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---
title: RC Circuit Model
order: 1
---

# RC Circuit

The Resistor-Capacitor circuit represents a linear first order differential equation. Mathematically:

```
i_c = C * dV_c/dt

i_r = V_r / R
```

Where

* `i_c` and `i_r` are the currents flowing through the capacitor and resistor,
* `C` is the capacitance,
* `R` is the resistance,
* `V_c` and `'V_r` are the voltages across the capacitor and resistor,
* `t` is time.

To make the model more realistic, a parasitic capacitance has been added in parallel to the resistor. This simulates electrical effects of conductors close by such that their electrical fields interact and affect current passing through them.

![Schematic](img/1_RC_schematic.png)

## Models

For this model, the inputs are the driving voltage. The measurements to be simulated/learned are the current through the circuit. Known parameters are capacitance and resistance. Parasitic effects are unaccounted for and must be learned from the data.

This is a first-order model. Therefore one parameter (Capacitance) is kept fixed while the other is free to be learned from the data.

Model parameters are learned from input/output measurements. In this case, the input is a ramped sine wave.

![Input Signal](img/1_input_signal.png)

### Grey box model

Using [Kirchoff's voltage law][2] and the above equations, the total voltage across the circuit can be written as:

```
V_s = 1/C ∫ i dt + i R

i = [- 1 / (R C)] ∫ i dt + [1 / R] V_s
```

Using the [modern control approach][1] and taking as input (`u`), state (`x`), and output (`y`) variables:

```
x = ∫ i dt, x' = i, u = V_s
y = x
```

We get the state-space equation:

```
x' = [- 1 / (R C)] x + [1 / R] u
y = [- 1 / (R C)] x + [1 / R] u
```

Which yeilds the state-space matrices:

```
A = C = [- 1 / (R C)]
B = D = [1 / R]
```

We set `R` as the learnable parameter under various conditions (noisy inputs, with/without parasitic effects). The state-space model tries to fit observed measurements with its estimate of the parameter value. The initial estimate is `R = 500`.

| SNR\Parasitic 	| Yes    	| No    	|
|---------------	|--------	|-------	|
| 25            	| 0.0034 	| 99.99 	|
| 250           	| 93.77  	| 99.99 	|

*Table: `SNR` is the signal-to-noise ratio in dB. Parasitic effects are simulated by a 1uF capacitance in parallel to the resistor. Values are percentage of fit with the training data.*

The results show that the model is highly susceptible to noise. The mathematical structure of the model does not account for a parasitic element, therefore the peak accuracy it is able to achieve theoretically tops out before reaching a 100%. However, the limited model still arrives at the correct estimate for `R`. The worst case for this grey-box model is under noisy training data and unaccounted-for electrical effects which reduce the model performance to 0.

[1]: https://en.wikibooks.org/wiki/Control_Systems#Modern_Control_Methods
[2]: https://en.wikipedia.org/wiki/Kirchhoff%27s_circuit_laws#Kirchhoff's_voltage_law_(KVL)