This post explains how to calculate resistor and capacitor values in transformerless power supply circuits using simple formulas like ohms law.

## Analyzing a Capactive Power Supply

Before we learn the formula for calculating and optimizing resistor and capacitor values in a transformerless power supply, it would be important to first summarize a standard transformerless power supply design.

Referring to the diagram, the various components involved are assigned with the following specific functions:

C1 is the nonopolar high voltage capacitor which is introduced for dropping the lethal mains current to the desired limits as per the load specification. This component thus becomes extremely crucial due to the assigned mains current limiting function.

D1 to D4 are configured as a bridge rectifier network for rectifying the stepped down AC from C1, in order to make the output suitable to any intended DC load.

Z1 is positioned for stabilizing the output to the required safe voltage limits.

C2 is installed to filter out any ripple in the DC and to create a perfectly clean DC for the connected load.

R2 may be optional but is recommended for tackling a switch ON surge from mains, although preferably this component must be replaced with a NTC thermistor.

## Using Ohm's Law

We all know how Ohm’s law works and how to use it for finding the unknown parameter when the other two are known. However, with a capacitive type of power supply having peculiar features and with LEDs connected to it, calculating current, voltage drop and LED resistor becomes a bit confusing.

How to Calculate and Deduce Current, Voltage Parameters in Transformerless Power Supplies.

After carefully studying the relevant patterns, I devised a simple and effective way of solving the above issues, especially when the power supply used is a transformerless one or incorporates PPC capacitors or reactance for controlling current.

## Evaluating Current in Capacitive Power Supplies

Typically, a transformerless power supply will produce an output with very low current values but with voltages equal to the applied AC mains (until it’s loaded).

For example, a 1 µF, 400 V (breakdown voltage) when connected to a 220 V x 1.4 = 308V (after bridge) mains supply will produce a maximum of 70 mA of current and an initial voltage reading of 308 Volts.

However this voltage will show a very linear drop as the output gets loaded and current is drawn from the “70 mA” reservoir.

We know that if the load consumes the whole 70 mA would mean the voltage dropping to almost zero.

Now since this drop is linear, we can simply divide the initial output voltage with the max current to find the voltage drops that would occur for different magnitudes of load currents.

Therefore dividing 308 volts by 70 mA gives 4.4V. This is the rate at which the voltage will drop for every 1 mA of current added with the load.

That implies if the load consumes 20 mA of current, the drop in voltage will be 20 × 4.4 = 88 volts, so the output now will show a voltage of 308 – 62.8 = 220 volts DC(after bridge).

For example with a 1 watt LED connected directly to this circuit without a resistor would show a voltage equal to forward voltage drop of the LED (3.3V), this is because the LED is sinking almost all the current available from the capacitor. However the voltage across the LED is not dropping to zero because the forward voltage is maximum specified voltage that can drop across it.

From the above discussion and analysis, it becomes clear that voltage in any power supply unit is immaterial if the current delivering capability of the power supply is "relatively" low.

For example if we consider an LED, it can withstand 30 to 40 mA current at voltages close to its "forward voltage drop", however at higher voltages this current can become dangerous for the LED, so it's all about keeping the maximum current equal to the maximum safe tolerable limit of the load.

## Calculating Resistor Values

**Resistor for the Load**: When an LED is used as the load, it is recommended to choose a capacitor whose reactance value allows only the maximum tolerable current to the LED, in which case a resistor can be totally avoided.

If the capacitor value is large with higher current outputs, then probably as discussed above we can incorporate a resistor to reduce the current to tolerable limits.

**Calculating Surge Limit Resistor**: The resistor R2 in the above diagram forms is included as the switch-ON surge limiter resistor. It basically protects the vulnerable load from the initial surge current.

During the initial switch ON periods, the capacitor C1 acts like a complete short circuit, although just for a few milliseconds, and may allow the whole 220V across the output.

This may be enough to blow the sensitive electronic circuits or LEDs connected with the supply, which also includes the stabilizing zener diode.

Since the zener diode forms the first electronic device in line which needs to be safeguarded from the initial surge, R2 can be calculated as per the zener diode specifications, and maximum zener current, or zener dissipation.

The maximum tolerable current by the zener for our example will be 1 watt / 12 V = 0.083 amps.

Therefore R2 should be = 12 / 0.083 = 144 Ohms

However, since the surge current is only for a milliseconds, this value could be much lower than this.

Here. we are not considering the 310V input for the zener calculation, since the current is limited to 70 mA by the C1.

Since R2 can unnecessarily restrict precious current for the load during the normal operations, it must be ideally an NTC type of resistor. An NTC will make sure that the current is restricted only during the initial switch ON period, and then the full 70 mA is allowed to pass unrestricted for the load.

**Calculating the Discharge Resistor**: Resistor R1 is used for discharging the stored high voltage charge inside C1, whenever the circuit is unplugged from the mains.

R1 value should be as low a possible for fast discharging of C1, yet dissipate minimum heat while being connected with the mains AC.

Since R1 can be a 1/4 watt resistor, its dissipation must be lower than 0.25 / 310 = 0.0008 amps or 0.8 mA.

Therefore R1 = 310 / 0.0008 = 387500 Ohms or 390 k approximately.

### Calculating a 20 mA LED Resistor

Example: In the shown diagram, the value of the capacitor produces 70 mA of max. current which is quite high for any LED to withstand. Using the standard LED/resistor formula:

R = (supply voltage VS – LED forward voltage VF) / LED current IL,

= (220 - 3.3)/0.02 = 10.83K,

However the 10.83K value looks pretty huge, and would substantially drop the illumination on the LED....none-the-less the calculations look absolutely legitimate....so are we missing something here??

I think here the voltage "220" might not be correct because ultimately the LED would be requiring just 3.3V....so why not apply this value in the above formula and check the results? In case you have used a zener diode, then the zener value could be applied here instead.

Ok, here we go again.

R = 3.3/0.02 = 165 ohms

Now this looks much better.

In case you used, let's say a 12V zener diode before the LED, the formula could be calculated as given below:

R = (supply voltage VS – LED forward voltage VF) / LED current IL,

= (12 - 3.3)/0.02 = 435 Ohms,

Therefore the value of the resistor for controlling one red LED safely would be around 400 ohm.

## Finding Capacitor Current

In the entire transformerless design discussed above, C1 is the one crucial component which must be dimensioned correctly so that the current output from it is optimized optimally as per the load specification.

Selecting a high value capacitor for a relatively smaller load may increase the risk of excessive surge current entering the load and damaging it sooner.

A properly calculated capacitor on the contrary ensures a controlled surge inrush and nominal dissipation maintaining adequate safety for the connected load.

**Using Ohm's Law**

The magnitude of current that may be optimally permissible through a transformerless power supply for a particular load may be calculated by using Ohm's law:

I = V/R

where I = current, V = Voltage, R = Resistance

However as we can see, in the above formula R is an odd parameter since we are dealing with a capacitor as the current limiting member.

In order to crack this we need to derive a method which will translate the capacitor's current limiting value in terms of Ohms or resistance unit, so that the Ohm's law formula could be solved.

**Calculating Capacitor Reactance**

To do this we first find out the reactance of the capacitor which may be considered as the resistance equivalent of a resistor.

The formula for reactance is:

**Xc = 1/2(pi) fC**

where Xc = reactance,

pi = 22/7

f = frequency

C = capacitor value in Farads

The result obtained from the above formula is in Ohms which can be directly substituted in our previously mentioned Ohm's law.

Let's solve an example for understanding the implementation of the above formulas:

Let's see how much current a 1uF capacitor can deliver to a particular load:

We have the following data in our hand:

pi = 22/7 = 3.14

f = 50 Hz (mains AC frequency)

and C= 1uF or 0.000001F

Solving the reactance equation using the above data gives:

Xc = 1 / (2 x 3.14 x 50 x 0.000001)

= 3184 ohms approximately

Substituting this equivalent resistance value in our Ohm's law formula, we get:

R = V/I

or I = V/R

Assuming V = 220V (since the capacitor is intended to work with the mains voltage.)

We get:

I = 220/3184

= 0.069 amps or 69 mA approximately

Similarly other capacitors can be calculated for knowing their maximum current delivering capacity or rating.

The above discussion comprehensively explains how a capacitor current may be calculated in any relevant circuit, particularly in transformerless capacitive power supplies.

WARNING: THE ABOVE DESIGN IS NOT ISOLATED FROM MAINS INPUT, THEREFORE THE WHOLE UNIT COULD BE FLOATING WITH LETHAL INPUT MAINS, BE EXTREMELY CAREFUL WHILE HANDLING IN SWITCHED ON POSITION.

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