Model behavioral representation of operational transconductance amplifier

Integrated Circuits

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**.

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.

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.

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.

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.

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}

The output voltage is also limited by the supply voltage:

*V*_{min} ≤ *v*_{o} ≤ *V*_{max}

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**.

**Transconductance**The transconductance,

*g*_{m}, when the control current is equal to the**Reference control current**. This is the ratio of the transconductance current,*i*_{gm}, to the voltage difference,*v*, across the`+`

and`-`

pins. The default value is`9600`

μS.**Input resistance**The input resistance,

*R*_{in}, 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,

*R*_{out}, 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**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,

*R*_{out}, and the output pin. The default value is`0`

A.

**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.

The block has the following ports:

`+`

Positive electrical voltage

`-`

Negative electrical voltage

`C`

Control current

`OUT`

Output current

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