This post will derive the transfer function for a bridged t-network, expressed using impedances, and will be the first of a series of posts I’ll be making as I’m learning about, and working on, a design for an analog drum machine similar to that of the famous Roland machines.
The post will begin with a short background for why the bridged t-network in particular will be examined. Then, the derivation of the transfer function will be given, followed by a simulation of the circuit to validate that the derived expression is correct.
In the service manual for the tr-808 drum machine, the following principal diagram for the basic pulse-triggered sound generator can be found:
While the circuitry in the tr-808 is not as straight forward as in Fig 1, it is an excellent starting point for a study. By understanding the principal schematic, it creates the foundation for learning about the more complex and nuanced circuits in the actual hardware.
The sound generator consists of an operational amplifier (op-amp) and a feedback circuit consisting of the bridged t-network, which forms a bandpass filter with a high resonance. The filter acts as a damped oscillator (see Fig 2), and will self-oscillate at the resonance-frequency for a short time after being stimulated by an input-pulse. This is makes up the basic drum sound.
The circuit above (Fig 1) will be able to create a tom-like sound or a kick-like sound depending on the resonant frequency’s value, which is determined by the valeus of the circuit components.
II. Transfer function for the bridged t-network
The transfer function describes the relation between the components and the circuits’ behavior, and will be derived from the following diagram:
The transfer function is an expression for the ratio between the output voltage and a given input voltage :
Deriving the transfer function from the schematic becomes a process of finding an expression for the output voltage as as a function of the input voltage, and factoring that expression.
The output voltage node and input voltage node are separated by the impedance . The current via causes a voltage drop, so that the output voltage can be expressed by:
can be expressed by introducing a common-voltage at the junction between , and :
To express the common-voltage we notice that , and form an equivalent resistor that together with form a voltage divider with as its output:
where the total impedance is:
The current can now be rewritten as:
and then factored:
By calculating the equivalent impedance and then , the expression for the current can be made explicit:
The impedance-quotient then becomes:
The current can now be expressed with the known impedances as:
Finally, we can return to the output voltage expression, and factor out :
and then calculate the difference inside the parentheses by expanding the leading 1-term:
With the original expression fully factorized, the transfer function can now be expressed:
And the parenthesises expanded:
The transfer function for the schematic has now been derived.
III. Validating the transfer function with a simulation
In order to validate that the transfer function is correct, it will be tested using resistive values and compared to the output from the simulation software PSpice.
A simple straight forward test is to simply define all the impedances to 1kΩ:
With the input voltage set to 1V, the output voltage is 0.8V as expected.
In order to make sure this isn’t just a fluke result, another set of test-values can be defined:
For this second test the result is also the expected one, with an input voltage of 1 giving 937.5 mV output.
A valid transfer function for the bridged t-network has been derived for impedances. In order to analyze its oscillation properties and bode diagram, those impedances need to be defined using resistors and capacitors .
This will be explored in the next part.