Convert Smith Chart to L and C

Converting Smith Chart Values into Physical Components

The Smith Chart provides normalized, unitless values of impedance and admittance. These normalized quantities are extremely useful for graphical impedance matching, but they cannot be directly used to select a physical inductor or capacitor. To implement the matching network in an actual RF or microwave circuit, the normalized reactance or susceptance obtained from the Smith Chart must be converted into its corresponding physical value. This process is known as denormalization.

Normalized Impedance and Admittance

In impedance form, a normalized impedance obtained from the Smith Chart is written as:

\[ \boxed{ z=r+jx } \]

where \(r\) is the normalized resistance and \(x\) is the normalized reactance. The normalized impedance is related to the actual impedance by:

\[ \boxed{ z=\frac{Z}{Z_0} } \]

Therefore, the actual impedance is obtained by multiplying the normalized impedance by the characteristic impedance \(Z_0\):

\[ \boxed{ Z=zZ_0 } \]

Substituting \(z=r+jx\):

\[ \boxed{ R+jX=(r+jx)Z_0 } \]

Therefore, the actual resistance and reactance are:

\[ \boxed{ R=rZ_0 } \]

\[ \boxed{ X=xZ_0 } \]

The reactance \(X\) is particularly important when selecting a series inductor or series capacitor.

Normalized Admittance and Denormalization

When dealing with shunt or parallel matching elements, it is more convenient to work with admittance. The normalized admittance is written as:

\[ \boxed{ y=g+jb } \]

where \(g\) is the normalized conductance and \(b\) is the normalized susceptance.

The normalized admittance is defined as:

\[ y=\frac{Y}{Y_0} \]

where \(Y_0\) is the characteristic admittance:

\[ \boxed{ Y_0=\frac{1}{Z_0} } \]

Therefore:

\[ Y=yY_0 \]

Substituting \(y=g+jb\):

\[ \boxed{ G+jB = (g+jb)\frac{1}{Z_0} } \]

Hence, the actual conductance and susceptance are:

\[ \boxed{ G=\frac{g}{Z_0} } \]

and:

\[ \boxed{ B=\frac{b}{Z_0} } \]

The susceptance \(B\) is used to determine the value of a shunt capacitor or shunt inductor.

Direct Conversion from Smith Chart Values

The two most important denormalization relationships are obtained directly from the definitions of normalized impedance and normalized admittance.

Series Components: Normalized Reactance to Actual Reactance

For a series component, the Smith Chart provides a normalized reactance \(x\). The actual reactance in ohms is:

\[ \boxed{ X=xZ_0 } \]

This actual reactance is then used to calculate the required inductance or capacitance.

Shunt Components: Normalized Susceptance to Actual Susceptance

For a shunt component, the Smith Chart provides a normalized susceptance \(b\). The actual susceptance in siemens is:

\[ \boxed{ B=\frac{b}{Z_0} } \]

This actual susceptance is then used to calculate the required capacitance or inductance.

Therefore, the basic denormalization rules are:

• Series component: \(X=xZ_0\)
• Shunt component: \(B=b/Z_0\)

Case 1: Positive Normalized Reactance \(+x\), Series Inductor

A positive normalized reactance on the impedance Smith Chart represents an inductive reactance. Therefore, if the required matching reactance is positive, the corresponding series component is an inductor.

Step 1: Read the Normalized Reactance

Suppose the Smith Chart gives a positive normalized reactance:

\[ x>0 \]

The corresponding normalized impedance component is therefore \(+jx\).

Step 2: Denormalize the Reactance

The actual reactance is obtained by multiplying the normalized reactance by the characteristic impedance:

\[ \boxed{ X_{\text{actual}} = xZ_0 } \]

Since \(x\) is positive, the actual reactance is also positive.

The impedance of the required series element is therefore:

\[ \boxed{ Z_{\text{actual}} = jX_{\text{actual}} = jxZ_0 } \]

Step 3: Relate Reactance to Inductance

The impedance of an ideal inductor is:

\[ \boxed{ Z_L=j\omega L } \]

where the angular frequency is:

\[ \boxed{ \omega=2\pi f } \]

Equating the physical inductor impedance to the required reactance:

\[ j\omega L=jxZ_0 \]

Canceling \(j\) from both sides gives:

\[ \omega L=xZ_0 \]

Therefore, the required inductance is:

\[ \boxed{ L=\frac{xZ_0}{\omega} } \]

Since \(\omega=2\pi f\), this can also be written as:

\[ \boxed{ L=\frac{xZ_0}{2\pi f} } \]

Thus, a positive normalized reactance \(+x\) obtained from the Smith Chart is converted into a physical series inductor using this relationship.

Case 2: Negative Normalized Reactance \(-x\), Series Capacitor

A negative normalized reactance represents a capacitive reactance. Therefore, when the Smith Chart indicates a negative reactance, the required series component is a capacitor.

Step 1: Read the Normalized Reactance

Suppose the normalized reactance obtained from the Smith Chart is:

\[ -x \]

where \(x\) represents the positive magnitude of the reactance.

The corresponding actual reactance is:

\[ \boxed{ X_{\text{actual}} = -xZ_0 } \]

Therefore, the impedance of the required series component is:

\[ \boxed{ Z_{\text{actual}} = -jxZ_0 } \]

Step 2: Relate Reactance to Capacitance

The impedance of an ideal capacitor is:

\[ \boxed{ Z_C = \frac{1}{j\omega C} = -\frac{j}{\omega C} } \]

Equating this to the required capacitive impedance:

\[ -\frac{j}{\omega C} = -jxZ_0 \]

Canceling the common \(-j\) factor gives:

\[ \frac{1}{\omega C} = xZ_0 \]

Therefore:

\[ \boxed{ C = \frac{1}{\omega xZ_0} } \]

Using \(\omega=2\pi f\):

\[ \boxed{ C = \frac{1}{2\pi f xZ_0} } \]

Thus, a negative normalized reactance \(-x\) corresponds to a series capacitor whose value is determined by the operating frequency, characteristic impedance, and magnitude of the normalized reactance.

Case 3: Positive Normalized Susceptance \(+b\), Shunt Capacitor

When the matching process is performed in the admittance domain, the imaginary part is called susceptance. A positive susceptance corresponds to a capacitive shunt element.

Step 1: Read the Normalized Susceptance

Suppose the Smith Chart gives a positive normalized susceptance:

\[ b>0 \]

The actual susceptance is obtained by denormalization:

\[ \boxed{ B_{\text{actual}} = bY_0 } \]

Since:

\[ Y_0=\frac{1}{Z_0} \]

we obtain:

\[ \boxed{ B_{\text{actual}} = \frac{b}{Z_0} } \]

The corresponding shunt admittance is:

\[ \boxed{ Y_{\text{actual}} = j\frac{b}{Z_0} } \]

Step 2: Relate Susceptance to Capacitance

The admittance of an ideal capacitor is:

\[ Y_C = \frac{1}{Z_C} \]

Since:

\[ Z_C = \frac{1}{j\omega C} \]

the capacitor admittance becomes:

\[ \boxed{ Y_C=j\omega C } \]

Therefore, the susceptance of the capacitor is:

\[ B_C=\omega C \]

Equating the required susceptance to the capacitor susceptance:

\[ \omega C = \frac{b}{Z_0} \]

Hence:

\[ \boxed{ C = \frac{b}{\omega Z_0} } \]

Using \(\omega=2\pi f\):

\[ \boxed{ C = \frac{b}{2\pi fZ_0} } \]

Therefore, a positive normalized susceptance \(+b\) corresponds to a shunt capacitor.

Case 4: Negative Normalized Susceptance \(-b\), Shunt Inductor

A negative normalized susceptance corresponds to an inductive shunt element. Therefore, when the Smith Chart gives a negative susceptance, a shunt inductor is required.

Step 1: Read the Normalized Susceptance

Suppose the normalized susceptance is:

\[ -b \]

where \(b\) represents its positive magnitude.

The actual susceptance is:

\[ \boxed{ B_{\text{actual}} = -bY_0 } \]

Since \(Y_0=1/Z_0\):

\[ \boxed{ B_{\text{actual}} = -\frac{b}{Z_0} } \]

The corresponding shunt admittance is:

\[ \boxed{ Y_{\text{actual}} = -j\frac{b}{Z_0} } \]

Step 2: Relate Susceptance to Inductance

The impedance of an ideal inductor is:

\[ Z_L=j\omega L \]

Therefore, its admittance is:

\[ Y_L = \frac{1}{j\omega L} \]

Since \(1/j=-j\):

\[ \boxed{ Y_L = -\frac{j}{\omega L} } \]

The corresponding susceptance is:

\[ B_L = -\frac{1}{\omega L} \]

Equating this to the required actual susceptance:

\[ -\frac{1}{\omega L} = -\frac{b}{Z_0} \]

Canceling the negative signs gives:

\[ \frac{1}{\omega L} = \frac{b}{Z_0} \]

Therefore:

\[ \boxed{ L = \frac{Z_0}{\omega b} } \]

Using \(\omega=2\pi f\):

\[ \boxed{ L = \frac{Z_0}{2\pi fb} } \]

Therefore, a negative normalized susceptance \(-b\) corresponds to a shunt inductor.

Smith Chart Value to Physical Component Conversion

The complete conversion process can be understood as a sequence of three steps. First, the required normalized reactance or susceptance is read from the Smith Chart. Second, the normalized quantity is denormalized using the characteristic impedance \(Z_0\). Finally, the resulting physical reactance or susceptance is converted into an inductance or capacitance using the operating frequency.

For a series element:

\[ x \longrightarrow X=xZ_0 \longrightarrow L\text{ or }C \]

For a shunt element:

\[ b \longrightarrow B=\frac{b}{Z_0} \longrightarrow L\text{ or }C \]

The sign of the Smith Chart quantity determines the type of reactive component required.

• \(+x\): positive series reactance, therefore use a series inductor.
• \(-x\): negative series reactance, therefore use a series capacitor.
• \(+b\): positive shunt susceptance, therefore use a shunt capacitor.
• \(-b\): negative shunt susceptance, therefore use a shunt inductor.

Final Component Conversion Formulas

For a transmission line with characteristic impedance \(Z_0\) operating at frequency \(f\), the required physical component values can be obtained directly from the normalized Smith Chart values.

Series Inductor

For positive normalized reactance \(+x\):

\[ \boxed{ L=\frac{xZ_0}{2\pi f} } \]

Series Capacitor

For negative normalized reactance \(-x\):

\[ \boxed{ C=\frac{1}{2\pi f xZ_0} } \]

Shunt Capacitor

For positive normalized susceptance \(+b\):

\[ \boxed{ C=\frac{b}{2\pi fZ_0} } \]

Shunt Inductor

For negative normalized susceptance \(-b\):

\[ \boxed{ L=\frac{Z_0}{2\pi fb} } \]

Important Relationship Between Normalized and Physical Values

The distinction between normalized and physical quantities is essential when using a Smith Chart for practical impedance matching. A normalized reactance \(x\) does not have units of ohms, while a physical reactance \(X\) is measured in ohms. Similarly, normalized susceptance \(b\) is dimensionless, whereas physical susceptance \(B\) is measured in siemens.

The conversion can therefore be written as:

\[ \boxed{ X=xZ_0 } \]

for series elements, and:

\[ \boxed{ B=\frac{b}{Z_0} } \]

for shunt elements.

Once these physical quantities are known, the required inductor or capacitor can be calculated from the operating frequency. This makes it possible to move directly from a graphical Smith Chart matching solution to an actual circuit containing physical \(L\) and \(C\) components.