is quoted on the datasheet.

• Rin0 is the input resistance measured at the reference control current ic0.

Therefore, increasing control current reduces input resistance.

### Limits

Because of the physical construction of an operational transconductance amplifier based on current mirrors, the transconductance current igm cannot exceed the control current. Hence the value of igm is limited by:

icigmic

The output voltage is also limited by the supply voltage:

VminvoVmax

where Vmin is the Minimum output voltage, and Vmax is the Maximum output voltage. Output voltage limiting is implemented by adding a low resistance to the output when the voltage limit is exceeded. The value of this resistance is set by the Additional output resistance at voltage swing limits parameter.

The transconductance current is also slew-rate limited, a value for slew rate limiting typically being given on datasheets:

$-\mu \le \frac{d{i}_{gm}}{dt}\le \mu$

where μ is the Maximum current slew rate.

## Dialog Box and Parameters

### Nominal Measurements Tab

Transconductance

The transconductance, gm, when the control current is equal to the Reference control current. This is the ratio of the transconductance current, igm, to the voltage difference, v, across the + and - pins. The default value is 9600 μS.

Input resistance

The input resistance, Rin, when the control current is equal to the Reference control current. The input resistance is the ratio of the voltage difference, v, across the + and - pins to the current flowing from the + to the - pin. The default value is 25 kOhm.

Output resistance

The output resistance, Rout, when the control current is equal to the Reference control current. See above for the equation defining output resistance. The default value is 3 MOhm.

Reference control current

The control current at which the Transconductance, Input resistance, and Output resistance are quoted. The default value is 500 μA.

### Dynamics Tab

Dynamics

Select one of the following options:

• No lag — Do not model the dynamics of the relationship between output current and input voltage. This is the default.

• Finite bandwidth with slew rate limiting — Model the dynamics of the relationship between output current and input voltage using a first-order lag. If you select this option, the Bandwidth, Maximum current slew rate, and Initial current parameters appear on the Dynamics tab.

Bandwidth

The bandwidth of the first-order lag used to model the dynamics of the relationship between output current and input voltage. The default value is 2 MHz.

Maximum current slew rate

The maximum rate-of-change of transconductance current when there is no feedback around the device. Note that datasheets sometimes quote slew rate as a maximum rate of change of voltage. In this case, the value depends on the particular test circuit. To get an accurate value for Maximum current slew rate, reproduce the test circuit in a Simscape™ Electronics™ model, and tune the parameter value to match the datasheet value. If the test circuit is open-loop, and the load resistance is quoted, you can obtain an approximate value for the Maximum current slew rate by dividing the voltage slew rate by the load resistance. The default value is 2 A/μs.

Initial current

The initial transconductance current (note, not the initial output current). This is the transconductance current sinking to both the internal output resistance, Rout, and the output pin. The default value is 0 A.

### Limits Tab

Minimum output voltage

The output voltage is limited to be greater than the value of this parameter. The default value is -15 V.

Maximum output voltage

The output voltage is limited to be less than the value of this parameter. The default value is 15 V.

Additional output resistance at voltage swing limits

To limit the output voltage swing, an additional output resistance is applied between output and the power rail when the output voltage exceeds the limit. The value of this resistance should be low compared to the output resistance and circuit load resistance. The default value is 1 Ohm.

Minimum control current for simulation

The control current measured at the control current pin C is limited to be greater than the value of this parameter. This prevents a potential divide-by-zero when calculating input and output resistance values based on the value of the control current. The default value is 0.001 μA.

## Ports

The block has the following ports:

+

Positive electrical voltage

-

Negative electrical voltage

C

Control current

OUT

Output current