  Using the techniques of optomechanics, a high-$Q$ mechanical oscillator may
serve as a link between electromagnetic modes of vastly different frequencies.
This approach has successfully been exploited for the frequency conversion of
classical signals and has the potential of performing quantum state transfer
between superconducting circuitry and a traveling optical signal. Such
transducers are often operated in a linear regime, where the hybrid system can
be described using linear response theory based on the Heisenberg-Langevin
equations. While mathematically straightforward to solve, this approach yields
little intuition about the dynamics of the hybrid system to aid the
optimization of the transducer. As an analysis and design tool for such
electro-optomechanical transducers, we introduce an equivalent circuit
formalism, where the entire transducer is represented by an electrical circuit.
Thereby we integrate the transduction functionality of optomechanical systems
into the toolbox of electrical engineering allowing the use of its
well-established design techniques. This unifying impedance description can be
applied both for static (DC) and harmonically varying (AC) drive fields,
accommodates arbitrary linear circuits, and is not restricted to the
resolved-sideband regime. Furthermore, by establishing the quantized
input-output formalism for the equivalent circuit, we obtain the scattering
matrix for linear transducers using circuit analysis, and thereby have a
complete quantum mechanical characterization of the transducer. Hence, this
mapping of the entire transducer to the language of electrical engineering both
sheds light on how the transducer performs and can at the same time be used to
optimize its performance by aiding the design of a suitable electrical circuit.
