# Operational Transconductance Amplifier

Behavioral representation of operational transconductance amplifier

**Library:**Simscape / Electrical / 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, *i*_{gm}, 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.*g*_{m}is the transconductance.*i*_{c}is the control current flowing into the control current pin`C`

.*i*_{c0}is the reference control current, that is, the control current at which transconductance is quoted on the datasheet.*g*_{m0}is the transconductance measured at the reference control current*i*_{c0}.

Therefore, increasing control current increases the transconductance.

### Output Resistance and Determining Output Current

The output resistance, *R*_{out}, 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:

*i*_{gm}is the transconductance current.*i*_{o}is the output current, defined as positive if flowing into the transconductance amplifier output pin.*i*_{c}is the control current flowing into the control current pin`C`

.*i*_{c0}is the reference control current, that is, the control current at which output resistance is quoted on the datasheet.*R*_{out0}is the output resistance measured at the reference control current*i*_{c0}.

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:

*i*_{c}is the control current flowing into the control current pin`C`

.*R*_{in}is the input resistance for the current control current value,*i*_{c}.*i*_{c0}is the reference control current, that is, the control current at which input resistance is quoted on the datasheet.*R*_{in0}is the input resistance measured at the reference control current*i*_{c0}.

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
*i*_{gm} cannot exceed the control current.
Hence the value of *i*_{gm} is limited
by:

–i_{c} ≤
i_{gm} ≤
i_{c} | (1) |

The output voltage is also limited by the supply voltage:

V_{min} ≤
v_{o} ≤
V_{max} | (2) |

where *V*_{min} is the **Minimum
output voltage**, and *V*_{max} 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**.

## Ports

### Conserving

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2011b**