Understanding Impedance Matching

In electrical and electronic engineering, there is a need to match the input resistance characteristic with that of the output. Achieving equilibrium of input and output resistance from the source to the load gives a stable system. The process of balancing the input and output resistances of a given electrical system is what we refer to as impedance matching.

 

Defining Impedance Matching

Impedance is defined as the total resistance of a given electric component or circuit to an alternating current originating from the reactance and resistance of the given system. Unlike resistance, which has magnitude only, impedance has both phase and magnitude in alternating current circuits. For direct current, there is no difference between resistance and impedance as the impedance has zero phase angle, just like resistance has no phase angle.

We can define impedance matching as the process where the input impedance and the output impedance of a given electrical load are designed to reduce signal reflection and maximize the power transferred to the electric load. For a better understanding of impedance matching, let’s look at the maximum power transfer theorem.

 

Maximum Power Transfer Theorem

 

Figure 1. Maximum Power Transfer Theorem Circuit. Image used courtesy of Simon Mugo

 

Suppose we have a system with a voltage source V of internal resistance R_i and powering an electric load of resistance R_L. The maximum power theorem will be used to determine the value of the load resistance R_{L, allowing} maximum power to be transferred from the source to the load. The maximum power transferred to the load depends on the size of the load resistance.

From the circuit above, power transferred to the load resistance is

\[P=I^{2}_{L}R_{L}=\frac{V^{2}R_{L}}{(R_{i}+R_{L})^{2}}\]

Figure 2. Equation of Power Transferred to the Load.

 

For maximum power, we differentiate the above equation to the load resistance R_L and equate the outcome to zero. We shall have

\[\frac{dP}{dR_{L}}=\frac{V^{2}(R_{i}+R_{L})^{2}-2R_{L}(R_{i}+R_{L})V^{2}}{(R_{i}+R_{L})^{4}}=0\Rightarrow R_{L}=R_{i}\]

Figure 3. Differentiated Equation and Equated to Zero.

 

Note that maximum power can only be transferred from the source to the load when the internal resistance of the voltage source is equal to the resistance of the load. Impedance matching ensures that the source resistance is equal to the load resistance. Another thing to note is that the load reactance should also be equal to the negative of the source reactance for maximum power to be reflected at the electric load side. This means the load power can only be at maximum when the load impedance is equal to the source impedance complex conjugate.

 

Impedance Matching Formulas and Circuits

 

Figure 4. Impedance Matching Circuit. Image used courtesy of Simon Mugo

 

For an available resistance R, we shall find the circuit that will match the resistance R’ at a certain frequency w₀ and develop a design of the L matching circuit displayed in Figure 4 above.

Let us start by finding the admittance Y_in of our circuit above.

From the figure, we can note that the resistance R and the inductor L are in series, and the combination of the two is in parallel with capacitor C. 

The impedance will be given by

\[Z=(R+j\omega L)//\frac{1}{j\omega C}\]

\[Z=\frac{[(R+j\omega L)\times\frac{1}{j\omega C}]}{(R+j\omega L)+\frac{1}{j\omega C}}\]

\[Z=\frac{R+j\omega L}{(R+j\omega L)(j\omega C)+1}\]

\[Y_{in}=\frac{1}{Z}=\frac{[(R+j\omega L)(j\omega C)+1]}{(R+j\omega L)}\]

\[Y_{in}=j\omega C+\frac{1}{(R+j\omega L)}\]

Let us use complex conjugates to separate the real and imaginary parts of the above equation.

\[Y_{in}=j\omega C+\frac{1}{(R+j\omega L)}\times\frac{(R-j\omega L)}{(R-j\omega L)}\]

\[Y_{in} = j\omega C+\frac{R}{(R^{2}+(\omega L)^{2})}-\frac{j\omega L}{(R^{2}(\omega L)^{2})}\]

Reorganizing the above equation, we get

\[Y_{in}=\frac{R}{(R^{2}+(\omega L)^{2})}+j[\omega C-\frac{\omega L}{(R^{2}+(\omega L)^{2})}]\]

Finally, we have

\(Y=\frac{R}{R^{2}+(\omega L)^{2}}\)                     (1)

And

\(\omega_{0}=\sqrt{\frac{1}{LC}}-\left(\frac{R}{L}\right)^{2}\)             (2)

At the frequency of ω= 0, the resistance of Yin should be set to R’.

\[R^{’}=\frac{1}{Y}=\frac{R^{2}+(\omega_{0}L)^{2}}{R}\]

\[R^{’}=R+\frac{\omega_{0}{^{2}}L^{2}}{R}\]

Separate R from the equation to get

\[R^{’}=R[1+\left(\frac{\omega_{0}L}{R}\right)^{2}]\]

Let \(\frac{\omega_{0}L}{R}\) be the Q-factor for L and R networks and our equation becomes

\(R^{’}=R[1+Q^{2}]\)                 (3)

From the above equations, it is easy to solve the impedance matching problems in any electrical circuit.

 

Why Impedance Matching Is Needed

Impedance matching has great use in high-frequency and high-speed devices. When designing printed circuit boards that require such characteristics, ensure that the impedance at the source matches the impedance at the load.

When designing applications of ultra-high frequencies, impedance matching becomes a difficult operation for designers. The challenge is also reflected while designing microwaves and radio frequency circuits. When you get a wrong impedance matching, expect distorted pulses and high signal reflections.

An increase in frequency decreases the window of errors. The electrical circuit works the best when we have a perfectly matched impedance. If the impedance matching is not done, expect the system to work abnormally because of the effects such as the signal reflections. The reflected waves cause data delays and distortion of the phase and minimize the ratio of signal to noise.

 

Application of Impedance Matching

The goal of any electrical and electronic design engineer is to ensure that there is maximum power delivery from the system source to the electric load. In almost all electrical and electronic applications, impedance matching is a necessity.

Let us have a look at a few applications of impedance matching below.

 

Transformer Impedance Matching

Transformers are one of the components used to match the impedance of the source to load. The power input of the transformer is similar to the power output by it. The transformer changes the electrical energy c\voltage level and does not affect the power level of the system.

To match the impedance of both sides, you have to set the turn ratio accordingly. Low voltage has fewer turns, and this tells us that the impedance of the low voltage winding is lower when compared to that of the high voltage winding.

Therefore, to ensure that we have matched our impedance, a transformer with proper winding turns is connected between the source and the load. The transformer is known as a matching transformer. 

We can define the matching transformer turn ratio as the square root of the ratio of the resistance of the source to the resistance of the load.

\[Turns\,Ratio=\sqrt{\frac{Source\,Resistance}{Load\,Resistance}}\]

Figure 5. Transformer Turn Ratio Calculation Formula. 

 

Impedance Matching Transmission Line

Transmission of electrical energy from the source to the load is done using a transmission line. While transferring this energy, it is important to zero or minimize energy losses that occur. For this to be possible, we should match the source and load impedances to the transmission line being used.

The characteristic impedance is defined as the voltage and current wave ratio at any given point along the transmission line. If the transmission line in discussion is long, then we expect to have a different characteristic impedance at different distances along this transmission line. If we fail to do the impedance matching, the signs reaching the load will be reflected in the source of the origin, giving rise to a standing wave. The amount of power reflected is measured using the coefficient of reflection, which is calculated using the equation below:

\[\Gamma=\frac{Z_{L}-Z_{0}}{Z_{L}+Z_{0}}\]

Figure 6. Transmission Line Reflection Coefficient Calculating Formula

 

With Z_L being Line Impedance and Z₀ Characteristic Impedance.

An ideal system has a load impedance similar to the characteristic impedance. The system is said to be ideal when the reflection coefficient is zero. In practice, it is difficult to achieve a zero-reflection coefficient; hence, in the transmission line, the reflective coefficient is kept close to zero.

 

Antenna Impedance Matching

The antenna needs to be coupled with a television. For this to be achieved, impedance matching is employed. Here, the antenna becomes our source as it has to give signals to the television, whereas the TV becomes the load because it gets signals from the antenna.

Let us take an example where the antenna and its cable have a resistance of 150 ohms, and the resistance of the TV is 600 ohms. These conditions indicate a different impedance between the source and the load, and hence it becomes impossible to have maximum power transmitted; therefore, poor signals are received by the TV.

 

Figure 7. Antenna Impedance Matching. Image used courtesy of Simon Mugo

 

To avoid the above condition, we have to use a transformer to achieve the antenna and television impedance matching. We had already introduced the formula for the turn ratio of the transformer calculation in this article, and will input the values accordingly, as shown below:

\[n=\sqrt{\frac{R_{L}}{R_{in}}}\]

\[n=\sqrt{\frac{600\Omega}{150\Omega}}\]

\[n=2\]

From the calculations above, our turn ratio is 1:2, and we have to connect our transformer as in the circuit below:

 

Figure 8. Impedance Matching of Antenna with Transformer. Image used courtesy of Simon Mugo

 

Headphone Impedance Matching

In the case of the headphone, the signal source is the device where the headphone is plugged. The headphone is the load. For the system to attain quality audio output, the source, and the load impedances must be matched. By matching the impedances, we make sure that there is maximum power transfer from the source of the audio to the headphone.

When building portable devices, ensure that low-impedance headphones are built. This makes the system work well with proper sound quality.

 

Key Takeaways of Impedance Matching

• Impedance matching is the balance between the source and load impedance to ensure the load receives maximized power.
• Maximum power theorem is important in transferring energy from the source to the load.
• Impedance matching is needed to allow maximum power transfer from the source to the load by reducing reflections.
• Impedance matching has several applications, such as matching the impedance of transformers, headphones, transmission lines, and antennas.

 

Featured image used courtesy of Adobe Stock